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line-through="false">уравнения поверхностей, ограничивающих </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">область V.</f> </text> </region> <region region-id="731" left="18" top="139.5" width="108" height="21" align-x="63" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">z1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="732" 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is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Поверхности x=a и<sp count="2"/>y=g</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>являются </f> <f family="Arial Cyr" charset="204" size="12">цилиндрическими</f> <f family="Arial Cyr" charset="204" size="12"> поверхностями, а в данном случае плоскостями, и, </f> <f family="Arial Cyr" charset="204" size="12">следовательно, одновременно проецирующими. </f> <f family="Arial Cyr" charset="204" size="12">В упомянутом выше файле проведено вычисление тройного интеграла и представлена</f> <f family="Arial Cyr" charset="204" size="12">необходимая графика выполненных рассчетов. Однако, она может показаться недоста- точной. В этой связи дадим для той же задачи графику в её расширенном виде.</f> </text> </region> <region region-id="586" left="18" top="258.75" width="489.75" height="84" align-x="18" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Начнем с указания о том, что в системах компьютерной математики пространственные</f> <f family="Arial Cyr" charset="204" size="12">графики строятся в правой системе координат, предполагающей по умолчанию следу-</f> <f family="Arial Cyr" charset="204" size="12">ющую последовательность осей: абсцисса, ордината и аппликата. Какие имена присва-</f> <f family="Arial Cyr" charset="204" size="12">ивать этим осям - не имеет значения. Если поверхность задается матрицей {M</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> , </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12">}, то</f> <f family="Arial Cyr" charset="204" size="12">первый индекс (номер строки матрицы) - это абсцисса, второй (номер столбца матрицы)</f> <f family="Arial Cyr" charset="204" size="12">- ордината, само значение компонента матрицы - будет аппликатой.</f> </text> </region> <region region-id="71" left="6" top="348.75" width="505.5" height="67.5" align-x="21.75" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Для определения и расстановки пределов </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">интегрирования в тройном интеграле </f> <f family="Arial Cyr" charset="204" size="12">в декар-</f> <f family="Arial Cyr" charset="204" size="12">товой системе координат</f> <f family="Arial Cyr" charset="204" size="12"> необходимо последовательно провести проецирование области </f> <f family="Arial Cyr" charset="204" size="12">V в</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> область интегрирования D на плоскости и далее D - спроецировать на одну из коорди-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">натных осей. Здесь удобно </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">D </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> получить на плоскости х0у. Найдем её границу как линию пересечения поверхностей (см. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">упомянутый выше файл</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">) .</inlineAttr> </f> </text> </region> <region region-id="291" left="12" top="426.75" width="489" height="27" align-x="12" align-y="438" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Уравнения поверхностей 2) для представления их н</f> <f family="Arial Cyr" charset="204" size="12">а графике проще записать в виде матриц, что и будет сделано ниже.</f> <f family="Arial Cyr" charset="204" size="12"> Для этого введем следующие параметры. </f> </text> </region> <region region-id="587" left="18" top="462.75" width="360" height="13.5" align-x="18" align-y="474" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Выберем дискрет, необходимый для представления переменных:</f> </text> </region> <region region-id="682" left="396" top="462.75" width="45" height="15.75" align-x="409.5" align-y="474" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff8080" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <ml:real>0.02</ml:real> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="838" left="18" top="486.75" width="460.5" height="27.75" align-x="18" align-y="498" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Укажем, что линия пересечения поверхностей z1(x,y) и </f> <f family="Arial Cyr" charset="204" size="12">z2(x,y) является окружностью радиуса </f> <f family="Arial Cyr" charset="204" size="12"> r </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">который определяется так:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="304" left="90" top="526.5" width="56.25" height="18" align-x="102" align-y="540" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">r</ml:id> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="688" left="24" top="558.75" width="468" height="57.75" align-x="24" align-y="570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Исходя из того, что поверхность </f> <f family="Arial Cyr" charset="204" size="12">z1(x,y) пересекается с плоскостью х0у по окружности</f> <f family="Arial Cyr" charset="204" size="12">радиуса Rx, </f> <f family="Arial Cyr" charset="204" size="12"> ограничивающей область, в которую попадает область интегрирования</f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12">, выберем поле значений матрицы размером<sp count="2"/>(Rx-R1x)</f> <f family="Arial Cyr" charset="204" size="9"> x </f> <f family="Arial Cyr" charset="204" size="12">(Ry-R1y). Зададим эти пара-</f> <f family="Arial Cyr" charset="204" size="12">метры:</f> </text> </region> <region region-id="684" left="24" top="624.75" width="37.5" height="15.75" align-x="45" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Rx</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="685" left="90" top="624.75" width="37.5" height="15.75" align-x="111" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Ry</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="686" left="150" top="624.75" width="50.25" height="15.75" align-x="177" align-y="636" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#0ff" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">R1x</ml:id> <ml:real>-2</ml:real> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="687" left="228" top="624.75" width="50.25" height="15.75" align-x="255" align-y="636" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#0ff" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">R1y</ml:id> <ml:real>-2</ml:real> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="690" left="24" top="654" width="484.5" height="41.25" align-x="24" align-y="666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">В интерактивном режиме параметры </f> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12">1</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12"> и<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12">1</f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12"> можно изменять от 0 до -2, разво-</f> <f family="Arial Cyr" charset="204" size="12">рачивая тем самым поле записанных ниже матриц, а, следовательно, область опре-</f> <f family="Arial Cyr" charset="204" size="12">деления задаваемых функций. </f> </text> </region> <region region-id="828" left="30" top="714.75" width="447" height="13.5" align-x="30" align-y="726" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Таким образом, диапазон изменения индексов матрицы должен быть следующим:</f> </text> </region> <region region-id="543" left="24" top="748.5" width="115.5" height="33.75" align-x="46.5" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Nx</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Rx</ml:id> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="544" left="180" top="748.5" width="115.5" height="33.75" align-x="202.5" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Ny</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Ry</ml:id> <ml:id xml:space="preserve">R1y</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="548" left="348" top="756.75" width="55.5" height="15.75" align-x="369.75" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Nx</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>200</ml:real> </result> </ml:eval> </math> <rendering item-idref="18"/> </region> <region region-id="547" left="432" top="756.75" width="55.5" height="15.75" align-x="453.75" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Ny</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>200</ml:real> </result> </ml:eval> </math> <rendering item-idref="19"/> </region> <region region-id="691" left="24" top="798.75" width="453" height="27" align-x="24" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Выберем имена </f> <f family="Arial Cyr" charset="204" size="12">индексов и присвоим </f> <f family="Arial Cyr" charset="204" size="12"> соответствующие </f> <f family="Arial Cyr" charset="204" size="12">им</f> <f family="Arial Cyr" charset="204" size="12"> диапазоны изменения </f> <f family="Arial Cyr" charset="204" size="12">значений: </f> </text> </region> <region region-id="839" left="72" top="834.75" width="54" height="15.75" align-x="85.5" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">Nx</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="840" left="162" top="834.75" width="57" height="15.75" align-x="178.5" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">Ny</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="356" left="30" top="864.75" width="210" height="13.5" align-x="30" align-y="876" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Перейдем к дискретным переменным:</f> </text> </region> <region region-id="337" left="30" top="888.75" width="81.75" height="17.25" align-x="51.75" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">R1y</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="336" left="150" top="888.75" width="75.75" height="17.25" align-x="168.75" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">R1x</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="335" left="276" top="888.75" width="78.75" height="17.25" align-x="332.25" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:id xml:space="preserve">R1y</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">Ry</ml:id> </ml:apply> </math> <rendering item-idref="24"/> </region> <region region-id="334" left="390" top="888.75" width="75.75" height="17.25" align-x="443.25" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:id xml:space="preserve">R1x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">Rx</ml:id> </ml:apply> </math> <rendering item-idref="25"/> </region> <region region-id="699" left="24" top="924.75" width="483" height="13.5" align-x="24" align-y="936" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">При этом имеет смысл распечатать числовые значения<sp count="2"/>некоторых из них для проверки: </f> </text> </region> <region region-id="692" left="36" top="960.75" width="46.5" height="17.25" align-x="54" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-2</ml:real> </result> </ml:eval> </math> <rendering item-idref="26"/> </region> <region region-id="693" left="96" top="960.75" width="48" height="17.25" align-x="122.25" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">Ny</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </result> </ml:eval> </math> <rendering item-idref="27"/> </region> <region region-id="694" left="156" top="952.5" width="47.25" height="33.75" align-x="175.5" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n2</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">Nx</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="695" left="222" top="960.75" width="45" height="17.25" align-x="245.25" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n2</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="29"/> </region> <region region-id="696" left="282" top="952.5" width="50.25" height="33.75" align-x="304.5" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m2</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">Ny</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="697" left="348" top="960.75" width="48" height="17.25" align-x="374.25" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">m2</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="31"/> </region> <region region-id="698" left="402" top="960.75" width="52.5" height="15.75" align-x="420.75" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n2</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>100</ml:real> </result> </ml:eval> </math> <rendering item-idref="32"/> </region> <region region-id="702" left="36" top="996.75" width="213.75" height="13.5" align-x="36" align-y="1008" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Введем необязательные обозначения:</f> </text> </region> <region region-id="414" left="36" top="1030.5" width="108" height="33.75" align-x="58.5" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m1</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">g</ml:id> <ml:id xml:space="preserve">R1y</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="413" left="168" top="1038.75" width="55.5" height="15.75" align-x="189.75" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m1</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>100</ml:real> </result> </ml:eval> </math> <rendering item-idref="34"/> </region> <region region-id="412" left="234" top="1038.75" width="92.25" height="15.75" align-x="250.5" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">max</ml:id> <ml:apply> <ml:id xml:space="preserve">z1</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="411" left="348" top="1038.75" width="37.5" height="15.75" align-x="363.75" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">C</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </result> </ml:eval> </math> <rendering item-idref="36"/> </region> <region region-id="700" left="402" top="1030.5" width="105" height="33.75" align-x="421.5" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n1</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="847" left="30" top="1080" width="241.5" height="14.25" align-x="30" align-y="1092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Представим плоскость </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> а </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">в виде матрицы:</inlineAttr> </f> </text> </region> <region region-id="705" left="312" top="1080.75" width="62.25" height="17.25" align-x="355.5" align-y="1092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">M</ml:id> <ml:sequence> <ml:id xml:space="preserve">n1</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">C</ml:id> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="848" left="30" top="1116" width="467.25" height="31.5" align-x="30" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Примечание </f> <f family="Arial Cyr" charset="204" size="12">. Первый индекс матрицы </f> <f family="Arial Cyr" charset="204" size="12"> М </f> <f family="Arial Cyr" charset="204" size="12"> выбирается из условия </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12">а </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и определяется из уравнения </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">=</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">1</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">x</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">+</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> h </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">*</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">. Значение </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">n</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">=</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">1 было получено выше.</inlineAttr> </f> </text> </region> <region region-id="849" left="30" top="1158" width="465.75" height="59.25" align-x="30" align-y="1170" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Аналогично получим плоскость </f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> g </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12">в виде матрицы </f> <f family="Arial Cyr" charset="204" size="12"> MY </f> <f family="Arial Cyr" charset="204" size="12">, где второй её индекс опре-</f> <f family="Arial Cyr" charset="204" size="12">деляется из уравнения </f> <f family="Arial Cyr" charset="204" size="12"> g </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12">1</f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12">+</f> <f family="Arial Cyr" charset="204" size="12"> h </f> <f family="Arial Cyr" charset="204" size="12">*</f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12">. Откуда находим индекс </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">: </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">m</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">=(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> g </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">1</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">)/</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> h </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">. Для прорисовки плоскости (её отличия от координатной плоскости) необходимо, чтобы</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">>0.<sp count="2"/>Поэтому </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">надо выбирать так:</inlineAttr> </f> </text> </region> <region region-id="741" left="54" top="1230.75" width="117" height="15.75" align-x="75.75" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">my</ml:id> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">m1</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> <ml:id xml:space="preserve">m1</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="740" left="192" top="1230.75" width="54.75" height="15.75" align-x="213" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">my</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>100</ml:real> </result> </ml:eval> </math> <rendering item-idref="40"/> </region> <region region-id="742" left="300" top="1230.75" width="70.5" height="17.25" align-x="351.75" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">MY</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">my</ml:id> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">C</ml:id> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="850" left="36" top="1260" width="419.25" height="14.25" align-x="36" align-y="1272" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Есть еще одна плоскость </f> <f family="Arial Cyr" charset="204" size="12"> z </f> <f family="Arial Cyr" charset="204" size="12">2(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12">)=b, для которой также зададим матрицу </f> <f family="Arial Cyr" charset="204" size="12"> Mb </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="745" left="60" top="1290.75" width="60.75" height="17.25" align-x="104.25" align-y="1302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Mb</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">b</ml:id> </ml:define> </math> <rendering item-idref="42"/> </region> <region region-id="756" left="36" top="1320" width="453.75" height="30.75" align-x="36" align-y="1332" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Остается построить на рис.1 саму поверхность </f> <f family="Arial Cyr" charset="204" size="12"> z </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">1</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12">), что сделаем задав матрицу</f> <f family="Arial Cyr" charset="204" size="12">{Z</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> , </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12">}.</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> </text> </region> <region region-id="758" left="180" top="1368.75" width="96.75" height="17.25" align-x="214.5" align-y="1380" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z1</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="43"/> </region> <region region-id="404" left="90" top="1452" width="365.25" height="337.5" align-x="271.5" align-y="1452" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="44" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">M</ml:id> <ml:id xml:space="preserve">MY</ml:id> <ml:id xml:space="preserve">Z</ml:id> <ml:id xml:space="preserve">Mb</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="45"/> </region> <region region-id="852" left="24" top="1806.75" width="474.75" height="41.25" align-x="24" align-y="1818" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Поверхности, изображенные на рис.1, можно просматривать в интерактивном режиме,</f> <f family="Arial Cyr" charset="204" size="12">убирая имя любой из них из титров. Из этого рисунка понятно по какому объему </f> <f family="Arial Cyr" charset="204" size="12"> V </f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">ведется интегрирование в тройном интеграле - верхний красный сегмент. </f> </text> </region> <region region-id="851" left="24" top="1860" width="463.5" height="45.75" align-x="24" align-y="1872" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Введем сложный индекс </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> v </f> <f family="Arial Cyr" charset="204" size="12"> s </f> <f family="Arial Cyr" charset="204" size="12"> для опредения направляющей (окружности) цилинд- рической поверхности, которая задаст одну из линий проецирования объема </f> <f family="Arial Cyr" charset="204" size="12"> V </f> <f family="Arial Cyr" charset="204" size="12"> в область </f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">на плоскости </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">х</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">0</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> у </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">. Две другие определяются плоскостями </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">=</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> и<sp count="3"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">y</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">=</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> g </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.</inlineAttr> </f> </text> </region> <region region-id="750" left="24" top="1929" width="226.5" height="41.25" align-x="36.75" align-y="1956" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">s</ml:id> <ml:range> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">g</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="751" left="294" top="1927.5" width="159" height="42.75" align-x="312" align-y="1956" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">R1y</ml:id> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="761" left="30" top="1986.75" width="447.75" height="49.5" align-x="30" align-y="2004" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Здесь должно выполняться<sp count="2"/> </f> <region region-id="783" left="187.5" top="1987.5" width="60" height="21" align-x="227.25" align-y="2004" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:greaterOrEqual/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">g</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="48"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. В противном случае объем </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false">V</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> не замкнут.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Сама же проецирующая цилиндрическая поверхность будет представлена в виде матрицы </f> <f family="Arial Cyr" charset="204" size="12"> МС </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и изображена </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">на рис.2</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">:</inlineAttr> </f> </text> </region> <region region-id="762" left="114" top="2052.75" width="110.25" height="22.5" align-x="161.25" align-y="2064" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">MC</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z1</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="49"/> </region> <region region-id="855" left="42" top="2112" width="445.5" height="28.5" align-x="42" align-y="2124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Укажем как должен изменяться индекс </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, который будет задавать значения пере-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">менной по оси ординат</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> и определять область </f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12"> и фрагменты плоскостей </f> <f family="Arial Cyr" charset="204" size="12">G</f> <f family="Arial Cyr" charset="204" size="12"> и </f> <f family="Arial Cyr" charset="204" size="12"> W </f> <f family="Arial Cyr" charset="204" size="12"> .</f> </text> </region> <region region-id="766" left="78" top="2157" width="228" height="41.25" align-x="92.25" align-y="2184" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">g</ml:id> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="50"/> </region> <region region-id="856" left="42" top="2208.75" width="451.5" height="49.5" align-x="42" align-y="2226" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Здесь по-прежнему должно выполняться </f> <region region-id="784" left="269.25" top="2209.5" width="60" height="21" align-x="309" align-y="2226" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:greaterOrEqual/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">g</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="51"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Зададим фрагмент плоскости </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> а </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, ограниченный в пространстве линиями пересечения с другими поверхностями</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="769" left="54" top="2274.75" width="102.75" height="17.25" align-x="92.25" align-y="2286" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">n1</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z1</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n1</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="52"/> </region> <region region-id="770" left="192" top="2274.75" width="54" height="17.25" align-x="215.25" align-y="2286" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n1</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.5</ml:real> </result> </ml:eval> </math> <rendering item-idref="53"/> </region> <region region-id="771" left="264" top="2274.75" width="52.5" height="15.75" align-x="282.75" align-y="2286" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n1</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>125</ml:real> </result> </ml:eval> </math> <rendering item-idref="54"/> </region> <region region-id="857" left="36" top="2316" width="234.75" height="14.25" align-x="36" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Аналогично для фрагмента плоскости </f> <f family="Arial Cyr" charset="204" size="12"> у </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> g </f> <f family="Arial Cyr" charset="204" size="12">: </f> </text> </region> <region region-id="830" left="294" top="2316.75" width="110.25" height="17.25" align-x="337.5" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">W</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">my</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z1</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">my</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="55"/> </region> <region region-id="829" left="426" top="2316.75" width="48" height="17.25" align-x="452.25" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">my</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="56"/> </region> <region region-id="858" left="36" top="2352" width="433.5" height="31.5" align-x="36" align-y="2364" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Тогда область </f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12"> и фрагмент поверхности над этой областью можно записать в</f> <f family="Arial Cyr" charset="204" size="12">виде следующих матриц<sp count="2"/>{</f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12"> s </f> <f family="Arial Cyr" charset="204" size="12"> , </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">} и {</f> <f family="Arial Cyr" charset="204" size="12">P</f> <f family="Arial Cyr" charset="204" size="12"> s </f> <f family="Arial Cyr" charset="204" size="12"> , </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">}</f> </text> </region> <region region-id="833" left="78" top="2410.5" width="165" height="33.75" align-x="110.25" align-y="2430" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">D</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">b</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="57"/> </region> <region region-id="832" left="282" top="2410.5" width="205.5" height="33.75" align-x="312.75" align-y="2430" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z1</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="58"/> </region> <region region-id="868" left="48" top="2454" width="458.25" height="28.5" align-x="48" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Покажем на рис.2<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">поверхность </f> <f family="Arial Cyr" charset="204" size="12"> МС </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, которая </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">проецирует на плоскость </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12">0</f> <f family="Arial Cyr" charset="204" size="12"> у </f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>про- странственную область </f> <f family="Arial Cyr" charset="204" size="12"> V </f> <f family="Arial Cyr" charset="204" size="12"> в </f> <f family="Arial Cyr" charset="204" size="12">область </f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.</inlineAttr> </f> </text> </region> <region region-id="866" left="54" top="2508" width="264.75" height="210" align-x="185.25" align-y="2508" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="59" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">MC</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="60"/> </region> <region region-id="869" left="30" top="2742" width="479.25" height="42" align-x="30" align-y="2754" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">На следующем графике </f> <f family="Arial Cyr" charset="204" size="12">рис.3</f> <f family="Arial Cyr" charset="204" size="12"> покажем все четыре поверхности </f> <f family="Arial Cyr" charset="204" size="12"> G </f> <f family="Arial Cyr" charset="204" size="12">, </f> <f family="Arial Cyr" charset="204" size="12"> W </f> <f family="Arial Cyr" charset="204" size="12">, </f> <f family="Arial Cyr" charset="204" size="12"> MC </f> <f family="Arial Cyr" charset="204" size="12">, </f> <f family="Arial Cyr" charset="204" size="12">P .</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> Любую </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">из этих поверхностей</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> можно также убрать из поля графика или добавить туда, напри- мер, область </f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12">. </f> </text> </region> <region region-id="834" left="48" top="2874" width="404.25" height="298.5" align-x="249" align-y="2874" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="61" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">G</ml:id> <ml:id xml:space="preserve">W</ml:id> <ml:id xml:space="preserve">MC</ml:id> <ml:id xml:space="preserve">P</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="62"/> </region> <region region-id="862" left="48" top="3210" width="424.5" height="56.25" align-x="48" align-y="3222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">На этом графике, если снять поверхность </f> <f family="Arial 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is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Поверхности x=a и<sp count="2"/>y=g</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>являются </f> <f family="Arial Cyr" charset="204" size="12">цилиндрическими</f> <f family="Arial Cyr" charset="204" size="12"> поверхностями, а в данном случае плоскостями, и, </f> <f family="Arial Cyr" charset="204" size="12">следовательно, одновременно проецирующими. </f> <f family="Arial Cyr" charset="204" size="12">В упомянутом выше файле проведено вычисление тройного интеграла и представлена</f> <f family="Arial Cyr" charset="204" size="12">необходимая графика выполненных рассчетов. Однако, она может показаться недоста- точной. В этой связи дадим для той же задачи графику в её расширенном виде.</f> </text> </region> <region region-id="586" left="18" top="258.75" width="489.75" height="84" align-x="18" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Начнем с указания о том, что в системах компьютерной математики пространственные</f> <f family="Arial Cyr" charset="204" size="12">графики строятся в правой системе координат, предполагающей по умолчанию следу-</f> <f family="Arial Cyr" charset="204" size="12">ющую последовательность осей: абсцисса, ордината и аппликата. Какие имена присва-</f> <f family="Arial Cyr" charset="204" size="12">ивать этим осям - не имеет значения. Если поверхность задается матрицей {M</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> , </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12">}, то</f> <f family="Arial Cyr" charset="204" size="12">первый индекс (номер строки матрицы) - это абсцисса, второй (номер столбца матрицы)</f> <f family="Arial Cyr" charset="204" size="12">- ордината, само значение компонента матрицы - будет аппликатой.</f> </text> </region> <region region-id="71" left="6" top="348.75" width="505.5" height="67.5" align-x="21.75" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Для определения и расстановки пределов </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">интегрирования в тройном интеграле </f> <f family="Arial Cyr" charset="204" size="12">в декар-</f> <f family="Arial Cyr" charset="204" size="12">товой системе координат</f> <f family="Arial Cyr" charset="204" size="12"> необходимо последовательно провести проецирование области </f> <f family="Arial Cyr" charset="204" size="12">V в</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> область интегрирования D на плоскости и далее D - спроецировать на одну из коорди-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">натных осей. Здесь удобно </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">D </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> получить на плоскости х0у. Найдем её границу как линию пересечения поверхностей (см. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">упомянутый выше файл</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">) .</inlineAttr> </f> </text> </region> <region region-id="291" left="12" top="426.75" width="489" height="27" align-x="12" align-y="438" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Уравнения поверхностей 2) для представления их н</f> <f family="Arial Cyr" charset="204" size="12">а графике проще записать в виде матриц, что и будет сделано ниже.</f> <f family="Arial Cyr" charset="204" size="12"> Для этого введем следующие параметры. </f> </text> </region> <region region-id="587" left="18" top="462.75" width="360" height="13.5" align-x="18" align-y="474" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Выберем дискрет, необходимый для представления переменных:</f> </text> </region> <region region-id="682" left="396" top="462.75" width="45" height="15.75" align-x="409.5" align-y="474" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff8080" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">h</ml:id> <ml:real>0.02</ml:real> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="838" left="18" top="486.75" width="460.5" height="27.75" align-x="18" align-y="498" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Укажем, что линия пересечения поверхностей z1(x,y) и </f> <f family="Arial Cyr" charset="204" size="12">z2(x,y) является окружностью радиуса </f> <f family="Arial Cyr" charset="204" size="12"> r </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">который определяется так:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="304" left="90" top="526.5" width="56.25" height="18" align-x="102" align-y="540" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">r</ml:id> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:real>4</ml:real> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="688" left="24" top="558.75" width="468" height="57.75" align-x="24" align-y="570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Исходя из того, что поверхность </f> <f family="Arial Cyr" charset="204" size="12">z1(x,y) пересекается с плоскостью х0у по окружности</f> <f family="Arial Cyr" charset="204" size="12">радиуса Rx, </f> <f family="Arial Cyr" charset="204" size="12"> ограничивающей область, в которую попадает область интегрирования</f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12">, выберем поле значений матрицы размером<sp count="2"/>(Rx-R1x)</f> <f family="Arial Cyr" charset="204" size="9"> x </f> <f family="Arial Cyr" charset="204" size="12">(Ry-R1y). Зададим эти пара-</f> <f family="Arial Cyr" charset="204" size="12">метры:</f> </text> </region> <region region-id="684" left="24" top="624.75" width="37.5" height="15.75" align-x="45" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Rx</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="685" left="90" top="624.75" width="37.5" height="15.75" align-x="111" align-y="636" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Ry</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="686" left="150" top="624.75" width="50.25" height="15.75" align-x="177" align-y="636" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#0ff" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">R1x</ml:id> <ml:real>-2</ml:real> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="687" left="228" top="624.75" width="50.25" height="15.75" align-x="255" align-y="636" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#0ff" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">R1y</ml:id> <ml:real>-2</ml:real> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="690" left="24" top="654" width="484.5" height="41.25" align-x="24" align-y="666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">В интерактивном режиме параметры </f> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12">1</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12"> и<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12">1</f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12"> можно изменять от 0 до -2, разво-</f> <f family="Arial Cyr" charset="204" size="12">рачивая тем самым поле записанных ниже матриц, а, следовательно, область опре-</f> <f family="Arial Cyr" charset="204" size="12">деления задаваемых функций. </f> </text> </region> <region region-id="828" left="30" top="714.75" width="447" height="13.5" align-x="30" align-y="726" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Таким образом, диапазон изменения индексов матрицы должен быть следующим:</f> </text> </region> <region region-id="543" left="24" top="748.5" width="115.5" height="33.75" align-x="46.5" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Nx</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Rx</ml:id> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="544" left="180" top="748.5" width="115.5" height="33.75" align-x="202.5" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Ny</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">Ry</ml:id> <ml:id xml:space="preserve">R1y</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="548" left="348" top="756.75" width="55.5" height="15.75" align-x="369.75" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Nx</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>200</ml:real> </result> </ml:eval> </math> <rendering item-idref="18"/> </region> <region region-id="547" left="432" top="756.75" width="55.5" height="15.75" align-x="453.75" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Ny</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>200</ml:real> </result> </ml:eval> </math> <rendering item-idref="19"/> </region> <region region-id="691" left="24" top="798.75" width="453" height="27" align-x="24" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Выберем имена </f> <f family="Arial Cyr" charset="204" size="12">индексов и присвоим </f> <f family="Arial Cyr" charset="204" size="12"> соответствующие </f> <f family="Arial Cyr" charset="204" size="12">им</f> <f family="Arial Cyr" charset="204" size="12"> диапазоны изменения </f> <f family="Arial Cyr" charset="204" size="12">значений: </f> </text> </region> <region region-id="839" left="72" top="834.75" width="54" height="15.75" align-x="85.5" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">Nx</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="840" left="162" top="834.75" width="57" height="15.75" align-x="178.5" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">Ny</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="356" left="30" top="864.75" width="210" height="13.5" align-x="30" align-y="876" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Перейдем к дискретным переменным:</f> </text> </region> <region region-id="337" left="30" top="888.75" width="81.75" height="17.25" align-x="51.75" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">R1y</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="22"/> </region> <region region-id="336" left="150" top="888.75" width="75.75" height="17.25" align-x="168.75" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">R1x</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">h</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="335" left="276" top="888.75" width="78.75" height="17.25" align-x="332.25" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:id xml:space="preserve">R1y</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">Ry</ml:id> </ml:apply> </math> <rendering item-idref="24"/> </region> <region region-id="334" left="390" top="888.75" width="75.75" height="17.25" align-x="443.25" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:id xml:space="preserve">R1x</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">Rx</ml:id> </ml:apply> </math> <rendering item-idref="25"/> </region> <region region-id="699" left="24" top="924.75" width="483" height="13.5" align-x="24" align-y="936" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">При этом имеет смысл распечатать числовые значения<sp count="2"/>некоторых из них для проверки: </f> </text> </region> <region region-id="692" left="36" top="960.75" width="46.5" height="17.25" align-x="54" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-2</ml:real> </result> </ml:eval> </math> <rendering item-idref="26"/> </region> <region region-id="693" left="96" top="960.75" width="48" height="17.25" align-x="122.25" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">Ny</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </result> </ml:eval> </math> <rendering item-idref="27"/> </region> <region region-id="694" left="156" top="952.5" width="47.25" height="33.75" align-x="175.5" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n2</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">Nx</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="695" left="222" top="960.75" width="45" height="17.25" align-x="245.25" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n2</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="29"/> </region> <region region-id="696" left="282" top="952.5" width="50.25" height="33.75" align-x="304.5" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m2</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">Ny</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="697" left="348" top="960.75" width="48" height="17.25" align-x="374.25" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">m2</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="31"/> </region> <region region-id="698" left="402" top="960.75" width="52.5" height="15.75" align-x="420.75" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n2</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>100</ml:real> </result> </ml:eval> </math> <rendering item-idref="32"/> </region> <region region-id="702" left="36" top="996.75" width="213.75" height="13.5" align-x="36" align-y="1008" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Введем необязательные обозначения:</f> </text> </region> <region region-id="414" left="36" top="1030.5" width="108" height="33.75" align-x="58.5" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m1</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">g</ml:id> <ml:id xml:space="preserve">R1y</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="413" left="168" top="1038.75" width="55.5" height="15.75" align-x="189.75" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">m1</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>100</ml:real> </result> </ml:eval> </math> <rendering item-idref="34"/> </region> <region region-id="412" left="234" top="1038.75" width="92.25" height="15.75" align-x="250.5" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">max</ml:id> <ml:apply> <ml:id xml:space="preserve">z1</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="411" left="348" top="1038.75" width="37.5" height="15.75" align-x="363.75" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">C</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </result> </ml:eval> </math> <rendering item-idref="36"/> </region> <region region-id="700" left="402" top="1030.5" width="105" height="33.75" align-x="421.5" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n1</ml:id> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="847" left="30" top="1080" width="241.5" height="14.25" align-x="30" align-y="1092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Представим плоскость </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> а </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">в виде матрицы:</inlineAttr> </f> </text> </region> <region region-id="705" left="312" top="1080.75" width="62.25" height="17.25" align-x="355.5" align-y="1092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">M</ml:id> <ml:sequence> <ml:id xml:space="preserve">n1</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">C</ml:id> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="848" left="30" top="1116" width="467.25" height="31.5" align-x="30" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> Примечание </f> <f family="Arial Cyr" charset="204" size="12">. Первый индекс матрицы </f> <f family="Arial Cyr" charset="204" size="12"> М </f> <f family="Arial Cyr" charset="204" size="12"> выбирается из условия </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12">а </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и определяется из уравнения </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">=</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">1</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">x</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">+</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> h </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">*</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">. Значение </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">n</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">=</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">1 было получено выше.</inlineAttr> </f> </text> </region> <region region-id="849" left="30" top="1158" width="465.75" height="59.25" align-x="30" align-y="1170" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Аналогично получим плоскость </f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> g </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12">в виде матрицы </f> <f family="Arial Cyr" charset="204" size="12"> MY </f> <f family="Arial Cyr" charset="204" size="12">, где второй её индекс опре-</f> <f family="Arial Cyr" charset="204" size="12">деляется из уравнения </f> <f family="Arial Cyr" charset="204" size="12"> g </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12">1</f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12">+</f> <f family="Arial Cyr" charset="204" size="12"> h </f> <f family="Arial Cyr" charset="204" size="12">*</f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12">. Откуда находим индекс </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">: </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">m</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">=(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> g </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">1</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">)/</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> h </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">. Для прорисовки плоскости (её отличия от координатной плоскости) необходимо, чтобы</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">>0.<sp count="2"/>Поэтому </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">надо выбирать так:</inlineAttr> </f> </text> </region> <region region-id="741" left="54" top="1230.75" width="117" height="15.75" align-x="75.75" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">my</ml:id> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">m1</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>1</ml:real> <ml:id xml:space="preserve">m1</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="740" left="192" top="1230.75" width="54.75" height="15.75" align-x="213" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">my</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>100</ml:real> </result> </ml:eval> </math> <rendering item-idref="40"/> </region> <region region-id="742" left="300" top="1230.75" width="70.5" height="17.25" align-x="351.75" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">MY</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">my</ml:id> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">C</ml:id> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="850" left="36" top="1260" width="419.25" height="14.25" align-x="36" align-y="1272" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Есть еще одна плоскость </f> <f family="Arial Cyr" charset="204" size="12"> z </f> <f family="Arial Cyr" charset="204" size="12">2(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12">)=b, для которой также зададим матрицу </f> <f family="Arial Cyr" charset="204" size="12"> Mb </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="745" left="60" top="1290.75" width="60.75" height="17.25" align-x="104.25" align-y="1302" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Mb</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:id xml:space="preserve">b</ml:id> </ml:define> </math> <rendering item-idref="42"/> </region> <region region-id="756" left="36" top="1320" width="453.75" height="30.75" align-x="36" align-y="1332" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Остается построить на рис.1 саму поверхность </f> <f family="Arial Cyr" charset="204" size="12"> z </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">1</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12">), что сделаем задав матрицу</f> <f family="Arial Cyr" charset="204" size="12">{Z</f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> , </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12">}.</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> </text> </region> <region region-id="758" left="180" top="1368.75" width="96.75" height="17.25" align-x="214.5" align-y="1380" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Z</ml:id> <ml:sequence> <ml:id xml:space="preserve">n</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z1</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="43"/> </region> <region region-id="404" left="90" top="1452" width="365.25" height="337.5" align-x="271.5" align-y="1452" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="44" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">M</ml:id> <ml:id xml:space="preserve">MY</ml:id> <ml:id xml:space="preserve">Z</ml:id> <ml:id xml:space="preserve">Mb</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="45"/> </region> <region region-id="852" left="24" top="1806.75" width="474.75" height="41.25" align-x="24" align-y="1818" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Поверхности, изображенные на рис.1, можно просматривать в интерактивном режиме,</f> <f family="Arial Cyr" charset="204" size="12">убирая имя любой из них из титров. Из этого рисунка понятно по какому объему </f> <f family="Arial Cyr" charset="204" size="12"> V </f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">ведется интегрирование в тройном интеграле - верхний красный сегмент. </f> </text> </region> <region region-id="851" left="24" top="1860" width="463.5" height="45.75" align-x="24" align-y="1872" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Введем сложный индекс </f> <f family="Arial Cyr" charset="204" size="12"> m </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> v </f> <f family="Arial Cyr" charset="204" size="12"> s </f> <f family="Arial Cyr" charset="204" size="12"> для опредения направляющей (окружности) цилинд- рической поверхности, которая задаст одну из линий проецирования объема </f> <f family="Arial Cyr" charset="204" size="12"> V </f> <f family="Arial Cyr" charset="204" size="12"> в область </f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">на плоскости </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">х</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">0</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> у </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">. Две другие определяются плоскостями </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">=</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> и<sp count="3"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">y</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">=</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> g </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.</inlineAttr> </f> </text> </region> <region region-id="750" left="24" top="1929" width="226.5" height="41.25" align-x="36.75" align-y="1956" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">s</ml:id> <ml:range> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">g</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="751" left="294" top="1927.5" width="159" height="42.75" align-x="312" align-y="1956" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">R1y</ml:id> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="761" left="30" top="1986.75" width="447.75" height="49.5" align-x="30" align-y="2004" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Здесь должно выполняться<sp count="2"/> </f> <region region-id="783" left="187.5" top="1987.5" width="60" height="21" align-x="227.25" align-y="2004" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:greaterOrEqual/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">g</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="48"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. В противном случае объем </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" underline="false" line-through="false">V</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> не замкнут.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Сама же проецирующая цилиндрическая поверхность будет представлена в виде матрицы </f> <f family="Arial Cyr" charset="204" size="12"> МС </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и изображена </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">на рис.2</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">:</inlineAttr> </f> </text> </region> <region region-id="762" left="114" top="2052.75" width="110.25" height="22.5" align-x="161.25" align-y="2064" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">MC</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z1</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="49"/> </region> <region region-id="855" left="42" top="2112" width="445.5" height="28.5" align-x="42" align-y="2124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Укажем как должен изменяться индекс </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, который будет задавать значения пере-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">менной по оси ординат</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> и определять область </f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12"> и фрагменты плоскостей </f> <f family="Arial Cyr" charset="204" size="12">G</f> <f family="Arial Cyr" charset="204" size="12"> и </f> <f family="Arial Cyr" charset="204" size="12"> W </f> <f family="Arial Cyr" charset="204" size="12"> .</f> </text> </region> <region region-id="766" left="78" top="2157" width="228" height="41.25" align-x="92.25" align-y="2184" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">g</ml:id> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">floor</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> </ml:apply> </ml:range> </ml:define> </math> <rendering item-idref="50"/> </region> <region region-id="856" left="42" top="2208.75" width="451.5" height="49.5" align-x="42" align-y="2226" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Здесь по-прежнему должно выполняться </f> <region region-id="784" left="269.25" top="2209.5" width="60" height="21" align-x="309" align-y="2226" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:greaterOrEqual/> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">g</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">a</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="51"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Зададим фрагмент плоскости </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> а </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, ограниченный в пространстве линиями пересечения с другими поверхностями</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">:</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="769" left="54" top="2274.75" width="102.75" height="17.25" align-x="92.25" align-y="2286" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">G</ml:id> <ml:sequence> <ml:id xml:space="preserve">n1</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z1</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n1</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="52"/> </region> <region region-id="770" left="192" top="2274.75" width="54" height="17.25" align-x="215.25" align-y="2286" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n1</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.5</ml:real> </result> </ml:eval> </math> <rendering item-idref="53"/> </region> <region region-id="771" left="264" top="2274.75" width="52.5" height="15.75" align-x="282.75" align-y="2286" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n1</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>125</ml:real> </result> </ml:eval> </math> <rendering item-idref="54"/> </region> <region region-id="857" left="36" top="2316" width="234.75" height="14.25" align-x="36" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Аналогично для фрагмента плоскости </f> <f family="Arial Cyr" charset="204" size="12"> у </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> g </f> <f family="Arial Cyr" charset="204" size="12">: </f> </text> </region> <region region-id="830" left="294" top="2316.75" width="110.25" height="17.25" align-x="337.5" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">W</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">my</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z1</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">my</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="55"/> </region> <region region-id="829" left="426" top="2316.75" width="48" height="17.25" align-x="452.25" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">my</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="56"/> </region> <region region-id="858" left="36" top="2352" width="433.5" height="31.5" align-x="36" align-y="2364" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Тогда область </f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12"> и фрагмент поверхности над этой областью можно записать в</f> <f family="Arial Cyr" charset="204" size="12">виде следующих матриц<sp count="2"/>{</f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12"> s </f> <f family="Arial Cyr" charset="204" size="12"> , </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">} и {</f> <f family="Arial Cyr" charset="204" size="12">P</f> <f family="Arial Cyr" charset="204" size="12"> s </f> <f family="Arial Cyr" charset="204" size="12"> , </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">}</f> </text> </region> <region region-id="833" left="78" top="2410.5" width="165" height="33.75" align-x="110.25" align-y="2430" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">D</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">b</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="57"/> </region> <region region-id="832" left="282" top="2410.5" width="205.5" height="33.75" align-x="312.75" align-y="2430" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">if</ml:id> <ml:sequence> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:lessOrEqual/> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">R1x</ml:id> </ml:apply> <ml:id xml:space="preserve">h</ml:id> </ml:apply> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">z1</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="58"/> </region> <region region-id="868" left="48" top="2454" width="458.25" height="28.5" align-x="48" align-y="2466" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Покажем на рис.2<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">поверхность </f> <f family="Arial Cyr" charset="204" size="12"> МС </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, которая </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">проецирует на плоскость </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12">0</f> <f family="Arial Cyr" charset="204" size="12"> у </f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>про- странственную область </f> <f family="Arial Cyr" charset="204" size="12"> V </f> <f family="Arial Cyr" charset="204" size="12"> в </f> <f family="Arial Cyr" charset="204" size="12">область </f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.</inlineAttr> </f> </text> </region> <region region-id="866" left="54" top="2508" width="264.75" height="210" align-x="185.25" align-y="2508" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="59" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">MC</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="60"/> </region> <region region-id="869" left="30" top="2742" width="479.25" height="42" align-x="30" align-y="2754" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">На следующем графике </f> <f family="Arial Cyr" charset="204" size="12">рис.3</f> <f family="Arial Cyr" charset="204" size="12"> покажем все четыре поверхности </f> <f family="Arial Cyr" charset="204" size="12"> G </f> <f family="Arial Cyr" charset="204" size="12">, </f> <f family="Arial Cyr" charset="204" size="12"> W </f> <f family="Arial Cyr" charset="204" size="12">, </f> <f family="Arial Cyr" charset="204" size="12"> MC </f> <f family="Arial Cyr" charset="204" size="12">, </f> <f family="Arial Cyr" charset="204" size="12">P .</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> Любую </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">из этих поверхностей</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> можно также убрать из поля графика или добавить туда, напри- мер, область </f> <f family="Arial Cyr" charset="204" size="12"> D </f> <f family="Arial Cyr" charset="204" size="12">. </f> </text> </region> <region region-id="834" left="48" top="2874" width="404.25" height="298.5" align-x="249" align-y="2874" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="61" disable-calc="false"> <inputs> <ml:sequence xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">G</ml:id> <ml:id xml:space="preserve">W</ml:id> <ml:id xml:space="preserve">MC</ml:id> <ml:id xml:space="preserve">P</ml:id> </ml:sequence> </inputs> <outputs/> </component> <rendering item-idref="62"/> </region> <region region-id="862" left="48" top="3210" width="424.5" height="56.25" align-x="48" align-y="3222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">На этом графике, если снять поверхность </f> <f family="Arial 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