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</units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="cm"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="869" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="474" left="12" top="6.75" width="486" height="161.25" align-x="25.5" align-y="18" 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">Задача 1</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Дано векторное поле<sp count="2"/> </inlineAttr> </f> <region region-id="5" left="134.25" top="21.75" width="145.5" height="15" align-x="171" align-y="33" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#000" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">P</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:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">Q</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:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="2"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">1. Проверить, что это поле является потенциальным.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">2. Найти потенциал поля </inlineAttr> </f> <region region-id="7" left="147.75" top="51" width="55.5" height="15" align-x="161.25" align-y="62.25" 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:equal/> <ml:id xml:space="preserve">u</ml:id> <ml:apply> <ml:id xml:space="preserve">u</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> </math> <rendering item-idref="3"/> </region> <inlineAttr line-through="false"> <f family="Arial Cyr" charset="204" size="12">3. Найти уравнение линий равного потенциала и изобразить линии равного потенциала на чертеже.</f> </inlineAttr> <inlineAttr line-through="false"> <f family="Arial Cyr" charset="204" size="12">4. Составить уравнение векторных линий поля и изобразить векторные линии на том же чертеже, указав стрелками направление векторных линий.</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">5. Вычислить линейный интеграл<sp count="2"/> </inlineAttr> </f> <region region-id="11" left="195.75" top="120.75" width="39.75" height="47.25" align-x="219" align-y="150" 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:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">l</ml:id> </ml:boundVars> <ml:apply> <ml:vectorize/> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">B</ml:id> </ml:bounds> </ml:apply> </math> <rendering item-idref="4"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> <sp count="2"/> </inlineAttr> </f> </text> </region> <region region-id="710" left="18" top="186.75" width="152.25" height="13.5" align-x="18" align-y="198" 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">Запишем исходные данные:</f> </text> </region> <region region-id="711" left="18" top="216.75" width="78.75" height="15" align-x="61.5" align-y="228" 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">P</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:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="712" left="120" top="216.75" width="66.75" height="15" align-x="164.25" align-y="228" 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">Q</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:mult style="auto-select"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="713" left="222" top="216.75" width="192" height="13.5" align-x="236.25" align-y="228" 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">Точка А(0,0)<sp count="11"/>Точка В(6,2)</inlineAttr> </f> </text> </region> <region region-id="742" left="456" top="216.75" width="16.5" height="15" align-x="463.5" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="7"/> </region> <region region-id="714" left="18" top="246.75" width="145.5" height="15" align-x="54.75" align-y="258" 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:equal/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">P</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:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">Q</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:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="8"/> </region> <region region-id="709" left="18" top="275.25" width="210.75" height="30" align-x="18" align-y="288" 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="14">Решение</f> <f family="Arial Cyr" charset="204" size="12">.</f> <f family="Arial Cyr" charset="204" size="12">1. </f> <f family="Arial Cyr" charset="204" size="12"> Проверка потенциальности поля </f> <f family="Arial Cyr" charset="204" size="12">.</f> </text> </region> <region region-id="480" left="18" top="324" width="483.75" height="57" align-x="18" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{00020906-0000-0000-C000-000000000046}" type="embedded" item-idref="9"/> <rendering item-idref="10"/> </region> <region region-id="717" left="18" top="384.75" width="450.75" height="13.5" align-x="18" align-y="396" 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">Для плоского векторного поля, если оно потенциально, справедливо утверждение:</f> </text> </region> <region region-id="733" left="24" top="405.75" width="116.25" height="35.25" align-x="81" align-y="426" 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:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">P</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:lambda> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Q</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:lambda> </ml:apply> </ml:apply> </math> <rendering item-idref="11"/> </region> <region region-id="743" left="336" top="414.75" width="16.5" height="15" align-x="343.5" align-y="426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </ml:parens> </math> <rendering item-idref="12"/> </region> <region region-id="772" left="18" top="456.75" width="80.25" height="13.5" align-x="18" align-y="468" 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">Проверим его:</f> </text> </region> <region region-id="769" left="114" top="447.75" width="75" height="35.25" align-x="171.75" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">P</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:lambda> </ml:apply> <ml:symResult> <ml:real>0</ml:real> </ml:symResult> </ml:symEval> </math> <rendering item-idref="13"/> </region> <region region-id="770" left="204" top="447.75" width="75.75" height="35.25" align-x="262.5" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Q</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:lambda> </ml:apply> <ml:symResult> <ml:real>0</ml:real> </ml:symResult> </ml:symEval> </math> <rendering item-idref="14"/> </region> <region region-id="788" left="294" top="447.75" width="220.5" height="35.25" align-x="459.75" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">evalb</ml:id> <ml:apply> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">P</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:lambda> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Q</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:lambda> </ml:apply> </ml:apply> </ml:apply> <ml:symResult> <ml:apply> <ml:id xml:space="preserve">evalb</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="15"/> </region> <region region-id="729" left="18" top="498.75" width="285" height="13.5" align-x="56.25" align-y="510" 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">Следовательно, поле является потенциальным.</inlineAttr> </f> </text> </region> <region region-id="749" left="18" top="528" width="487.5" height="167.25" align-x="49.5" align-y="540" 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">2. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Проведем вычисление потенциала поля</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Выполнение условия (2) равносильно тому, что выражение </inlineAttr> </f> <region region-id="744" left="343.5" top="543" width="112.5" height="15" align-x="398.25" align-y="554.25" 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:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:id xml:space="preserve">P</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:id xml:space="preserve">dx</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:id xml:space="preserve">Q</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:id xml:space="preserve">dy</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="16"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> есть </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">полный </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">дифференциал </inlineAttr> </f> <region region-id="745" left="147.75" top="558.75" width="128.25" height="15" align-x="170.25" align-y="570" 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:equal/> <ml:id xml:space="preserve">dU</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> <ml:id xml:space="preserve">dx</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">dy</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="17"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.<sp count="2"/>Таким образом, интегрируя последнее </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">выражение, мы получим значение разности потенциалов в двух точках, выраженное </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">через криволинейный интеграл 2-го рода, не зависящий от пути интегрирования. В </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">качестве одной из точек можно </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">взять начало координат, а другую - произвольно, равно </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">как и сам путь интегрирования, соединяющий эти точки. Путь интегрирования составим </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">из двух прямолинейных отрезков, один </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">из которых соединяет точку (0,0) с точкой (Х,0), </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">а другой - соединяет точку (Х,0) с произвольной точкой (X,Y). Как видно это отрезки </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">прымых, параллельных соответствующим координатным </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">осям. Для каждого отрезка </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">найдем значение криволинейного интеграла 2-го рода, который </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">легко сводится к </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">определенному интегралу и вычисляется.</f> </inlineAttr> </text> </region> <region region-id="755" left="24" top="724.5" width="217.5" height="43.5" align-x="72" align-y="750" 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">U</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:plus/> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">X</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Q</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:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">Y</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="754" left="252" top="731.25" width="159.75" height="32.25" align-x="300" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">U</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:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">X</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">X</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">Y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="19"/> </region> <region region-id="763" left="444" top="738.75" width="16.5" height="15" align-x="451.5" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </ml:parens> </math> <rendering item-idref="20"/> </region> <region region-id="756" left="30" top="786.75" width="452.25" height="27.75" align-x="30" align-y="798" 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">Вычисленное значение криволинейного интеграла 2-го рода запишем как функцию </f> <f family="Arial Cyr" charset="204" size="12">двух </f> <f family="Arial Cyr" charset="204" size="12">переменных </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12"> и </f> <f family="Arial Cyr" charset="204" size="12"> у </f> </text> </region> <region region-id="764" left="192" top="828" width="148.5" height="37.5" align-x="233.25" align-y="852" 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">u</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:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="765" left="444" top="840.75" width="16.5" height="15" align-x="451.5" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </ml:parens> </math> <rendering item-idref="22"/> </region> <region region-id="760" left="36" top="870.75" width="204.75" height="13.5" align-x="36" align-y="882" 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">и представим её графически на рис.1</f> </text> </region> <region region-id="767" left="156" top="918" width="273" height="249.75" align-x="291.75" align-y="918" 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="23" disable-calc="false"> <inputs> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">u</ml:id> </inputs> <outputs/> </component> <rendering item-idref="24"/> </region> <region region-id="773" left="24" top="1182.75" width="477.75" height="194.25" align-x="24" align-y="1194" 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">Криволинейный интеграл 2-го рода можно интерпретировать как р</f> <f family="Arial Cyr" charset="204" size="12">аботу поля, созда- ваемого вектором </f> <f family="Arial Cyr" charset="204" size="12"> a </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> <f family="Arial Cyr" charset="204" size="12"> по перемещению заряда из точки (</f> <f family="Arial Cyr" charset="204" size="12"> 0 </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> 0 </f> <f family="Arial Cyr" charset="204" size="12">) в точку (</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">). Таким образом, каждой точке (</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">) плоскости (</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">0</f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12">) можно поставить в соответсвие так называемый её потенциал (или работу). При этом работа не будет зависеть от<sp count="3"/>выбранного пути перемещения заряда, так как она совершается в потенциальном поле. Тогда работа, совершаемая полем по перемещению заряда из одной точки в другую, будет определяться как разность потенциалов этих точек. Можно привести простое тому доказательство. Выберем любые две точки </f> <f family="Arial Cyr" charset="204" size="12">А</f> <f family="Arial Cyr" charset="204" size="12"> и </f> <f family="Arial Cyr" charset="204" size="12"> В </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">поля</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и соединим их</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">линией двумя способами: в первом - линией проходящей через начало координат, во</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">втором - нет. Таким образом, мы получим замкнутый контур </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> АВ0А </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, работа по которо-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">му в потенциальном поле рана нулю. А теперь получим работу на каждом участке </inlineAttr> </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">этого контура. Из чего будет следовать, что работа на участке </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> АВ </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12"> равна работе на участке </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> 0 </f> <f family="Arial Cyr" charset="204" size="12"> В </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">минус работа на участке </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> 0А </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, которые определяются как потенциалы в точках </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> В </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> и </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">А</f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">.</f> </inlineAttr> </text> </region> <region region-id="774" left="24" top="1410" width="437.25" height="14.25" align-x="41.25" align-y="1422" 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">3. Построим на</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="11"> <inlineAttr line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">плоскости </f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">0</f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">линии равного потенциала </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> u </inlineAttr> </f> <inlineAttr line-through="false"> <f family="Arial Cyr" charset="204" size="12">(</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">y</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">).</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> <sp/> </inlineAttr> </f> </text> </region> <region region-id="775" left="24" top="1440" width="465.75" height="69.75" align-x="24" align-y="1452" 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">. Это можно увидеть и на рис.1, если включить опцию </f> <f family="Arial Cyr" charset="204" size="12"> контур </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">. Однако,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">имеет смысл это сделать на отдельном рисунке.<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Линии семейства </f> <f family="Arial Cyr" charset="204" size="12"> u </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">)=</f> <f family="Arial Cyr" charset="204" size="12"> C </f> <f family="Arial Cyr" charset="204" size="12">эллипсов</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, представленные на </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">рис.2</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> называются<sp count="2"/>эквипо- тенциальными линиями. Приведем (4) к стандартной фор</f> <f family="Arial Cyr" charset="204" size="12">ме записи эллипса, полагая </f> <f family="Arial Cyr" charset="204" size="12"> u </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">)=</f> <f family="Arial Cyr" charset="204" size="12"> c </f> </text> </region> <region region-id="785" left="162" top="1524" width="132.75" height="51.75" align-x="279" align-y="1548" 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:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="25"/> </region> <region region-id="802" left="420" top="1536.75" width="16.5" height="15" align-x="427.5" align-y="1548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>5</ml:real> </ml:parens> </math> <rendering item-idref="26"/> </region> <region region-id="787" left="24" top="1590" width="462" height="27.75" align-x="24" align-y="1602" 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, считая </f> <f family="Arial Cyr" charset="204" size="12"> с </f> <f family="Arial Cyr" charset="204" size="12"> параметром, уравнение эллипса можно записать в следующем пара-</f> <f family="Arial Cyr" charset="204" size="12">метрическом виде, которое использовано при построении графиков рис.2,:</f> </text> </region> <region region-id="780" left="42" top="1637.25" width="180" height="17.25" align-x="93.75" align-y="1650" 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">x1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">c</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="779" left="240" top="1629.75" width="150" height="33.75" align-x="291.75" align-y="1650" 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">y1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">c</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="803" left="438" top="1638.75" width="16.5" height="15" align-x="445.5" align-y="1650" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>6</ml:real> </ml:parens> </math> <rendering item-idref="29"/> </region> <region region-id="569" left="24" top="1674.75" width="207.75" height="13.5" align-x="51" align-y="1686" 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">4. Векторные линии.</inlineAttr> </f> </text> </region> <region region-id="790" left="18" top="1704.75" width="482.25" height="40.5" align-x="18" align-y="1716" 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">Векторная линия задает геометрическое место точек, в каждой из которых касательная</f> <f family="Arial Cyr" charset="204" size="12">к этой линии совпадает с направлением вектора </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12">. Для плоского поля векторную линию</f> <f family="Arial Cyr" charset="204" size="12">определяют из дифференциального уравнения</f> </text> </region> <region region-id="793" left="42" top="1755.75" width="105" height="35.25" align-x="84" align-y="1776" 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:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="30"/> </region> <region region-id="805" left="312" top="1764.75" width="16.5" height="15" align-x="319.5" align-y="1776" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>7</ml:real> </ml:parens> </math> <rendering item-idref="31"/> </region> <region region-id="795" left="24" top="1806.75" width="449.25" height="27.75" 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">правая часть которого задает направление вектора </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, а левая - есть тангенс угла</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">наклона касательной к </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> у </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">) и между ними стоит знак равенства. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="797" left="24" top="1842" width="481.5" height="122.25" align-x="24" align-y="1854" 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">Для линий равного потенциала имеем уравнение </f> <f family="Arial Cyr" charset="204" size="12"> u </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">)=</f> <f family="Arial Cyr" charset="204" size="12"> c </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и должно выполняться </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> du </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">(</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"> y </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">=0. С другой стороны </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> du </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">)=u</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">*dx+u</f> <f family="Arial Cyr" charset="204" size="12">y</f> <f family="Arial Cyr" charset="204" size="12">*dy=0 и, следовательно, исходя из</f> <f family="Arial Cyr" charset="204" size="12">определения потенциального поля </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12">=grad(</f> <f family="Arial Cyr" charset="204" size="12"> u </f> <f family="Arial Cyr" charset="204" size="12">) и полного дифференциала d</f> <f family="Arial Cyr" charset="204" size="12"> u </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">d</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">Q</f> <f family="Arial Cyr" charset="204" size="12">d</f> <f family="Arial Cyr" charset="204" size="12">y</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">будем иметь </inlineAttr> </f> <region region-id="798" left="96.75" top="1888.5" width="94.5" height="48.75" align-x="155.25" align-y="1909.5" 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:equal/> <ml:apply> <ml:equal/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">dy</ml:id> <ml:id xml:space="preserve">dx</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">Q</ml:id> </ml:apply> <ml:id xml:space="preserve">P</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="32"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Из чего следует вывод о том, что линии равного потенциала </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">и векторные линии пересекаются под прямым углом. Аналогичные рас-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">суждения справедливы и для пространственного поля.</inlineAttr> </f> </text> </region> <region region-id="799" left="24" top="1980" width="477.75" height="123" align-x="24" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{00020906-0000-0000-C000-000000000046}" type="embedded" item-idref="33"/> <rendering item-idref="34"/> </region> <region region-id="807" left="30" top="2112.75" width="422.25" height="27" align-x="30" 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">Таким образом, решая дифференциальное уравнение (7) с разделяющимися переменными, получим</f> </text> </region> <region region-id="584" left="36" top="2155.5" width="108.75" height="20.25" align-x="84" align-y="2172" 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">y2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">c</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> <ml:id xml:space="preserve">c</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="829" left="282" top="2160.75" width="16.5" height="15" align-x="289.5" align-y="2172" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>8</ml:real> </ml:parens> </math> <rendering item-idref="36"/> </region> <region region-id="837" left="24" top="2190.75" width="462.75" height="27" align-x="68.25" align-y="2202" 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">Cемейство кривых (8), о</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">ртогональных семейству эллипсов (6), представим также на рис.2.</inlineAttr> </f> </text> </region> <region region-id="838" left="6" top="2244" width="519" height="281.25" align-x="6" align-y="2244" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="37"/> <rendering item-idref="38"/> </region> <region region-id="839" left="30" top="2544.75" width="457.5" height="41.25" align-x="30" align-y="2556" 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 векторных линиях стрелки должны указывать направление вектора </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">, определя-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">емое его проекциями </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">P</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal"> <sp count="2"/>и </inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12"> Q </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal"> на соответствующие оси координат. Исходя из этого,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">в верхней полуплоскости они направлены вверх, а в нижней - вниз.</inlineAttr> </f> </text> </region> <region region-id="841" left="30" top="2604.75" width="177" height="13.5" align-x="30" align-y="2616" 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">5. Вычислим линейный интеграл.</f> </text> </region> <region region-id="842" left="24" top="2634.75" width="462" height="54" align-x="32.25" align-y="2646" 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">Так как поле является потенциальным, то криволинейный интеграл можно<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">представить как два определенных интеграла по двум участкам интегрирования рис.3 в виде отрезков прямых, параллельных соответствующим координатным осям, и вычислить их по формуле (3).</inlineAttr> </f> </text> </region> <region region-id="845" left="30" top="2704.5" width="190.5" height="43.5" align-x="53.25" align-y="2730" 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">U1</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:real>6</ml:real> </ml:bounds> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:sequence> <ml:real>6</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:real>2</ml:real> </ml:bounds> </ml:apply> </ml:apply> 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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">Рис.3.</f> </text> </region> <region region-id="868" left="276" top="2904.75" width="123" height="15" align-x="299.25" align-y="2916" 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">U2</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:real>6</ml:real> <ml:real>2</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="43"/> </region> <region region-id="867" left="432" top="2904.75" width="52.5" height="15" align-x="454.5" align-y="2916" 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">U2</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>12</ml:real> </result> </ml:eval> </math> <rendering item-idref="44"/> </region> <region region-id="858" left="42" top="2964" width="444" height="28.5" align-x="42" align-y="2976" 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">Или же вычислить, как было сказано выше, разность потенциалов в точках </f> <f family="Arial Cyr" charset="204" size="12"> В </f> <f family="Arial Cyr" charset="204" size="12"> и </f> <f family="Arial Cyr" 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font-charset="204" font-size="12" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Constants" font-family="Arial Cyr" font-charset="204" font-size="12" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 1" font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 2" font-family="Courier New" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 3" font-family="Arial" font-charset="0" font-size="10" font-weight="bold" font-style="normal" underline="false"/> <mathStyle name="User 4" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="italic" underline="false"/> <mathStyle name="User 5" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 6" font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="User 7" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Math Text Font" font-family="Arial Cyr" font-charset="204" font-size="11" font-weight="normal" font-style="normal" underline="false"/> </mathStyles> <dimensionNames mass="mass" length="length" time="time" current="charge" thermodynamic-temperature="temperature" luminous-intensity="luminosity" amount-of-substance="substance"/> <symbolics derivation-steps-style="vertical-insert" show-comments="false" evaluate-in-place="false"/> <results> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" exponential-threshold="3" imaginary-value="i" zero-threshold="15"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </results> </mathRendering> <pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="9" orientation="portrait" print-single-page-width="false" page-width="595.5" page-height="841.5"> <margins left="56.66457" right="19.81417" top="70.86614" bottom="70.86614"/> <header use-full-page-width="true"> <left>{\rtf1\ansi\ansicpg1251\deff0\deflang1049{\fonttbl{\f0\fnil\fprq15\fcharset204{\*\fname Arial;}Arial CYR;}{\f1\fswiss\fprq2\fcharset204{\*\fname Arial;}Arial CYR;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\f0\fs18\{p\}\par \cf1\f1\fs16\{f\}\cf0\f0\fs18\par }</left> </header> <footer use-full-page-width="false"> <left>{\rtf1\ansi\ansicpg1251\deff0\deflang1049{\fonttbl{\f0\fswiss\fprq2\fcharset204{\*\fname Arial;}Arial CYR;}{\f1\fnil\fprq15\fcharset204{\*\fname Arial;}Arial CYR;}} {\colortbl ;\red0\green0\blue0;} \viewkind4\uc1\pard\cf1\f0\fs16\{d\}\cf0\f1\fs18\par }</left> <center>{\rtf1\ansi\ansicpg1251\deff0\deflang1049{\fonttbl{\f0\fnil\fprq15\fcharset204{\*\fname Arial;}Arial CYR;}} \viewkind4\uc1\pard\qc\f0\fs18\'d2\'e8\'f2\'ee\'e2 \'ca.\'c2.\par }</center> <right>{\rtf1\ansi\ansicpg1251\deff0\deflang1049{\fonttbl{\f0\fnil\fprq15\fcharset204{\*\fname Arial;}Arial CYR;}} \viewkind4\uc1\pard\qr\f0\fs18\{n\}\par }</right> </footer> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="EN" UI="EN"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="true" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="cm"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="869" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="474" left="12" top="6.75" width="486" height="161.25" align-x="25.5" align-y="18" 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">Задача 1</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Дано векторное поле<sp count="2"/> </inlineAttr> </f> <region region-id="5" left="134.25" top="21.75" width="145.5" height="15" align-x="171" align-y="33" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#000" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">P</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:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">Q</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:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="2"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">1. Проверить, что это поле является потенциальным.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">2. Найти потенциал поля </inlineAttr> </f> <region region-id="7" left="147.75" top="51" width="55.5" height="15" align-x="161.25" align-y="62.25" 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:equal/> <ml:id xml:space="preserve">u</ml:id> <ml:apply> <ml:id xml:space="preserve">u</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> </math> <rendering item-idref="3"/> </region> <inlineAttr line-through="false"> <f family="Arial Cyr" charset="204" size="12">3. Найти уравнение линий равного потенциала и изобразить линии равного потенциала на чертеже.</f> </inlineAttr> <inlineAttr line-through="false"> <f family="Arial Cyr" charset="204" size="12">4. Составить уравнение векторных линий поля и изобразить векторные линии на том же чертеже, указав стрелками направление векторных линий.</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">5. Вычислить линейный интеграл<sp count="2"/> </inlineAttr> </f> <region region-id="11" left="195.75" top="120.75" width="39.75" height="47.25" align-x="219" align-y="150" 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:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">l</ml:id> </ml:boundVars> <ml:apply> <ml:vectorize/> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">B</ml:id> </ml:bounds> </ml:apply> </math> <rendering item-idref="4"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> <sp count="2"/> </inlineAttr> </f> </text> </region> <region region-id="710" left="18" top="186.75" width="152.25" height="13.5" align-x="18" align-y="198" 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">Запишем исходные данные:</f> </text> </region> <region region-id="711" left="18" top="216.75" width="78.75" height="15" align-x="61.5" align-y="228" 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">P</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:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="712" left="120" top="216.75" width="66.75" height="15" align-x="164.25" align-y="228" 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">Q</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:mult style="auto-select"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="713" left="222" top="216.75" width="192" height="13.5" align-x="236.25" align-y="228" 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">Точка А(0,0)<sp count="11"/>Точка В(6,2)</inlineAttr> </f> </text> </region> <region region-id="742" left="456" top="216.75" width="16.5" height="15" align-x="463.5" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="7"/> </region> <region region-id="714" left="18" top="246.75" width="145.5" height="15" align-x="54.75" align-y="258" 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:equal/> <ml:apply> <ml:id xml:space="preserve">a</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">P</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:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">Q</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:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="8"/> </region> <region region-id="709" left="18" top="275.25" width="210.75" height="30" align-x="18" align-y="288" 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="14">Решение</f> <f family="Arial Cyr" charset="204" size="12">.</f> <f family="Arial Cyr" charset="204" size="12">1. </f> <f family="Arial Cyr" charset="204" size="12"> Проверка потенциальности поля </f> <f family="Arial Cyr" charset="204" size="12">.</f> </text> </region> <region region-id="480" left="18" top="324" width="483.75" height="57" align-x="18" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{00020906-0000-0000-C000-000000000046}" type="embedded" item-idref="9"/> <rendering item-idref="10"/> </region> <region region-id="717" left="18" top="384.75" width="450.75" height="13.5" align-x="18" align-y="396" 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">Для плоского векторного поля, если оно потенциально, справедливо утверждение:</f> </text> </region> <region region-id="733" left="24" top="405.75" width="116.25" height="35.25" align-x="81" align-y="426" 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:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">P</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:lambda> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Q</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:lambda> </ml:apply> </ml:apply> </math> <rendering item-idref="11"/> </region> <region region-id="743" left="336" top="414.75" width="16.5" height="15" align-x="343.5" align-y="426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </ml:parens> </math> <rendering item-idref="12"/> </region> <region region-id="772" left="18" top="456.75" width="80.25" height="13.5" align-x="18" align-y="468" 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">Проверим его:</f> </text> </region> <region region-id="769" left="114" top="447.75" width="75" height="35.25" align-x="171.75" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">P</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:lambda> </ml:apply> <ml:symResult> <ml:real>0</ml:real> </ml:symResult> </ml:symEval> </math> <rendering item-idref="13"/> </region> <region region-id="770" left="204" top="447.75" width="75.75" height="35.25" align-x="262.5" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Q</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:lambda> </ml:apply> <ml:symResult> <ml:real>0</ml:real> </ml:symResult> </ml:symEval> </math> <rendering item-idref="14"/> </region> <region region-id="788" left="294" top="447.75" width="220.5" height="35.25" align-x="459.75" align-y="468" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">evalb</ml:id> <ml:apply> <ml:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">P</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:lambda> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Q</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:lambda> </ml:apply> </ml:apply> </ml:apply> <ml:symResult> <ml:apply> <ml:id xml:space="preserve">evalb</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="15"/> </region> <region region-id="729" left="18" top="498.75" width="285" height="13.5" align-x="56.25" align-y="510" 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">Следовательно, поле является потенциальным.</inlineAttr> </f> </text> </region> <region region-id="749" left="18" top="528" width="487.5" height="167.25" align-x="49.5" align-y="540" 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">2. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Проведем вычисление потенциала поля</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Выполнение условия (2) равносильно тому, что выражение </inlineAttr> </f> <region region-id="744" left="343.5" top="543" width="112.5" height="15" align-x="398.25" align-y="554.25" 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:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:id xml:space="preserve">P</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:id xml:space="preserve">dx</ml:id> </ml:apply> <ml:apply> <ml:mult style="auto-select"/> <ml:apply> <ml:id xml:space="preserve">Q</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:id xml:space="preserve">dy</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="16"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> есть </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">полный </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">дифференциал </inlineAttr> </f> <region region-id="745" left="147.75" top="558.75" width="128.25" height="15" align-x="170.25" align-y="570" 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:equal/> <ml:id xml:space="preserve">dU</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> <ml:id xml:space="preserve">dx</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">dy</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="17"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.<sp count="2"/>Таким образом, интегрируя последнее </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">выражение, мы получим значение разности потенциалов в двух точках, выраженное </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">через криволинейный интеграл 2-го рода, не зависящий от пути интегрирования. В </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">качестве одной из точек можно </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">взять начало координат, а другую - произвольно, равно </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">как и сам путь интегрирования, соединяющий эти точки. Путь интегрирования составим </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">из двух прямолинейных отрезков, один </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">из которых соединяет точку (0,0) с точкой (Х,0), </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">а другой - соединяет точку (Х,0) с произвольной точкой (X,Y). Как видно это отрезки </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">прымых, параллельных соответствующим координатным </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">осям. Для каждого отрезка </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">найдем значение криволинейного интеграла 2-го рода, который </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">легко сводится к </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">определенному интегралу и вычисляется.</f> </inlineAttr> </text> </region> <region region-id="755" left="24" top="724.5" width="217.5" height="43.5" align-x="72" align-y="750" 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">U</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:plus/> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">X</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Q</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:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">Y</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="754" left="252" top="731.25" width="159.75" height="32.25" align-x="300" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">U</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:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">X</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">X</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">Y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="19"/> </region> <region region-id="763" left="444" top="738.75" width="16.5" height="15" align-x="451.5" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </ml:parens> </math> <rendering item-idref="20"/> </region> <region region-id="756" left="30" top="786.75" width="452.25" height="27.75" align-x="30" align-y="798" 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">Вычисленное значение криволинейного интеграла 2-го рода запишем как функцию </f> <f family="Arial Cyr" charset="204" size="12">двух </f> <f family="Arial Cyr" charset="204" size="12">переменных </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12"> и </f> <f family="Arial Cyr" charset="204" size="12"> у </f> </text> </region> <region region-id="764" left="192" top="828" width="148.5" height="37.5" align-x="233.25" align-y="852" 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">u</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:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="765" left="444" top="840.75" width="16.5" height="15" align-x="451.5" align-y="852" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </ml:parens> </math> <rendering item-idref="22"/> </region> <region region-id="760" left="36" top="870.75" width="204.75" height="13.5" align-x="36" align-y="882" 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">и представим её графически на рис.1</f> </text> </region> <region region-id="767" left="156" top="918" width="273" height="249.75" align-x="291.75" align-y="918" 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="23" disable-calc="false"> <inputs> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">u</ml:id> </inputs> <outputs/> </component> <rendering item-idref="24"/> </region> <region region-id="773" left="24" top="1182.75" width="477.75" height="194.25" align-x="24" align-y="1194" 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">Криволинейный интеграл 2-го рода можно интерпретировать как р</f> <f family="Arial Cyr" charset="204" size="12">аботу поля, созда- ваемого вектором </f> <f family="Arial Cyr" charset="204" size="12"> a </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> <f family="Arial Cyr" charset="204" size="12"> по перемещению заряда из точки (</f> <f family="Arial Cyr" charset="204" size="12"> 0 </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> 0 </f> <f family="Arial Cyr" charset="204" size="12">) в точку (</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">). Таким образом, каждой точке (</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">) плоскости (</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">0</f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12">) можно поставить в соответсвие так называемый её потенциал (или работу). При этом работа не будет зависеть от<sp count="3"/>выбранного пути перемещения заряда, так как она совершается в потенциальном поле. Тогда работа, совершаемая полем по перемещению заряда из одной точки в другую, будет определяться как разность потенциалов этих точек. Можно привести простое тому доказательство. Выберем любые две точки </f> <f family="Arial Cyr" charset="204" size="12">А</f> <f family="Arial Cyr" charset="204" size="12"> и </f> <f family="Arial Cyr" charset="204" size="12"> В </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">поля</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и соединим их</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">линией двумя способами: в первом - линией проходящей через начало координат, во</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">втором - нет. Таким образом, мы получим замкнутый контур </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> АВ0А </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, работа по которо-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">му в потенциальном поле рана нулю. А теперь получим работу на каждом участке </inlineAttr> </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">этого контура. Из чего будет следовать, что работа на участке </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> АВ </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12"> равна работе на участке </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> 0 </f> <f family="Arial Cyr" charset="204" size="12"> В </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">минус работа на участке </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> 0А </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, которые определяются как потенциалы в точках </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> В </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> и </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">А</f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">.</f> </inlineAttr> </text> </region> <region region-id="774" left="24" top="1410" width="437.25" height="14.25" align-x="41.25" align-y="1422" 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">3. Построим на</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="11"> <inlineAttr line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">плоскости </f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">0</f> <f family="Arial Cyr" charset="204" size="12"> y </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">линии равного потенциала </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> u </inlineAttr> </f> <inlineAttr line-through="false"> <f family="Arial Cyr" charset="204" size="12">(</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">x</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">y</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">).</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> <sp/> </inlineAttr> </f> </text> </region> <region region-id="775" left="24" top="1440" width="465.75" height="69.75" align-x="24" align-y="1452" 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">. Это можно увидеть и на рис.1, если включить опцию </f> <f family="Arial Cyr" charset="204" size="12"> контур </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">. Однако,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">имеет смысл это сделать на отдельном рисунке.<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Линии семейства </f> <f family="Arial Cyr" charset="204" size="12"> u </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">)=</f> <f family="Arial Cyr" charset="204" size="12"> C </f> <f family="Arial Cyr" charset="204" size="12">эллипсов</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, представленные на </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">рис.2</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> называются<sp count="2"/>эквипо- тенциальными линиями. Приведем (4) к стандартной фор</f> <f family="Arial Cyr" charset="204" size="12">ме записи эллипса, полагая </f> <f family="Arial Cyr" charset="204" size="12"> u </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">)=</f> <f family="Arial Cyr" charset="204" size="12"> c </f> </text> </region> <region region-id="785" left="162" top="1524" width="132.75" height="51.75" align-x="279" align-y="1548" 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:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="25"/> </region> <region region-id="802" left="420" top="1536.75" width="16.5" height="15" align-x="427.5" align-y="1548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>5</ml:real> </ml:parens> </math> <rendering item-idref="26"/> </region> <region region-id="787" left="24" top="1590" width="462" height="27.75" align-x="24" align-y="1602" 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, считая </f> <f family="Arial Cyr" charset="204" size="12"> с </f> <f family="Arial Cyr" charset="204" size="12"> параметром, уравнение эллипса можно записать в следующем пара-</f> <f family="Arial Cyr" charset="204" size="12">метрическом виде, которое использовано при построении графиков рис.2,:</f> </text> </region> <region region-id="780" left="42" top="1637.25" width="180" height="17.25" align-x="93.75" align-y="1650" 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">x1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">c</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> </ml:parens> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="27"/> </region> <region region-id="779" left="240" top="1629.75" width="150" height="33.75" align-x="291.75" align-y="1650" 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">y1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">c</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:sqrt/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:plus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">c</ml:id> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">φ</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="803" left="438" top="1638.75" width="16.5" height="15" align-x="445.5" align-y="1650" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>6</ml:real> </ml:parens> </math> <rendering item-idref="29"/> </region> <region region-id="569" left="24" top="1674.75" width="207.75" height="13.5" align-x="51" align-y="1686" 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">4. Векторные линии.</inlineAttr> </f> </text> </region> <region region-id="790" left="18" top="1704.75" width="482.25" height="40.5" align-x="18" align-y="1716" 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">Векторная линия задает геометрическое место точек, в каждой из которых касательная</f> <f family="Arial Cyr" charset="204" size="12">к этой линии совпадает с направлением вектора </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12">. Для плоского поля векторную линию</f> <f family="Arial Cyr" charset="204" size="12">определяют из дифференциального уравнения</f> </text> </region> <region region-id="793" left="42" top="1755.75" width="105" height="35.25" align-x="84" align-y="1776" 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:equal/> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="30"/> </region> <region region-id="805" left="312" top="1764.75" width="16.5" height="15" align-x="319.5" align-y="1776" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>7</ml:real> </ml:parens> </math> <rendering item-idref="31"/> </region> <region region-id="795" left="24" top="1806.75" width="449.25" height="27.75" 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">правая часть которого задает направление вектора </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, а левая - есть тангенс угла</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">наклона касательной к </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> у </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">) и между ними стоит знак равенства. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="797" left="24" top="1842" width="481.5" height="122.25" align-x="24" align-y="1854" 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">Для линий равного потенциала имеем уравнение </f> <f family="Arial Cyr" charset="204" size="12"> u </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">)=</f> <f family="Arial Cyr" charset="204" size="12"> c </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и должно выполняться </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> du </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">(</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"> y </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">=0. С другой стороны </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> du </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">)=u</f> <f family="Arial Cyr" charset="204" size="12"> x </f> <f family="Arial Cyr" charset="204" size="12">*dx+u</f> <f family="Arial Cyr" charset="204" size="12">y</f> <f family="Arial Cyr" charset="204" size="12">*dy=0 и, следовательно, исходя из</f> <f family="Arial Cyr" charset="204" size="12">определения потенциального поля </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12">=grad(</f> <f family="Arial Cyr" charset="204" size="12"> u </f> <f family="Arial Cyr" charset="204" size="12">) и полного дифференциала d</f> <f family="Arial Cyr" charset="204" size="12"> u </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">d</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">Q</f> <f family="Arial Cyr" charset="204" size="12">d</f> <f family="Arial Cyr" charset="204" size="12">y</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">будем иметь </inlineAttr> </f> <region region-id="798" left="96.75" top="1888.5" width="94.5" height="48.75" align-x="155.25" align-y="1909.5" 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:equal/> <ml:apply> <ml:equal/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">dy</ml:id> <ml:id xml:space="preserve">dx</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">Q</ml:id> </ml:apply> <ml:id xml:space="preserve">P</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="32"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Из чего следует вывод о том, что линии равного потенциала </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">и векторные линии пересекаются под прямым углом. Аналогичные рас-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">суждения справедливы и для пространственного поля.</inlineAttr> </f> </text> </region> <region region-id="799" left="24" top="1980" width="477.75" height="123" align-x="24" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{00020906-0000-0000-C000-000000000046}" type="embedded" item-idref="33"/> <rendering item-idref="34"/> </region> <region region-id="807" left="30" top="2112.75" width="422.25" height="27" align-x="30" 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">Таким образом, решая дифференциальное уравнение (7) с разделяющимися переменными, получим</f> </text> </region> <region region-id="584" left="36" top="2155.5" width="108.75" height="20.25" align-x="84" align-y="2172" 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">y2</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">c</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> <ml:id xml:space="preserve">c</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <region region-id="829" left="282" top="2160.75" width="16.5" height="15" align-x="289.5" align-y="2172" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>8</ml:real> </ml:parens> </math> <rendering item-idref="36"/> </region> <region region-id="837" left="24" top="2190.75" width="462.75" height="27" align-x="68.25" align-y="2202" 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">Cемейство кривых (8), о</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">ртогональных семейству эллипсов (6), представим также на рис.2.</inlineAttr> </f> </text> </region> <region region-id="838" left="6" top="2244" width="519" height="281.25" align-x="6" align-y="2244" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="37"/> <rendering item-idref="38"/> </region> <region region-id="839" left="30" top="2544.75" width="457.5" height="41.25" align-x="30" align-y="2556" 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 векторных линиях стрелки должны указывать направление вектора </f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">, определя-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">емое его проекциями </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">P</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal"> <sp count="2"/>и </inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12"> Q </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal"> на соответствующие оси координат. Исходя из этого,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">в верхней полуплоскости они направлены вверх, а в нижней - вниз.</inlineAttr> </f> </text> </region> <region region-id="841" left="30" top="2604.75" width="177" height="13.5" align-x="30" align-y="2616" 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">5. Вычислим линейный интеграл.</f> </text> </region> <region region-id="842" left="24" top="2634.75" width="462" height="54" align-x="32.25" align-y="2646" 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">Так как поле является потенциальным, то криволинейный интеграл можно<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">представить как два определенных интеграла по двум участкам интегрирования рис.3 в виде отрезков прямых, параллельных соответствующим координатным осям, и вычислить их по формуле (3).</inlineAttr> </f> </text> </region> <region region-id="845" left="30" top="2704.5" width="190.5" height="43.5" align-x="53.25" align-y="2730" 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">U1</ml:id> <ml:apply> <ml:plus/> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">P</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:real>6</ml:real> </ml:bounds> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Q</ml:id> <ml:sequence> <ml:real>6</ml:real> <ml:id xml:space="preserve">y</ml:id> </ml:sequence> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:real>2</ml:real> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="847" left="264" top="2718.75" width="52.5" height="15" align-x="286.5" align-y="2730" 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">U1</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>12</ml:real> </result> </ml:eval> </math> <rendering item-idref="40"/> </region> <region region-id="856" left="30" top="2814" width="154.5" height="136.5" align-x="30" align-y="2814" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <metafile mapping-mode="8" x-extent="153.0142" y-extent="135.2126" item-idref="41"/> <rendering item-idref="42"/> </region> <region region-id="869" left="210" top="2904.75" width="33" height="13.5" align-x="210" align-y="2916" 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">Рис.3.</f> </text> </region> <region region-id="868" left="276" top="2904.75" width="123" height="15" align-x="299.25" align-y="2916" 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">U2</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:real>6</ml:real> <ml:real>2</ml:real> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">U</ml:id> <ml:sequence> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="43"/> </region> <region region-id="867" left="432" top="2904.75" width="52.5" height="15" align-x="454.5" align-y="2916" 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">U2</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>12</ml:real> </result> </ml:eval> </math> <rendering item-idref="44"/> </region> <region region-id="858" left="42" top="2964" width="444" height="28.5" align-x="42" align-y="2976" 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">Или же вычислить, как было сказано выше, разность потенциалов в точках </f> <f family="Arial Cyr" charset="204" size="12"> В </f> <f family="Arial Cyr" charset="204" size="12"> и </f> <f family="Arial Cyr" charset="204" size="12"> А: </f> <f family="Arial Cyr" charset="204" size="12"> U </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">2=</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">U</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> B </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">)-</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> U </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">(</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> A </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">).</inlineAttr> </f> </text> </region> <region region-id="671" left="546" top="3372.75" width="14.25" height="15" align-x="552" align-y="3384" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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