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</ml:apply> </ml:lambda> </ml:apply> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">R</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="17"/> </region> <region region-id="1122" left="18" top="787.5" width="193.5" height="20.25" align-x="87" align-y="804" 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">divb</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id 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</region> <region region-id="1152" left="462" top="792.75" width="16.5" height="15" align-x="469.5" align-y="804" 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="1215" left="24" top="825.75" width="474" height="33" align-x="24" align-y="840" 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"> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">При этом, если div</f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>b</i> </b> </f> <f family="Arial Cyr" charset="204" size="12">>0, то<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">в данной области пространства, где х</f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">+у</f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">+z</f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">>1/3, </f> <f family="Arial Cyr" charset="204" size="12">находятся источники. А при </f> <f family="Arial Cyr" charset="204" size="12">div</f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>b</i> </b> </f> <inlineAttr font-weight="normal" font-style="normal"> <f family="Arial Cyr" charset="204" size="12"><</f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">0, то есть</f> <f family="Arial Cyr" charset="204" size="12"> в области </f> <f family="Arial Cyr" charset="204" size="12">х</f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">+у</f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">+z</f> <sup> <f family="Arial Cyr" charset="204" size="12">2</f> </sup> <f family="Arial Cyr" charset="204" size="12"><</f> <f family="Arial Cyr" charset="204" size="12">1/3, будут </f> <f family="Arial Cyr" charset="204" size="12">находиться</f> <f family="Arial Cyr" charset="204" size="12"> стоки.</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal"> <sp/> </inlineAttr> </f> </p> </text> </region> <region region-id="1211" left="30" top="882" width="469.5" height="14.25" align-x="30" align-y="894" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <b>1.1</b> </f> <f family="Arial Cyr" charset="204" size="12">. </f> <f family="Arial Cyr" charset="204" size="12"> <i>Найдем поток векторного поля через замкнутую поверхность</i> </f> <f family="Arial Cyr" charset="204" size="12"> <i> G</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.</inlineAttr> </f> </p> </text> </region> <region region-id="1216" left="24" top="915" width="453" height="59.25" align-x="24" align-y="930" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Вычислим поток векторного поля </f> <f family="Arial Cyr" charset="204" size="12"> <i>b </i> </f> <f family="Arial Cyr" charset="204" size="12">через замкнутую поверхность G: </f> <f family="Arial Cyr" charset="204" size="12">х</f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">+у</f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">+z</f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">=r</f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">,</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">являющуюся сферой радиуса </f> <i> <f family="Arial Cyr" charset="204" size="12">r</f> </i> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12">. Для этого используем формулу Остроградского</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">и вычислим тройной интеграл от </f> <f family="Arial Cyr" charset="204" size="12">div</f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>b </i> </b> </f> <f family="Arial Cyr" charset="204" size="12">по объему </f> <f family="Arial Cyr" charset="204" size="12"> <i>V</i> </f> <f family="Arial Cyr" charset="204" size="12">, границей которого является по-</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">верхность G:</f> </p> </text> </region> <region region-id="1172" left="120" top="996" width="168" height="30" align-x="120" align-y="996" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="21"/> <rendering item-idref="22"/> </region> <region region-id="1306" left="420" top="1008.75" width="16.5" height="15" align-x="427.5" align-y="1020" 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="23"/> </region> <region region-id="392" left="162" top="1026.75" width="288" height="13.5" align-x="163.5" align-y="1038" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> <sp/> </inlineAttr> </f> </p> </text> </region> <region region-id="1175" left="30" top="1050.75" width="467.25" height="137.25" align-x="30" align-y="1062" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Интеграл (4) от дивергенции векторного поля </f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>b </i> </b> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">, распространенный по всему объему</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> <i>V </i> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">, равен [13] потоку вектора через поверхность, ограничивающую данный объем,</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">в направлении внешней нормали </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>n</i> </b> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">.</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <inlineAttr font-weight="normal" font-style="normal"> <f family="Arial Cyr" charset="204" size="12">Если вектор<sp count="2"/> </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <i> <b>b </b> </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">ассоциировать со скоростью жидкости, протекающей через область</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <i> <inlineAttr font-weight="normal">V</inlineAttr> </i> </f> <inlineAttr font-weight="normal" font-style="normal"> <f family="Arial Cyr" charset="204" size="12">, то интеграл </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <i> <inlineAttr font-weight="normal">vg </inlineAttr> </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">дает количество жидкости, втекающей (</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i>vg</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"><0</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">) в область или вытека-</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">ющей (</inlineAttr> </f> <i> <f family="Arial Cyr" charset="204" size="12">vg</f> </i> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">></f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">0</f> <inlineAttr font-weight="normal" font-style="normal"> <f family="Arial Cyr" charset="204" size="12">) через поверхность </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <i> <inlineAttr font-weight="normal">G</inlineAttr> </i> </f> <f family="Arial Cyr" charset="204" size="12"> <i> <inlineAttr font-weight="normal"> <sp/> </inlineAttr> </i> </f> <inlineAttr font-weight="normal" font-style="normal"> <f family="Arial Cyr" charset="204" size="12">в единицу времени. Если </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">div</f> <i> <b> <f family="Arial Cyr" charset="204" size="12">b</f> </b> </i> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">=0, то двойной интег-</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">рал по любой замкнутой поверхности равен нулю, то есть количество </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">вытекающей</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">(втекающей) через любую замкнутую поверхность жидкости будет равно нулю. Точ-</f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">нее говоря, </f> <f family="Arial Cyr" charset="204" size="12">количество </f> <f family="Arial Cyr" charset="204" size="12">вытекающей </f> <f family="Arial Cyr" charset="204" size="12">жидкости </f> <f family="Arial Cyr" charset="204" size="12">из этой области</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12">равно количеству </f> <f family="Arial Cyr" charset="204" size="12">втекающей<sp count="2"/>жидкости в область.</f> </p> </text> </region> <region region-id="1176" left="36" top="1200" width="449.25" height="31.5" align-x="36" align-y="1212" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Вычисление интеграла </f> <f family="Arial Cyr" charset="204" size="12"> <i>v</i> </f> <f family="Arial Cyr" charset="204" size="12"> <i>g</i> </f> <f family="Arial Cyr" charset="204" size="12"> <i> <sp/> </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">проведем в сферической системе координат, в которой</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">якобиан равен </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i>r</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> <sup>2</sup> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">*cos(</inlineAttr> </f> <f family="Symbol" charset="2" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">q</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">) и </inlineAttr> </f> <f family="Arial" charset="0" size="12"> <inlineAttr font-style="normal"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">х</f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">+у</f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">+z</f> <sup> <f family="Arial Cyr" charset="204" size="12">2</f> </sup> <f family="Arial Cyr" charset="204" size="12">=r</f> <f family="Arial Cyr" charset="204" size="12"> <sup>2</sup> </f> <f family="Arial Cyr" charset="204" size="12">:</f> </p> </text> </region> <region region-id="1154" left="42" top="1237.5" width="295.5" height="100.5" align-x="79.5" align-y="1284" 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">vg</ml:id> <ml:boundVars> <ml:id xml:space="preserve">rr</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">φ</ml:id> </ml:boundVars> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">θ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">θ</ml:id> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">rr</ml:id> </ml:bounds> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:apply> <ml:div/> <ml:apply> <ml:neg/> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:bounds> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="24"/> </region> <region region-id="1218" left="480" top="1272.75" width="16.5" height="15" align-x="487.5" align-y="1284" 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="25"/> </region> <region region-id="1217" left="42" top="1349.25" width="148.5" height="32.25" align-x="79.5" align-y="1368" 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">vg</ml:id> <ml:id xml:space="preserve">rr</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>12</ml:real> <ml:real>5</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">rr</ml:id> <ml:real>5</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:real>3</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">rr</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="26"/> </region> <region region-id="1221" left="36" top="1414.5" width="466.5" height="61.5" align-x="36" align-y="1434" 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"> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Очевидно, в области </f> <f family="Arial Cyr" charset="204" size="12"> <i>V</i> </f> <f family="Arial Cyr" charset="204" size="12">1, ограниченной сферой </f> <f family="Arial Cyr" charset="204" size="12"> <i>G</i> </f> <f family="Arial Cyr" charset="204" size="12">1 </f> <f family="Arial Cyr" charset="204" size="12">радиуса</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/> </f> <region region-id="1219" left="366" top="1415.25" width="45.75" height="33.75" align-x="384" align-y="1434" 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">r1</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="27"/> </region> <f family="Arial Cyr" charset="204" size="12"> , будут нахо- диться, </f> <f family="Arial Cyr" charset="204" size="12">как было показано выше, </f> <f family="Arial Cyr" charset="204" size="12">только стоки и поток векторного поля через эту область </f> <f family="Arial Cyr" charset="204" size="12">будет отрицательным. Проверим это численно:<sp count="2"/> </f> </p> </text> </region> <region region-id="1168" left="42" top="1491.75" width="120" height="47.25" align-x="95.25" align-y="1524" 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">vg</ml:id> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:sqrt/> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>-8</ml:real> <ml:real>135</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:pow/> <ml:real>3</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="28"/> </region> <region region-id="1170" left="36" top="1554" width="456.75" height="42" align-x="36" align-y="1566" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Расширяя область </f> <f family="Arial Cyr" charset="204" size="12"> <i>V</i> </f> <f family="Arial Cyr" charset="204" size="12">1 путем увеличения rr в<sp count="2"/>(5), мы тем самым в эту область </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">будем вовлекать все новые источники, и, следовательно, наступит момент, когда</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <i>vg</i> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> <i>rr</i> </f> <f family="Arial Cyr" charset="204" size="12">)=0.</f> </p> </text> </region> <region region-id="1223" left="36" top="1608" width="456.75" height="62.25" align-x="36" align-y="1620" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Иными словами, можно найти такое </f> <f family="Arial Cyr" charset="204" size="12"> <i>rr</i> </f> <f family="Arial Cyr" charset="204" size="12">, когда мощность стоков будет равна мощ-</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">ности источников - источники поглащаются стоками и возникает баланс между ними.</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Таковым значением </f> <f family="Arial Cyr" charset="204" size="12"> <i>rr</i> </f> <f family="Arial Cyr" charset="204" size="12"> будет<sp count="2"/> </f> <region region-id="1179" left="198" top="1629.75" width="45.75" height="33.75" align-x="216" 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:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">r2</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>5</ml:real> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="29"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> см. график рис.1.</inlineAttr> </f> </p> </text> </region> <region region-id="1202" left="54" top="1691.25" width="150.75" height="32.25" align-x="88.5" align-y="1710" 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">vr</ml:id> <ml:boundVars> <ml:id xml:space="preserve">rr</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">rr</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>5</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">rr</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="1201" left="234" top="1689.75" width="48" height="33.75" align-x="252.75" align-y="1710" 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">r2</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:sqrt/> <ml:real>5</ml:real> </ml:apply> <ml:real>3</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="1200" left="306" top="1698.75" width="64.5" height="15" align-x="324" align-y="1710" 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">r2</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.7453559924999299</ml:real> </result> </ml:eval> </math> <rendering item-idref="32"/> </region> <region region-id="1199" left="384" top="1698.75" width="60" height="15" align-x="420.75" align-y="1710" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">vr</ml:id> <ml:id xml:space="preserve">r2</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="33"/> </region> <region region-id="1203" left="462" top="1698.75" width="16.5" height="15" align-x="469.5" align-y="1710" 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="34"/> </region> <region region-id="1197" left="42" top="1758" width="384.75" height="192" align-x="42" align-y="1758" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="35"/> <rendering item-idref="36"/> </region> <region region-id="1224" left="30" top="1998" width="452.25" height="27.75" align-x="30" align-y="2010" 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"> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">На рис.1 в зависимости от радиуса </f> <f family="Arial Cyr" charset="204" size="12"> <i>rr</i> </f> <f family="Arial Cyr" charset="204" size="12"> сферы </f> <f family="Arial Cyr" charset="204" size="12"> <i>G</i> </f> <f family="Arial Cyr" charset="204" size="12"> видно как изменяется поток вектора </f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>b</i> </b> </f> <f family="Arial Cyr" charset="204" size="12"> с отрицательного значения на положительное .</f> </p> </text> </region> <region region-id="1229" left="18" top="2064" width="416.25" height="29.25" align-x="18" align-y="2076" 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"> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <b>2</b> </f> <f family="Arial Cyr" charset="204" size="12">. </f> <f family="Arial Cyr" charset="204" size="12"> <i>Найдем поток векторного поля через замкнутую поверхность</i> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <i>б</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">,<sp count="29"/> </inlineAttr> </f> <i> <f family="Arial Cyr" charset="204" size="12"> <sp count="4"/>ограничивающую пространственную область</f> </i> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <i>Vb</i> </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> </p> </text> </region> <region region-id="1356" left="18" top="2112.75" width="480" height="41.25" align-x="25.5" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Times New Roman CYR" charset="204" size="12"> <inlineAttr line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Построим поверхности двух фигур - тел вращения : сферы и конуса рис.3. Для этого</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">с</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">интезируем профиль тела вращения, которое будем</f> <f family="Arial Cyr" charset="204" size="12"> проводить </f> <f family="Arial Cyr" charset="204" size="12">вокруг</f> <f family="Arial Cyr" charset="204" size="12"> оси абсцисс,</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">и</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>з</f> <f family="Arial Cyr" charset="204" size="12">ададим его параметры, исходя из (2):</f> </p> </text> </region> <region region-id="1266" left="18" top="2178.75" 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optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="e3367df2-6cf4-4c5c-bb5e-6bca819bd0dc" version-id="9c56af4f-11d0-46b9-b57a-7922bd091bb0" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="true" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\КратИнтТеорПоля\СферичГрафика.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="e3367df2-6cf4-4c5c-bb5e-6bca819bd0dc" version-id="9c56af4f-11d0-46b9-b57a-7922bd091bb0" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="true" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\КратИнтТеорПоля\СферичГрафика.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment 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background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Построим функцию самого профиля рис.2</f> </p> </text> </region> <region region-id="1278" left="24" top="2250" width="250.5" height="16.5" align-x="63.75" align-y="2262" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="e3367df2-6cf4-4c5c-bb5e-6bca819bd0dc" version-id="9c56af4f-11d0-46b9-b57a-7922bd091bb0" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="true" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют 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<ml:lessOrEqual/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x1</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">q</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">y1</ml:id> <ml:id xml:space="preserve">x1</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">y2</ml:id> <ml:id xml:space="preserve">x1</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="42"> <element-image-map> <box left="1.5" top="0.75" width="249" height="15" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="1277" left="36" top="2298" width="245.25" height="151.5" align-x="36" align-y="2298" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="43"/> <rendering item-idref="44"/> </region> <region region-id="1288" left="42" top="2484" width="460.5" height="56.25" align-x="42" align-y="2496" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">Вращение </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i>f</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">1(</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i>u</i> </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">) </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">рис</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.2</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i> <sp/> </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">вокруг оси абсцисс з</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">адаст пространственную область (2) </f> <f family="Arial Cyr" charset="204" size="12"> <i>Vb</i> </f> <f family="Arial Cyr" charset="204" size="12">, ограниченную замкнутой повехностью </f> <f family="Arial Cyr" charset="204" size="12"> <i>б</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, через которую требуется определить поток векторного поля. Запишем необходиме функции, задающие конструкцию </inlineAttr> </f> <i> <f family="Arial Cyr" charset="204" size="12">Vb</f> </i> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, </inlineAttr> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и изобразим её на графике рис.3.</inlineAttr> </f> </p> </text> </region> <region region-id="1301" left="42" top="2562.75" width="39" height="15" align-x="63.75" align-y="2574" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="e3367df2-6cf4-4c5c-bb5e-6bca819bd0dc" version-id="9c56af4f-11d0-46b9-b57a-7922bd091bb0" branch-id="00000000-0000-0000-0000-000000000000" 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version-id="9c56af4f-11d0-46b9-b57a-7922bd091bb0" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="true" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\КратИнтТеорПоля\СферичГрафика.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">5f9a9a26e70150d7def157cb0dddb59b</contentHash> <ml:define> <ml:function> <ml:id xml:space="preserve">G1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">v</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f1</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">v</ml:id> </ml:apply> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="49"> <element-image-map> <box left="1.5" top="0.75" width="124.5" height="13.5" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="1291" left="264" top="2592.75" width="121.5" height="15" align-x="315" align-y="2604" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="e3367df2-6cf4-4c5c-bb5e-6bca819bd0dc" version-id="9c56af4f-11d0-46b9-b57a-7922bd091bb0" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="true" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\КратИнтТеорПоля\СферичГрафика.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="e3367df2-6cf4-4c5c-bb5e-6bca819bd0dc" version-id="9c56af4f-11d0-46b9-b57a-7922bd091bb0" branch-id="00000000-0000-0000-0000-000000000000" revision-num="13" is-modified="true" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\КратИнтТеорПоля\СферичГрафика.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">6492ea4f89f02f6a81e66748dff073bf</contentHash> <ml:define warning="WarnRedefinedBIFunction"> <ml:function> <ml:id xml:space="preserve">H1</ml:id> <ml:boundVars> <ml:id xml:space="preserve">u</ml:id> <ml:id xml:space="preserve">v</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">f1</ml:id> <ml:id xml:space="preserve">u</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">v</ml:id> </ml:apply> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="50"> <element-image-map> <box left="1.5" top="0.75" width="120" height="13.5" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="1292" left="42" top="2628" width="283.5" height="16.5" align-x="65.25" align-y="2640" 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">S1</ml:id> <ml:apply> <ml:id xml:space="preserve">CreateMesh</ml:id> <ml:sequence> <ml:id xml:space="preserve">H1</ml:id> <ml:id xml:space="preserve">G1</ml:id> <ml:id xml:space="preserve">F1</ml:id> <ml:id xml:space="preserve">a1</ml:id> <ml:id xml:space="preserve">b1</ml:id> <ml:real>0</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">mesh</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="51"/> </region> <region region-id="1305" left="42" top="2664" width="448.5" height="27.75" align-x="42" align-y="2676" 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"> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <i>Примечание</i> </f> <f family="Arial Cyr" charset="204" size="12">. Перичисление H1, G1, F1 в этой последовательности соответствует совпадению оси вращения с аппликатой.</f> </p> </text> </region> <region region-id="1336" left="48" top="2706" width="435.75" height="28.5" align-x="48" align-y="2718" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Теперь можно провести вычисление потока векторного поля </f> <f family="Arial Cyr" charset="204" size="12"> <i>b </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">через замкнутую</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">поверхность </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i>б </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">согласно его определению (4)</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.</inlineAttr> </f> </p> </text> </region> <region region-id="1300" left="36" top="2808" width="231" height="241.5" align-x="150.75" align-y="2808" 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="52" disable-calc="false"> <inputs> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">S1</ml:id> </inputs> <outputs/> </component> <rendering item-idref="53"/> </region> <region region-id="1307" left="30" top="3078.75" width="488.25" height="27" align-x="30" align-y="3090" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Воспользуемся для этого формулой (5), изменив соответствующие пределы интегри- рования таким образом:</f> </p> </text> </region> <region region-id="1310" left="36" top="3121.5" width="290.25" height="100.5" align-x="73.5" align-y="3168" 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">vb</ml:id> <ml:boundVars> <ml:id xml:space="preserve">rr</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">φ</ml:id> </ml:boundVars> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">θ</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">θ</ml:id> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">rr</ml:id> </ml:bounds> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>4</ml:real> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">π</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:bounds> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="54"/> </region> <region region-id="1311" left="366" top="3135.75" width="138.75" height="45.75" align-x="402.75" align-y="3168" 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">vb</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>8</ml:real> <ml:real>15</ml:real> </ml:apply> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:real>15</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:pow/> <ml:real>2</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="55"/> </region> <region region-id="1337" left="30" top="3240.75" width="258.75" height="13.5" align-x="30" align-y="3252" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Итак, поток векторного поля вычислен и равен:</f> </p> </text> </region> <region region-id="1338" left="306" top="3240.75" width="82.5" height="15" align-x="342" align-y="3252" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">vb</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.49074729840565889</ml:real> </result> </ml:eval> </math> <rendering item-idref="56"/> </region> <region region-id="1339" left="30" top="3270.75" width="174.75" height="13.5" align-x="30" align-y="3282" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Для сравнения укажем, что:</f> </p> </text> </region> <region region-id="1357" left="252" top="3270.75" width="79.5" height="15" align-x="285" align-y="3282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">vr</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>3.3510321638291125</ml:real> </result> </ml:eval> </math> <rendering item-idref="57"/> </region> <region region-id="1341" left="30" top="3294" width="465.75" height="27.75" align-x="30" align-y="3306" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <i>Примечание</i> </f> <f family="Arial Cyr" charset="204" size="12">. Приведем вычисление того же интеграла </f> <f family="Arial Cyr" charset="204" size="12"> <i>vb</i> </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> <i>rr</i> </f> <f family="Arial Cyr" charset="204" size="12">), но в цилиндрической системе координат:</f> </p> </text> </region> <region region-id="1334" left="54" top="3335.25" width="338.25" height="68.25" align-x="278.25" align-y="3378" 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:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">r</ml:id> </ml:boundVars> <ml:apply> <ml:integral auto-algorithm="true"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">z</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">r</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">r</ml:id> <ml:apply> <ml:sqrt/> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">r</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:bounds> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> <ml:symResult> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">π</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:div/> <ml:real>4</ml:real> <ml:real>15</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>2</ml:real> <ml:real>15</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:real>2</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="58"/> </region> <region region-id="944" left="30" top="3432" width="459" height="50.25" align-x="30" align-y="3432" 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="59"/> <rendering item-idref="60"/> </region> <region region-id="1343" left="24" top="3510.75" width="318" height="13.5" align-x="24" align-y="3522" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Ротор можно записать и определить следующим образом:</f> </p> </text> </region> <region region-id="951" left="6" top="3546" width="518.25" height="51.75" align-x="6" align-y="3546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE02-0000-0000-C000-000000000046}" type="embedded" item-idref="61"/> <rendering item-idref="62"/> </region> <region region-id="1346" left="36" top="3618.75" width="330" height="13.5" align-x="36" align-y="3630" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Проведем соответствующие вычисления компонент ротора:</f> </p> </text> </region> <region region-id="1345" left="42" top="3651.75" width="162" height="35.25" align-x="186.75" align-y="3672" 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:minus/> <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">R</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">z</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:id xml:space="preserve">z</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> <ml:symResult> <ml:real>2</ml:real> </ml:symResult> </ml:symEval> </math> <rendering item-idref="63"/> </region> <region region-id="1348" left="234" top="3651.75" width="168.75" height="35.25" align-x="385.5" align-y="3672" 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:plus/> <ml:apply> <ml:neg/> <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">R</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">y</ml:id> <ml:id xml:space="preserve">z</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">z</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:id xml:space="preserve">z</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> <ml:symResult> <ml:real>5</ml:real> </ml:symResult> </ml:symEval> </math> <rendering item-idref="64"/> </region> <region region-id="1350" left="48" top="3705.75" width="168.75" height="35.25" align-x="192.75" align-y="3726" 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:minus/> <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:id xml:space="preserve">z</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> <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:id xml:space="preserve">z</ml:id> </ml:sequence> </ml:apply> </ml:lambda> </ml:apply> </ml:apply> <ml:symResult> <ml:real>-2</ml:real> </ml:symResult> </ml:symEval> </math> <rendering item-idref="65"/> </region> <region region-id="1589" left="24" top="3768" width="188.25" height="14.25" align-x="62.25" align-y="3780" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">Следовательно, р</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">отор </f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>b</i> </b> </f> <f family="Arial Cyr" charset="204" size="12"> <i> <sp/> </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" line-through="false">равен</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">: </inlineAttr> </f> </p> </text> </region> <region region-id="1588" left="216" top="3768.75" width="146.25" height="15" align-x="279.75" align-y="3780" 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">rotb</ml:id> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">k</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="66"/> </region> <region region-id="1590" left="384" top="3748.5" width="69.75" height="55.5" align-x="412.5" align-y="3780" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">rotb</ml:id> <ml:matrix rows="3" cols="1"> <ml:real>2</ml:real> <ml:real>5</ml:real> <ml:real>-2</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="67"/> </region> <region region-id="616" left="24" top="3816" width="458.25" height="46.5" align-x="24" align-y="3816" 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="68"/> <rendering item-idref="69"/> </region> <region region-id="1361" left="24" top="3870.75" width="450.75" height="56.25" align-x="24" align-y="3882" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Приведем формулу Стокса [13], которая устанавливает зависимость между интег-</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">ралом по поверхности </f> <f family="Arial Cyr" charset="204" size="12"> <i>Б </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и криволинейным интегралом по границе </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i>L </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">этой поверх-</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">ности. Причем обход по замкнутому контуру </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i>L </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">совершается в положительном на-</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">правлении относительно выбранной </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">к поверхности </f> <f family="Arial Cyr" charset="204" size="12"> <i>Б </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">положительной</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> нормали </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>n</i> </b> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.</inlineAttr> </f> </p> </text> </region> <region region-id="1363" left="66" top="3942" width="180" height="29.25" align-x="66" align-y="3942" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="70"/> <rendering item-idref="71"/> </region> <region region-id="1365" left="402" top="3948.75" width="16.5" height="15" align-x="409.5" align-y="3960" 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="72"/> </region> <region region-id="1535" left="30" top="3984.75" width="468.75" height="68.25" align-x="30" align-y="3996" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">В данном случае положительная нормаль определяется как вектор, образующий с</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">осью 0z острый угол. После чего выбирается положительный обход контура </f> <f family="Arial Cyr" charset="204" size="12"> <i>L</i> </f> <f family="Arial Cyr" charset="204" size="12">. Он</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">должен происходить против часовой стрелки, если смотреть из конца вектора </f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>n</i> </b> </f> <f family="Arial Cyr" charset="204" size="12">.</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">В</f> <f family="Arial Cyr" charset="204" size="12">ектор </f> <i> <f family="Arial Cyr" charset="204" size="12"> <b>n </b> </f> </i> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">можно задать как векторное произведение двух векторов </inlineAttr> </f> <b> <i> <f family="Arial Cyr" charset="204" size="12">АС</f> </i> </b> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal"> и </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>ВА </i> </b> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">рис.4.</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">Зададим каждый из этих векторов в отдельности:</inlineAttr> </f> </p> </text> </region> <region region-id="1546" left="30" top="4060.5" width="66.75" height="55.5" align-x="55.5" align-y="4092" 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">AC</ml:id> <ml:matrix rows="3" cols="1"> <ml:real>0</ml:real> <ml:real>-1</ml:real> <ml:real>0</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="73"/> </region> <region region-id="1547" left="132" top="4060.5" width="66" height="55.5" align-x="156.75" align-y="4092" 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">BA</ml:id> <ml:matrix rows="3" cols="1"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>-1</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="74"/> </region> <region region-id="1548" left="234" top="4080.75" width="70.5" height="15" align-x="248.25" align-y="4092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:crossProduct/> <ml:id xml:space="preserve">AC</ml:id> <ml:id xml:space="preserve">BA</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="75"/> </region> <region region-id="1549" left="336" top="4060.5" width="54" height="55.5" align-x="349.5" align-y="4092" 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">n</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="76"/> </region> <region region-id="1550" left="408" top="4059.75" width="51" height="36" align-x="432.75" align-y="4092" 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:absval/> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:pow/> <ml:real>2</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="77"/> </region> <region region-id="1558" left="30" top="4152.75" width="243" height="13.5" align-x="30" align-y="4164" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Запишем единичный нормальный вектор </f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>n1</i> </b> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">:</inlineAttr> </f> </p> </text> </region> <region region-id="1559" left="300" top="4145.25" width="51" height="32.25" align-x="321" align-y="4164" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n1</ml:id> <ml:apply> <ml:div/> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="78"/> </region> <region region-id="1560" left="378" top="4132.5" width="84" height="55.5" align-x="398.25" align-y="4164" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n1</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:real>0.70710678118654746</ml:real> <ml:real>0</ml:real> <ml:real>0.70710678118654746</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="79"/> </region> <region region-id="1576" left="24" top="4212.75" width="470.25" height="27.75" align-x="24" align-y="4224" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Найденный единичный нормальный вектор </f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>n1 </i> </b> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">позволяет записать уравнение плос-</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">кости, в которой лежит контур </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i> <inlineAttr font-weight="normal">L </inlineAttr> </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">рис.4</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">:</inlineAttr> </f> </p> </text> </region> <region region-id="1574" left="72" top="4253.25" width="83.25" height="33.75" align-x="139.5" align-y="4272" 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:id xml:space="preserve">x</ml:id> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">z</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="80"/> </region> <region region-id="1573" left="186" top="4260.75" width="21" height="13.5" align-x="186" align-y="4272" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">или </f> </p> </text> </region> <region region-id="1572" left="252" top="4253.25" width="83.25" height="33.75" align-x="310.5" align-y="4272" 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:id xml:space="preserve">x</ml:id> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">z</ml:id> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="81"/> </region> <region region-id="1575" left="24" top="4294.5" width="462" height="17.25" align-x="24" align-y="4308" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Из уравнения следует, что расстояние плоскости до начала координат равно </f> <region region-id="1567" left="450" top="4295.25" width="16.5" height="16.5" align-x="453" align-y="4308" 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:sqrt/> <ml:real>2</ml:real> </ml:apply> </math> <rendering item-idref="82"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.</inlineAttr> </f> </p> </text> </region> <region region-id="1367" left="24" top="4320" width="318" height="14.25" align-x="24" align-y="4332" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <i>Приведем графическую иллюстрацию контура</i> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <i>L </i> </f> <f family="Arial Cyr" charset="204" size="12"> <i>рис</i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.4</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.</inlineAttr> </f> </p> </text> </region> <region region-id="1385" left="24" top="4350.75" width="217.5" height="13.5" align-x="24" align-y="4362" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Зададим поверхность z(x,y) - плоскость:</f> </p> </text> </region> <region region-id="1386" left="294" top="4350.75" width="77.25" height="15" align-x="336" align-y="4362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="64b2f7ca-8cac-44d5-8e1b-f0fffbd30a10" version-id="c6a9a43f-b98d-4cea-afd0-70004a63352c" branch-id="00000000-0000-0000-0000-000000000000" revision-num="8" 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xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:real>2</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="83"> <element-image-map> <box left="1.5" top="0.75" width="75.75" height="13.5" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="1387" left="24" top="4398" width="99.75" height="14.25" align-x="24" align-y="4410" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Зададим контур </f> <f family="Arial Cyr" charset="204" size="12"> <i>L</i> </f> <f family="Arial Cyr" charset="204" size="12"> 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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">curveC</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:matrix rows="3" cols="1"> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:real>1</ml:real> <ml:id xml:space="preserve">t</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="86"/> </region> <region region-id="1391" left="24" top="4446" width="459" height="14.25" align-x="24" align-y="4458" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" 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text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Зададим н</f> <f family="Arial Cyr" charset="204" size="12">ормаль:</f> </p> </text> </region> <region region-id="1406" left="156" top="4546.5" width="126" height="55.5" align-x="216" align-y="4578" 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">curven</ml:id> <ml:boundVars> <ml:id xml:space="preserve">w</ml:id> </ml:boundVars> </ml:function> <ml:matrix rows="3" cols="1"> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">w</ml:id> <ml:real>0.3</ml:real> </ml:apply> <ml:real>0.2</ml:real> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">w</ml:id> <ml:real>1.7</ml:real> </ml:apply> </ml:matrix> </ml:define> </math> <rendering item-idref="92"/> </region> <region 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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="94" disable-calc="false"> <inputs> <ml:matrix rows="1" cols="5" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:sequence> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">SB</ml:id> <ml:id xml:space="preserve">SA</ml:id> <ml:id xml:space="preserve">SC</ml:id> <ml:id xml:space="preserve">Sn</ml:id> </ml:sequence> <ml:placeholder/> <ml:placeholder/> <ml:placeholder/> <ml:placeholder/> </ml:matrix> </inputs> <outputs/> </component> <rendering item-idref="95"/> </region> <region region-id="1401" left="12" top="4914.75" width="457.5" height="42" align-x="12" align-y="4926" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Координатная сетка рисунка достаточно просто позволяет найти вершины тре-</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">угольника </f> <f family="Arial Cyr" charset="204" size="12"> <i>АВС, </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">стороны которого имеют черный цвет.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> На этом же риунке изо-</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">бражена также нормаль к плоскости контура </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i>L</i> </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">.</f> </inlineAttr> </p> </text> </region> <region region-id="1408" left="18" top="4974" width="432.75" height="42.75" align-x="18" align-y="4986" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">4а). </f> <f family="Arial Cyr" charset="204" size="12"> <i>Вычислим циркуляцию вектора b, используя левую часть формулы</i> </f> <f family="Arial Cyr" charset="204" size="12"> (7), </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">преобразовав криволинейный интеграл по замкнутому контуру </f> <f family="Arial Cyr" charset="204" size="12"> <i>L</i> </f> <f family="Arial Cyr" charset="204" size="12"> в сумму трех определенных интегралов по каждой стороне треугольника </f> <f family="Arial Cyr" charset="204" size="12"> <i>АВС</i> </f> <f family="Arial Cyr" charset="204" size="12"> отдельно.</f> </p> </text> </region> <region region-id="1409" left="18" top="5034.75" width="470.25" height="13.5" align-x="24" align-y="5046" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">На<sp count="2"/>участке АВ имеем: dy=0, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">y=0,</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> z=-x+2, 0<x<1. 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</ml:apply> <ml:symResult> <ml:apply> <ml:div/> <ml:real>3</ml:real> <ml:real>4</ml:real> </ml:apply> </ml:symResult> </ml:symEval> </ml:define> </math> <rendering item-idref="106"/> </region> <region region-id="1496" left="312" top="5292.75" width="61.5" height="15" align-x="333.75" align-y="5304" 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">uB</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.75</ml:real> </result> </ml:eval> </math> <rendering item-idref="107"/> </region> <region region-id="1597" left="18" top="5334.75" width="470.25" height="13.5" align-x="24" align-y="5346" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">На<sp count="2"/>участке CA имеем: dz=dx=0, x=0</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> z=2, 1>y>0. Поэтому:</inlineAttr> </f> </p> </text> </region> <region region-id="1515" left="48" top="5371.5" width="81" height="20.25" align-x="89.25" align-y="5388" 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">QC</ml:id> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">y</ml:id> <ml:real>3</ml:real> </ml:apply> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="108"/> </region> <region region-id="1514" left="156" top="5362.5" width="110.25" height="43.5" align-x="179.25" align-y="5388" 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">uC</ml:id> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">y</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:id xml:space="preserve">QC</ml:id> <ml:id xml:space="preserve">y</ml:id> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:real>0</ml:real> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="109"/> </region> <region region-id="1513" left="318" top="5376.75" width="62.25" height="15" align-x="340.5" align-y="5388" 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">uC</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.25</ml:real> </result> </ml:eval> </math> <rendering item-idref="110"/> </region> <region region-id="1506" left="30" top="5424" width="261.75" height="14.25" align-x="68.25" align-y="5436" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" line-through="false">Следовательно, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">циркуляцию вектора</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i> b </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">равна:</inlineAttr> </f> </p> </text> </region> <region region-id="1517" left="312" top="5424.75" width="120" height="15" align-x="326.25" align-y="5436" 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">u</ml:id> <ml:symEval style="default" hide-keywords="false" hide-lhs="false"> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">uA</ml:id> <ml:id xml:space="preserve">uB</ml:id> </ml:apply> <ml:id xml:space="preserve">uC</ml:id> </ml:apply> <ml:symResult> <ml:real>0</ml:real> </ml:symResult> </ml:symEval> </ml:define> </math> <rendering item-idref="111"/> </region> <region region-id="1584" left="30" top="5454" width="433.5" height="27.75" align-x="30" align-y="5466" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">4б). </f> <f family="Arial Cyr" charset="204" size="12"> <i>Вычислим циркуляцию вектора b, используя правую часть формулы</i> </f> <f family="Arial Cyr" charset="204" size="12"> (7)</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <sp count="8"/>или теорему Стокса.</f> </p> </text> </region> <region region-id="1585" left="36" top="5496.75" width="385.5" height="13.5" align-x="36" align-y="5508" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Для этого найдем скалярное произведение векторов </f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>n1</i> </b> </f> <f family="Arial Cyr" charset="204" size="12"> и rot</f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>b</i> </b> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">. Так как</inlineAttr> </f> </p> </text> </region> <region region-id="1592" left="240" top="5526.75" width="68.25" height="15" align-x="285" align-y="5538" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">n1</ml:id> <ml:id xml:space="preserve">rotb</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="112"/> </region> <region region-id="1594" left="36" top="5556" width="441" height="27.75" align-x="36" align-y="5568" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">то поток вихря через плоскость треугольника </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i>АВС</i> </f> <f family="Arial Cyr" charset="204" size="12"> <i> <sp/> </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">будет равен нулю, также как и</inlineAttr> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">циркуляция, что и следовало ожидать.</inlineAttr> </f> </p> </text> </region> <region region-id="1598" left="18" top="5616.75" width="489.75" height="54.75" align-x="54" align-y="5628" 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"> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">5. Максимальное значение циркуляции вектора </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> <b> <i>b</i> </b> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> по замкнотому контуру получается в </inlineAttr> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">том случае, когда угол между векторами </inlineAttr> </f> <i> <inlineAttr line-through="false"> <f family="Arial Cyr" charset="204" size="12"> <b>n1</b> </f> </inlineAttr> </i> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> и rot</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i> <inlineAttr line-through="false"> <b>b</b> </inlineAttr> </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false"> равен 0, т.е. плоскость треуголь-</inlineAttr> </f> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr line-through="false">ника </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <i> <inlineAttr line-through="false">АВС </inlineAttr> </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal" line-through="false">надо развернуть так, чтобы её нормаль была коллинеарна вектору </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">rot</f> <f family="Arial Cyr" charset="204" size="12"> <b> <i>b</i> </b> </f> <inlineAttr font-weight="normal" font-style="normal"> <f family="Arial Cyr" charset="204" size="12">. </f> </inlineAttr> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">Согласно формуле Стокса (7) её правой части получим:</inlineAttr> </f> </p> </text> </region> <region region-id="1599" left="150" top="5688" width="92.25" height="41.25" align-x="150" align-y="5688" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="113"/> <rendering item-idref="114"/> </region> <region region-id="1600" left="408" top="5694.75" width="16.5" height="15" align-x="415.5" align-y="5706" 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="115"/> </region> <region region-id="1606" left="30" top="5748" width="469.5" height="62.25" align-x="30" align-y="5760" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Учитывая, что модуль ротора </f> <f family="Arial Cyr" charset="204" size="12"> <i>b</i> </f> <f family="Arial Cyr" charset="204" size="12"> есть костанта, а площадь треугольника </f> <f family="Arial Cyr" charset="204" size="12"> <i>АВС </i> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">равна</inlineAttr> </f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <region region-id="1608" left="30" top="5753.25" width="19.5" height="33.75" align-x="36" align-y="5772" 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:div/> <ml:real>1</ml:real> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:apply> </math> <rendering item-idref="116"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, по формуле (8) получим максимальное значение циркуляции для данного контура или потока через<sp count="2"/>поверхность, натянутую на этот контур.</inlineAttr> </f> </p> </text> </region> <region region-id="1617" left="60" top="5831.25" width="63.75" height="33.75" align-x="79.5" align-y="5850" 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">Pt</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">rotb</ml:id> </ml:apply> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="117"/> </region> <region region-id="1616" left="168" top="5838.75" width="65.25" height="15" align-x="186.75" align-y="5850" 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">Pt</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>4.0620192023179795</ml:real> </result> </ml:eval> </math> <rendering item-idref="118"/> </region> <region region-id="1620" left="36" top="5874.75" width="457.5" height="27.75" align-x="36" align-y="5886" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">Этот результат можно проверить непосредственным вычислением циркуляции век-</f> </p> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f family="Arial Cyr" charset="204" size="12">тора </f> <f family="Arial Cyr" charset="204" size="12"> <i>b</i> </f> <f family="Arial Cyr" charset="204" size="12"> по новому расположению контура </f> <f family="Arial Cyr" charset="204" size="12"> <i>L</i> </f> <f family="Arial Cyr" charset="204" size="12">. </f> </p> </text> </region> </regions> <binaryContent> <item item-id="1" 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