数学冲刺班（理工类）补充试题

lingq2080

<P >例<FONT face="Times New Roman">4 </FONT>确定函数<v:shapetype><FONT face="Times New Roman"> <v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path o:connecttype="rect" gradientshapeok="t" o:extrusionok="f"></v:path><o:lock aspectratio="t" v:ext="edit"></o:lock></FONT></v:shapetype><v:shape><v:imagedata></v:imagedata></v:shape>，使曲线积分<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>对于任何闭曲线<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>的积分都等于<FONT face="Times New Roman">0</FONT>，且<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><FONT face="Times New Roman">,</FONT>试计算沿曲线<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>从点<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>的曲线积分<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>。<o:p></o:p></P>
<P >解：由斯氏公式<o:p></o:p></P>
<P ><v:shape><v:imagedata></v:imagedata></v:shape><v:shape><v:imagedata></v:imagedata></v:shape><o:p></o:p></P>
<P >è<FONT face="Times New Roman"> </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman">   </FONT>（<FONT face="Times New Roman">1</FONT>）<o:p></o:p></P>
<P ><v:shape><v:imagedata></v:imagedata></v:shape><v:shape><v:imagedata></v:imagedata></v:shape><FONT face="Times New Roman">  </FONT>（<FONT face="Times New Roman">2</FONT>）<o:p></o:p></P>
<P ><v:shape><v:imagedata></v:imagedata></v:shape><FONT face="Times New Roman">    </FONT>（<FONT face="Times New Roman">3</FONT>）<o:p></o:p></P>
<P >显然（<FONT face="Times New Roman">1</FONT>），（<FONT face="Times New Roman">2</FONT>）满足。将<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>代入（<FONT face="Times New Roman">3</FONT>）得<o:p></o:p></P>
<P ><FONT face="Times New Roman">  </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><o:p></o:p></P>
<P >è<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><o:p></o:p></P>
<P >è<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman"> </FONT>è<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><o:p></o:p></P>
<P ><FONT face="Times New Roman">   </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><o:p></o:p></P>
<P ><FONT face="Times New Roman">   </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><o:p></o:p></P>
<P >将<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape>代入，得<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><o:p></o:p></P>
<P ><v:group><v:line><v:stroke endarrow="block"><FONT face="Times New Roman"></FONT></v:stroke></v:line><v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:path o:connecttype="rect" gradientshapeok="t"></v:path></v:shapetype><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1032">
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<P ><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape></P>
<P ><FONT face="Times New Roman">y </FONT></P></DIV></TD></TR></TABLE></v:textbox></v:shape><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1033">
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<TD >
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<P ><FONT face="Times New Roman">            z  </FONT></P>
<P ><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape></P>
<P ><o:p><FONT face="Times New Roman"> </FONT></o:p></P>
<P ><o:p><FONT face="Times New Roman"> </FONT></o:p></P>
<P ><FONT face="Times New Roman">              o</FONT></P>
<P ><o:p><FONT face="Times New Roman"> </FONT></o:p></P>
<P ><FONT face="Times New Roman">x</FONT></P></DIV></TD></TR></TABLE></v:textbox></v:shape></v:group>è<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><FONT face="Times New Roman">  </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><o:p></o:p></P>
<P ><v:line><v:stroke startarrow="block" dashstyle="dash"><FONT face="Times New Roman"></FONT></v:stroke></v:line><v:line><v:stroke startarrow="block"><FONT face="Times New Roman"></FONT></v:stroke></v:line><v:line><v:stroke endarrow="block"><FONT face="Times New Roman"></FONT></v:stroke></v:line><v:line><v:stroke endarrow="block" startarrow="block"><FONT face="Times New Roman"></FONT></v:stroke></v:line><FONT face="Times New Roman">                                        <o:p></o:p></FONT></P>
<P ><o:p><FONT face="Times New Roman"> </FONT></o:p></P>
<P ><FONT face="Times New Roman">                                         </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape></P><BR  clear=all>
<P >例<FONT face="Times New Roman">5</FONT>．设<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>是以<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>为边界的光滑曲面，试求连续可微函数<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>使曲面积分<o:p></o:p></P>
<P ><FONT face="Times New Roman">  </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape>与曲面<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>的形状无关。<o:p></o:p></P>
<P >解：以<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>为边界任作两个光滑曲面<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>的法向量指向同一侧。记<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>为<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>所围闭曲面，取外侧，<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>所围区域为口。依题意<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，<FONT face="Times New Roman">(</FONT><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>的反向<FONT face="Times New Roman">)<o:p></o:p></FONT></P>
<P >由高斯定理<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><o:p></o:p></P>
<P >è<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><FONT face="Times New Roman"> </FONT><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><o:p></o:p></P>
<P ><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape>代入上式<o:p></o:p></P>
<P ><FONT face="Times New Roman">==</FONT>〉<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><o:p></o:p></P>
<P ><FONT face="Times New Roman">==</FONT>〉<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman"> ==</FONT>〉<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><FONT face="Times New Roman"> </FONT><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape></P>