P=2xy³-y²cosx,Q=1-2ysinx+3x²y²
æéªè¯ï¼∂Q/∂x=∂P/∂y=6xy²-2ycosx
å æ¤æ¬é¢ç§¯åä¸è·¯å¾æ å ³,å¯èªé积å路线
éä»(0,0)å°(Ï/2,1)çæ线,
L1ï¼y=0,xï¼0--->Ï/2
L2ï¼x=Ï/2,yï¼0--->1
åå积å=â«L1 (2xy^3-y^2cosx)dx+(1-2ysinx+3x^2y^2)dy
+â«L2 (2xy^3-y^2cosx)dx+(1-2ysinx+3x^2y^2)dy
=0+â«[0--->1] [1-2y+(3ϲ/4)y²]dy
=y-y²+(ϲ/4)y³ |[0--->1]
=ϲ/4
æéªè¯ï¼∂Q/∂x=∂P/∂y=6xy²-2ycosx
å æ¤æ¬é¢ç§¯åä¸è·¯å¾æ å ³,å¯èªé积å路线
éä»(0,0)å°(Ï/2,1)çæ线,
L1ï¼y=0,xï¼0--->Ï/2
L2ï¼x=Ï/2,yï¼0--->1
åå积å=â«L1 (2xy^3-y^2cosx)dx+(1-2ysinx+3x^2y^2)dy
+â«L2 (2xy^3-y^2cosx)dx+(1-2ysinx+3x^2y^2)dy
=0+â«[0--->1] [1-2y+(3ϲ/4)y²]dy
=y-y²+(ϲ/4)y³ |[0--->1]
=ϲ/4
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