因为有任意闭路积分的问题，故先验证积分是否与路径无关，即验证[tex=1.5x2.214]V9fVXReHUrcmKJSTnoNlS57WSpaHGvSXFI+U9WJ8cmYi2lsP/u2Vp3up8rIK0/Pv[/tex]与[tex=1.5x2.071]V9fVXReHUrcmKJSTnoNlS51lpt21UdUZGENLx9+ScffrwD7RK0Gv41izkGK8DLMq[/tex]是否处处相等。[tex=2.571x2.214]V9fVXReHUrcmKJSTnoNlS1KS/fQXa8GdYgRXmH87+Az/Kb1rIie/u5/nNuu7VrjI[/tex][tex=11.071x2.643]JMZdUyCFun86L13G+9+Lt3iYCRqzhkNKEusb0OLjEkl5mLb722orBT4pVAPuKARc0gT47Bm7UTCduL8xo+fBpaQXIZh9lx8WxR/+Wfnfz1Y=[/tex][tex=6.071x2.571]C0oEe9Iq/dWgXfgRRJJnLZsvfqDmSKZWe+PmhzdhxwcQSiuEAn4kMIKez8T+LFprDJfr9ZlvOV4p/DQxCqEurw==[/tex]，[tex=2.643x2.071]V9fVXReHUrcmKJSTnoNlS+m+sqsVguAfUlcXLQaFRyRzA0hwaCdgDmLqWz8zNe2U[/tex][tex=11.571x2.643]JMZdUyCFun86L13G+9+Lt3iYCRqzhkNKEusb0OLjEklLaXmBdPnTi9EKOjtTlZQ/z8BRHh6mspCICXC2/AO8sBR05c/gyNfB17QCZO6NDvzR0qMmQeFGFTt0dK9P+oDK[/tex][tex=7.071x2.5]o+le9nTrwLQxD6oOl0H1VomgY2CawdGRcVhway2KSaq6GqgxnuhQ0hL3trAtM0arVBmPIy9ssMH7SrB3FC3yHQ==[/tex]，所以在全平面上除掉原点[tex=2.143x1.286]q8d9ecMZwZI3gbdeOe+7AA==[/tex]的复连通域内，有[tex=4.357x2.214]V9fVXReHUrcmKJSTnoNlS1KS/fQXa8GdYgRXmH87+AzVjx8r+mraCk3Er/zPigNPImdyT0ngPeA6cQNmfijt+G79GlFAD9c8VNU6k70dtIA=[/tex]，故（1）在不包围也不经过原点的任意闭曲线[tex=1.071x1.286]znnbcDroem+0Djr3hBPwXg==[/tex]上（如图所示）[img=295x355]1787bea62378613.png[/img][tex=13.0x2.357]T3Nt5gkMrd3cell2fNGv8knM5KlaEKD40F5ZOJDwaQ2sixw5mNe36q7M+TKKChPkbNRgrSsBIR98rS/CgERpO+P65w2K3yTV6FzqL2WKHKU=[/tex]，因为由[tex=1.071x1.286]znnbcDroem+0Djr3hBPwXg==[/tex]所围域[tex=1.214x1.214]Ho8mAPpdke4daIdB3oO8tA==[/tex]是单连通域，且有[tex=4.143x2.643]aoAtmkWSHYklGULM9bBrEok+OCW9H9iIUjaJolgWGZfTO10gzFl54iBNVNJpKggK7pL0trOHcm5AuYzp6HMuew==[/tex]处处成立，由曲线积分与路径无关的等价条件知[tex=13.0x2.357]T3Nt5gkMrd3cell2fNGv8knM5KlaEKD40F5ZOJDwaQ2sixw5mNe36q7M+TKKChPkbNRgrSsBIR98rS/CgERpO+P65w2K3yTV6FzqL2WKHKU=[/tex]（2）设[tex=1.071x1.286]ViYfYChGNyO6YfIJTH5iMg==[/tex]是以原点为中心的单位圆，方向取为正向，由于[tex=1.071x1.286]ViYfYChGNyO6YfIJTH5iMg==[/tex]所围的域包围有原点，是复连通域，此积分不一定为零，可利用参数方程直接计算[tex=1.071x1.286]ViYfYChGNyO6YfIJTH5iMg==[/tex]：[tex=5.143x2.786]fnpmC2J6JmQBLyo5NmGAz3ydYJr5HNPwSdEPxdwncUc9s1UdWT0V4tmejPdK5MONl2OUltanfhZKSp85Q6NNnA==[/tex]，[tex=4.571x1.286]OiGH0k5KxfGmLHkL/3aHG6GgZuezAp0mtFmTOsS6JMVre4nxhci657IGf9KJLCDS[/tex]，[tex=12.286x2.357]/bmxXiv0OQCAiBO07amnRz59+PbxMNMZAIxpRUoVQ5V/DQVCsmOCafwUjff9eXfuxi0l8jBi+0fWPHgAOqJ1BmF5YapbimvoMvLu7F4L7/E=[/tex][tex=19.214x2.429]43DHJW5hNMMx+YjFVGcc+jySmtwfNTuvpkRyJe28PjcSk+dUzizGTrPJYAj0hiqxHEPyo/010D78krL4oFtcW7jWdgC8IJVerbFz+2moDFls0ol90PvgB4G8FVxmsqIAY+g3v45TQhhUkAGcTPaM3A==[/tex][tex=9.071x2.429]4Cd+cFRG7vN36wpB3nnACMQqNiCsMaswscOCREUMY3hUHDLeXwkCSnI7Tz+i2YAC[/tex][tex=12.143x1.286]ga/qhmJysEU1NpSvHlXlZ72spxWF2G1bhw/amsFvRh1ygWZ40bMaxwmZGTVL9O0tC/GJpp9Y3kMiS2YlH2DclQ==[/tex][tex=8.929x2.429]19hOqotq8LBSNBwiShBCBbsak3mGKObnJk4xSVMLeTrIS8xq7Dw3VfaNJYebBoH4[/tex]（3）如上图所示，由于[tex=1.071x1.214]rpdZzEcni+HXXpShcnbq7Q==[/tex]包围原点，故是复连通域，又[tex=1.071x1.214]rpdZzEcni+HXXpShcnbq7Q==[/tex]是任意闭曲线（包围原点）直接积分不现实。为了除去原点，在[tex=1.071x1.214]rpdZzEcni+HXXpShcnbq7Q==[/tex]和单位圆[tex=1.071x1.214]k9Aohr7/cYPv99wZ5Rg2rA==[/tex]（当[tex=1.071x1.214]rpdZzEcni+HXXpShcnbq7Q==[/tex]不能完全包含[tex=1.071x1.214]k9Aohr7/cYPv99wZ5Rg2rA==[/tex]时，在[tex=1.071x1.214]rpdZzEcni+HXXpShcnbq7Q==[/tex]内任作一个中心在原点，半径为充分小正数[tex=0.5x1.0]g3C024VcW5lWpceJ6ZrB4A==[/tex]的小圆即可解决）之间作辅助线[tex=1.571x1.286]cHJ4KDAad01mWuGaiQQpfA==[/tex]（如图所示），使连接[tex=1.071x1.214]rpdZzEcni+HXXpShcnbq7Q==[/tex]和[tex=1.071x1.286]6yqW8fcpVF9OkaZv4Gb/5A==[/tex]，则[tex=9.143x1.429]FVOrZJ1hU2QhmVaSUL69R9fkLTqodHSq3hiWf7HceQnYYEGKBCziIaPulBPrXw0UWsEY9e6Az+nInmtbDBnD7w==[/tex]成为一条闭曲线（其中[tex=1.857x1.214]YeiKhvSMq5U4dAPAXFLO/Q==[/tex]表示[tex=1.071x1.214]k9Aohr7/cYPv99wZ5Rg2rA==[/tex]的负向闭曲线），这条闭曲线不包围原点，所以在以 [tex=0.929x1.143]rhHRFIHsX3I90ujZbbcvjQ==[/tex]为边界曲线的单连通域上，恒有[tex=4.357x2.214]V9fVXReHUrcmKJSTnoNlS1KS/fQXa8GdYgRXmH87+AzVjx8r+mraCk3Er/zPigNPImdyT0ngPeA6cQNmfijt+G79GlFAD9c8VNU6k70dtIA=[/tex]，故[tex=12.929x2.286]GXpil/8Thm/KbL5EeAF5uwGN6Poxeayp80BVrHlHk9RSTbubpNLsiTCXM0zz1+wHA7kt6cyaPG4/Mc+agu9GOJ2XAEfckLK1uA6DZToixSM=[/tex]，即[tex=12.143x2.286]GXpil/8Thm/KbL5EeAF5uwGN6Poxeayp80BVrHlHk9RSTbubpNLsiTCXM0zz1+wHA7kt6cyaPG4/Mc+agu9GONAZGico3T3nBpOOL1LoGXw=[/tex][tex=12.0x2.429]tXCYbBkMN60J31NUrgpQXMGJ04N/C2otzhnXETHPf8Kp9YZNYkrxPhvTbENECMRigNEPDDZ5TThlygtsV8UNxL4LI2TirJfsK4baQ3vuF3I=[/tex][tex=12.286x2.286]5tM6qzGZGZa00VZRE3srZrWgW7LtqPPKPkw6gVk/54db52ygc77CmATcSHW/bnKgQuWkM/5J5RMORmocENDtssQwMsHOVvclnXc2httx6Y4=[/tex][tex=12.571x2.357]zXv9LcPEL0LodaURVb3U0aes17z/LkmlikJKDzEsm0e/AlaNQuT5swsbjHSGoTXX3F5BgBzP9Sc/pfiKe/jf4uAKe73C3i6f7m58f+t5WFs=[/tex][tex=13.286x2.286]5tM6qzGZGZa00VZRE3srZnff2MNWYyEcKJUv8fViG1+u1+a6piFbBRKkpEhcOo1D2Cs+lgvelZU6qPRIFnCtXaVOm6rxpAZ7CT8IKfkBhqc=[/tex]因为[tex=12.571x2.286]5tM6qzGZGZa00VZRE3srZrWgW7LtqPPKPkw6gVk/54db52ygc77CmATcSHW/bnKgQuWkM/5J5RMORmocENDtsoX2kkxU7vAYMqpAIErT+lM=[/tex][tex=12.429x2.286]kDG7s2Xk3xa0crwU0hCMIUVnfqvZXThxFVKyAQwEvofa15X+7/u766/iE+3BlXAQAq2fGjjOWjuMaBPsB6WFSjODBSpH8ojDl7qbv4JQcRE=[/tex]，所以[tex=12.0x2.429]tXCYbBkMN60J31NUrgpQXMGJ04N/C2otzhnXETHPf8Kp9YZNYkrxPhvTbENECMRi3TtdrMC87p24q6WTVJH60VCFZJq/M38Jt189ulR+FSU=[/tex][tex=13.643x2.357]zXv9LcPEL0LodaURVb3U0aes17z/LkmlikJKDzEsm0e/AlaNQuT5swsbjHSGoTXX3F5BgBzP9Sc/pfiKe/jf4oFU4BRjlSx0UpmoiBjNMxQ=[/tex]，[tex=12.286x2.429]tXCYbBkMN60J31NUrgpQXMGJ04N/C2otzhnXETHPf8Kp9YZNYkrxPhvTbENECMRiYtzY2eafAT2dAMusNfpgoGIRfGWcdfiuqUHi3/mDs5M=[/tex][tex=12.714x2.357]Lw4VLcOpMm4iLOaCLSxdcPxi3+9REoecHKx9MGtNKE4lJNAfx98xxQFg+RwlDgRKVgrgWKz8B66JdilGk5a1kWqVIlpLZhhu3C2GYEeRZ0c=[/tex][tex=12.214x2.357]RBtSQ0xcYmeKy+HnvXAGkyBh3dinuu/zwjpgvSQdLJd/qWpyFg/pd/YOqMz7PlLvFJOr62FmYJndai6wEOwseLJQV8VRSiyD+gOQwaFi9eo=[/tex]，由此推出了包围原点的任意正向闭路[tex=1.071x1.286]CByAIXOj9Gclpix4UeG1Dw==[/tex]上的积分等于包围原点的正向单位圆的积分，故[tex=11.214x2.429]tXCYbBkMN60J31NUrgpQXMGJ04N/C2otzhnXETHPf8Kp9YZNYkrxPhvTbENECMRiWnwjGFuaJjA6A88HzUXxOZy7l7CirfO3fP2Wm2/VJe8=[/tex][tex=2.857x1.286]OWO+nkAJzpTPxLtwrlUMIg==[/tex]。