.<\/span> \u5047\u8bbe$f$\u662f\u5b9a\u4e49\u5728\u590d\u5e73\u9762\u4e0a\u4e00\u5355\u8fde\u901a\u5f00\u533a\u57df$\\Omega$\u4e0a\u7684\u89e3\u6790(\u5168\u7eaf)\u51fd\u6570, \u5219\u5bf9\u4efb\u4f55$\\Omega$\u4e2d\u7684\u95ed\u66f2\u7ebf$C$,
\n$$
\n\\oint_C f(z)dz=0.
\n$$
\n<\/div>
\n\u6211\u4eec\u60f3\u8981\u8bf4\u660e, \u5b83\u5176\u5b9e\u53ef\u4ee5\u7531\u5b9e\u5206\u6790\u4e2d\u7684Stokes\u516c\u5f0f\u5f97\u5230.
\n<\/p>\n\u4e8b\u5b9e\u4e0a, \u5047\u8bbe$f(z)=u(x,y)+iv(x,y)$, \u76f4\u63a5\u8ba1\u7b97\u5f97\u5230(\u8bb0$C$\u56f4\u6210\u7684\u533a\u57df\u4e3a$D$)
\n\\begin{align*}
\n2\\int_D\\frac{\\partial }{\\partial \\bar z}f(z)dx\\wedge dy&=\\int_D(\\partial_x+i\\partial _y)(u+iv)dx\\wedge dy\\\\
\n&=\\int_D\\bigl(u_x-v_y+i(u_y+v_x)\\bigr)dx\\wedge dy\\\\
\n&=\\int_Dd\\bigl(udy+vdx+i(-udx+vdy)\\bigr)\\\\
\n&=\\int_Dd\\bigl(v(dx+idy)-iu(dx+idy)\\bigr)\\\\
\n&=\\int_Dd\\bigl(-i(u+iv)dz\\bigr)=-i\\int_D d \\bigl(f(z)dz\\bigr)\\\\
\n&=-i\\oint_Cf(z)dz,
\n\\end{align*}
\n\u8fd9\u5373\u8868\u660e
\n$$
\n\\int_D\\bar\\partial f(z):=\\int_D\\frac{\\partial }{\\partial \\bar z}f(z)dx\\wedge dy=-\\frac{i}{2}\\oint_C f(z)dz.
\n$$
\n\u4e0a\u8ff0\u516c\u5f0f\u4e00\u4e2a\u76f4\u63a5\u7684\u63a8\u8bba\u5c31\u662f, \u5f53$f$\u5168\u7eaf\u65f6(\u5373$\\bar\\partial f=0$), \u6709
\n$$
\n\\oint_C f(z)dz=0,
\n$$
\n\u5373\u5f97\u5230Cauchy\u79ef\u5206\u5b9a\u7406.<\/p>\n","protected":false},"excerpt":{"rendered":"
\u56de\u5fc6, \u5355\u590d\u53d8\u4e2dCauchy\u79ef\u5206\u5b9a\u7406\u8bf4\u7684\u662f: \u5b9a\u7406 1 (Cauchy integral theorem). … \u7ee7\u7eed\u9605\u8bfb\u5e94\u7528Stokes\u516c\u5f0f\u63a8\u5bfcCauchy\u79ef\u5206\u5b9a\u7406<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[70,72,71],"class_list":["post-386","post","type-post","status-publish","format-standard","hentry","category-math","tag-cauchy","tag-stokesgongshi","tag-jifendingli","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/386","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=386"}],"version-history":[{"count":2,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/386\/revisions"}],"predecessor-version":[{"id":388,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/386\/revisions\/388"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=386"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=386"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=386"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}