{"id":92,"date":"2026-08-23T19:30:01","date_gmt":"2026-08-23T11:30:01","guid":{"rendered":"https:\/\/wdlobin.com\/blog\/?p=92"},"modified":"2026-08-23T19:30:01","modified_gmt":"2026-08-23T11:30:01","slug":"%e5%a4%8d%e5%8f%98%e5%87%bd%e6%95%b0%e9%97%ad%e5%90%88%e8%b7%af%e5%be%84%e7%a7%af%e5%88%86%e7%9a%84%e5%88%86%e8%a7%a3%e5%85%ac%e5%bc%8f","status":"publish","type":"post","link":"https:\/\/wdlobin.com\/blog\/%e5%a4%8d%e5%8f%98%e5%87%bd%e6%95%b0%e9%97%ad%e5%90%88%e8%b7%af%e5%be%84%e7%a7%af%e5%88%86%e7%9a%84%e5%88%86%e8%a7%a3%e5%85%ac%e5%bc%8f\/","title":{"rendered":"\u590d\u53d8\u51fd\u6570\u95ed\u5408\u8def\u5f84\u79ef\u5206\u7684\u5206\u89e3\u516c\u5f0f"},"content":{"rendered":"\n

\u7ed9\u5b9a\u590d\u51fd\u6570 f<\/mi>(<\/mo>z<\/mi>)<\/mo>=<\/mo>u<\/mi>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo>+<\/mo>i<\/mi>v<\/mi>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo><\/mrow>f(z) = u(x,y) + iv(x,y)<\/annotation><\/semantics><\/math> \u548c\u5fae\u5206 d<\/mi><\/mrow>z<\/mi>=<\/mo>d<\/mi><\/mrow>x<\/mi>+<\/mo>i<\/mi>d<\/mi><\/mrow>y<\/mi><\/mrow>\\mathrm{d}z = \\mathrm{d}x + i\\mathrm{d}y<\/annotation><\/semantics><\/math>\uff0c\u95ed\u5408\u8def\u5f84\u79ef\u5206\u53ef\u5206\u89e3\u4e3a\uff1a<\/p>\n\n\n\n
\u222e<\/mo>C<\/mi><\/msub>f<\/mi>(<\/mo>z<\/mi>)<\/mo>d<\/mi><\/mrow>z<\/mi>=<\/mo>\u222e<\/mo>C<\/mi><\/msub>[<\/mo>u<\/mi>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo>+<\/mo>i<\/mi>v<\/mi>(<\/mo>x<\/mi>,<\/mo>y<\/mi>)<\/mo>]<\/mo>(<\/mo>d<\/mi><\/mrow>x<\/mi>+<\/mo>i<\/mi>d<\/mi><\/mrow>y<\/mi>)<\/mo><\/mrow>\\oint_{C}f(z)\\mathrm{d}z = \\oint_{C}[u(x,y) + iv(x,y)](\\mathrm{d}x + i\\mathrm{d}y)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

\u5206\u89e3\u6b65\u9aa4\uff1a<\/h4>\n\n\n\n
    \n
  1. \u5c55\u5f00\u88ab\u79ef\u8868\u8fbe\u5f0f<\/strong>\uff1a<\/li>\n<\/ol>\n\n\n\n
    [<\/mo>u<\/mi>+<\/mo>i<\/mi>v<\/mi>]<\/mo>(<\/mo>d<\/mi><\/mrow>x<\/mi>+<\/mo>i<\/mi>d<\/mi><\/mrow>y<\/mi>)<\/mo><\/mrow><\/mtd>=<\/mo>u<\/mi>\u22c5<\/mo>d<\/mi><\/mrow>x<\/mi>+<\/mo>u<\/mi>\u22c5<\/mo>(<\/mo>i<\/mi>d<\/mi><\/mrow>y<\/mi>)<\/mo>+<\/mo>(<\/mo>i<\/mi>v<\/mi>)<\/mo>\u22c5<\/mo>d<\/mi><\/mrow>x<\/mi>+<\/mo>(<\/mo>i<\/mi>v<\/mi>)<\/mo>\u22c5<\/mo>(<\/mo>i<\/mi>d<\/mi><\/mrow>y<\/mi>)<\/mo><\/mrow><\/mtd><\/mtr><\/mrow><\/mtd>=<\/mo>u<\/mi>d<\/mi><\/mrow>x<\/mi>+<\/mo>i<\/mi>u<\/mi>d<\/mi><\/mrow>y<\/mi>+<\/mo>i<\/mi>v<\/mi>d<\/mi><\/mrow>x<\/mi>+<\/mo>i<\/mi>2<\/mn><\/msup>v<\/mi>d<\/mi><\/mrow>y<\/mi><\/mrow><\/mtd><\/mtr><\/mtable>\\begin{aligned}\n [u + iv](\\mathrm{d}x + i\\mathrm{d}y) &= u\\cdot\\mathrm{d}x + u\\cdot(i\\mathrm{d}y) + (iv)\\cdot\\mathrm{d}x + (iv)\\cdot(i\\mathrm{d}y) \\\\\n &= u\\mathrm{d}x + iu\\mathrm{d}y + iv\\mathrm{d}x + i^2v\\mathrm{d}y\n \\end{aligned}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
      \n
    1. \u7b80\u5316\u865a\u6570\u5355\u4f4d<\/strong>\uff08\u5229\u7528 i<\/mi>2<\/mn><\/msup>=<\/mo>\u2212<\/mo>1<\/mn><\/mrow>i^2 = -1<\/annotation><\/semantics><\/math>\uff09\uff1a<\/li>\n<\/ol>\n\n\n\n
      i<\/mi>2<\/mn><\/msup>v<\/mi>d<\/mi><\/mrow>y<\/mi>=<\/mo>(<\/mo>\u2212<\/mo>1<\/mn>)<\/mo>v<\/mi>d<\/mi><\/mrow>y<\/mi>=<\/mo>\u2212<\/mo>v<\/mi>d<\/mi><\/mrow>y<\/mi><\/mrow>i^2v\\mathrm{d}y = (-1)v\\mathrm{d}y = -v\\mathrm{d}y<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

      \u4ee3\u5165\u5f97\uff1a<\/p>\n\n\n\n

      u<\/mi>d<\/mi><\/mrow>x<\/mi>+<\/mo>i<\/mi>u<\/mi>d<\/mi><\/mrow>y<\/mi>+<\/mo>i<\/mi>v<\/mi>d<\/mi><\/mrow>x<\/mi>\u2212<\/mo>v<\/mi>d<\/mi><\/mrow>y<\/mi><\/mrow>u\\mathrm{d}x + iu\\mathrm{d}y + iv\\mathrm{d}x – v\\mathrm{d}y<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
        \n
      1. \u5206\u7ec4\u5b9e\u90e8\u4e0e\u865a\u90e8<\/strong>\uff1a<\/li>\n<\/ol>\n\n\n\n
          \n
        • \u5b9e\u90e8\u7ec4\u5408<\/strong>\uff1au<\/mi>d<\/mi><\/mrow>x<\/mi>\u2212<\/mo>v<\/mi>d<\/mi><\/mrow>y<\/mi><\/mrow>u\\mathrm{d}x – v\\mathrm{d}y<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
        • \u865a\u90e8\u7ec4\u5408<\/strong>\uff1ai<\/mi>u<\/mi>d<\/mi><\/mrow>y<\/mi>+<\/mo>i<\/mi>v<\/mi>d<\/mi><\/mrow>x<\/mi>=<\/mo>i<\/mi>(<\/mo>u<\/mi>d<\/mi><\/mrow>y<\/mi>+<\/mo>v<\/mi>d<\/mi><\/mrow>x<\/mi>)<\/mo><\/mrow>iu\\mathrm{d}y + iv\\mathrm{d}x = i(u\\mathrm{d}y + v\\mathrm{d}x)<\/annotation><\/semantics><\/math>
          \u5408\u5e76\u4e3a\uff1a<\/li>\n<\/ul>\n\n\n\n
          (<\/mo>u<\/mi>d<\/mi><\/mrow>x<\/mi>\u2212<\/mo>v<\/mi>d<\/mi><\/mrow>y<\/mi>)<\/mo>+<\/mo>i<\/mi>(<\/mo>u<\/mi>d<\/mi><\/mrow>y<\/mi>+<\/mo>v<\/mi>d<\/mi><\/mrow>x<\/mi>)<\/mo><\/mrow>(u\\mathrm{d}x – v\\mathrm{d}y) + i(u\\mathrm{d}y + v\\mathrm{d}x)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n
            \n
          1. \u5206\u79bb\u79ef\u5206<\/strong>\uff08\u5229\u7528\u79ef\u5206\u7ebf\u6027\u6027\u8d28\uff09\uff1a<\/li>\n<\/ol>\n\n\n\n
            \u222e<\/mo>C<\/mi><\/msub>[<\/mo>(<\/mo>u<\/mi>d<\/mi><\/mrow>x<\/mi>\u2212<\/mo>v<\/mi>d<\/mi><\/mrow>y<\/mi>)<\/mo>+<\/mo>i<\/mi>(<\/mo>u<\/mi>d<\/mi><\/mrow>y<\/mi>+<\/mo>v<\/mi>d<\/mi><\/mrow>x<\/mi>)<\/mo>]<\/mo><\/mrow>=<\/mo>\u222e<\/mo>C<\/mi><\/msub>(<\/mo>u<\/mi>d<\/mi><\/mrow>x<\/mi>\u2212<\/mo>v<\/mi>d<\/mi><\/mrow>y<\/mi>)<\/mo>+<\/mo>i<\/mi>\u222e<\/mo>C<\/mi><\/msub>(<\/mo>u<\/mi>d<\/mi><\/mrow>y<\/mi>+<\/mo>v<\/mi>d<\/mi><\/mrow>x<\/mi>)<\/mo><\/mrow>\\oint_{C} \\left[(u\\mathrm{d}x – v\\mathrm{d}y) + i(u\\mathrm{d}y + v\\mathrm{d}x)\\right] = \\oint_{C}(u\\mathrm{d}x – v\\mathrm{d}y) + i\\oint_{C}(u\\mathrm{d}y + v\\mathrm{d}x)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

            \u6700\u7ec8\u5206\u89e3\u5f0f\uff1a<\/h4>\n\n\n\n
            <\/mspace>\u222e<\/mo>C<\/mi><\/msub>f<\/mi>(<\/mo>z<\/mi>)<\/mo>d<\/mi><\/mrow>z<\/mi>=<\/mo>\u222e<\/mo>C<\/mi><\/msub>(<\/mo>u<\/mi>d<\/mi><\/mrow>x<\/mi>\u2212<\/mo>v<\/mi>d<\/mi><\/mrow>y<\/mi>)<\/mo>+<\/mo>i<\/mi>\u222e<\/mo>C<\/mi><\/msub>(<\/mo>u<\/mi>d<\/mi><\/mrow>y<\/mi>+<\/mo>v<\/mi>d<\/mi><\/mrow>x<\/mi>)<\/mo><\/mrow><\/mspace><\/mrow>\\boxed{\n\\oint_{C}f(z)\\mathrm{d}z = \\oint_{C}(u\\mathrm{d}x – v\\mathrm{d}y) + i\\oint_{C}(u\\mathrm{d}y + v\\mathrm{d}x)\n}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

            \u51e0\u4f55\u89e3\u91ca<\/h3>\n\n\n\n

            \u8be5\u5206\u89e3\u5c06\u590d\u79ef\u5206\u8f6c\u5316\u4e3a\u4e24\u4e2a\u5b9e\u7b2c\u4e8c\u7c7b\u66f2\u7ebf\u79ef\u5206<\/strong>\uff1a<\/p>\n\n\n\n

              \n
            • \u5b9e\u90e8\u79ef\u5206<\/strong>\uff1a\u222e<\/mo>C<\/mi><\/msub>(<\/mo>u<\/mi>d<\/mi><\/mrow>x<\/mi>\u2212<\/mo>v<\/mi>d<\/mi><\/mrow>y<\/mi><\/mrow>\\oint_C (u\\mathrm{d}x – v\\mathrm{d}y<\/annotation><\/semantics><\/math> \u2192 \u5411\u91cf\u573a (<\/mo>u<\/mi>,<\/mo>\u2212<\/mo>v<\/mi>)<\/mo><\/mrow>(u, -v)<\/annotation><\/semantics><\/math> \u7684\u73af\u91cf<\/li>\n\n\n\n
            • \u865a\u90e8\u79ef\u5206<\/strong>\uff1a(<\/mo>u<\/mi>,<\/mo>\u2212<\/mo>v<\/mi>)<\/mo><\/mrow>(u, -v)<\/annotation><\/semantics><\/math> \u2192 \u5411\u91cf\u573a (<\/mo>v<\/mi>,<\/mo>u<\/mi>)<\/mo><\/mrow>(v, u)<\/annotation><\/semantics><\/math> \u7684\u73af\u91cf<\/li>\n<\/ul>\n\n\n\n

              \u4e0e\u67ef\u897f-\u9ece\u66fc\u65b9\u7a0b\u7684\u8054\u7cfb<\/h3>\n\n\n\n

              \u5f53 f<\/mi>(<\/mo>z<\/mi>)<\/mo><\/mrow>f(z)<\/annotation><\/semantics><\/math> \u89e3\u6790\u65f6\uff0c\u6ee1\u8db3\uff1a<\/p>\n\n\n\n
              \u2202<\/mi>u<\/mi><\/mrow>\u2202<\/mi>x<\/mi><\/mrow><\/mfrac>=<\/mo>\u2202<\/mi>v<\/mi><\/mrow>\u2202<\/mi>y<\/mi><\/mrow><\/mfrac>,<\/mo><\/mspace>\u2202<\/mi>u<\/mi><\/mrow>\u2202<\/mi>y<\/mi><\/mrow><\/mfrac>=<\/mo>\u2212<\/mo>\u2202<\/mi>v<\/mi><\/mrow>\u2202<\/mi>x<\/mi><\/mrow><\/mfrac><\/mrow>\\frac{\\partial u}{\\partial x} = \\frac{\\partial v}{\\partial y}, \\quad \n\\frac{\\partial u}{\\partial y} = -\\frac{\\partial v}{\\partial x}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n

              \u6b64\u65f6\u4e24\u4e2a\u5b9e\u79ef\u5206\u5747\u4e3a\u96f6\uff0c\u7b26\u5408\u67ef\u897f\u79ef\u5206\u5b9a\u7406 \u222e<\/mo>C<\/mi><\/msub>f<\/mi>(<\/mo>z<\/mi>)<\/mo>d<\/mi><\/mrow>z<\/mi>=<\/mo>0<\/mn><\/mrow>\\oint_C f(z)\\mathrm{d}z = 0<\/annotation><\/semantics><\/math>\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

              balabala<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17],"tags":[18,19],"class_list":["post-92","post","type-post","status-publish","format-standard","hentry","category-17","tag-18","tag-19"],"_links":{"self":[{"href":"https:\/\/wdlobin.com\/blog\/wp-json\/wp\/v2\/posts\/92","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wdlobin.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wdlobin.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wdlobin.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wdlobin.com\/blog\/wp-json\/wp\/v2\/comments?post=92"}],"version-history":[{"count":1,"href":"https:\/\/wdlobin.com\/blog\/wp-json\/wp\/v2\/posts\/92\/revisions"}],"predecessor-version":[{"id":93,"href":"https:\/\/wdlobin.com\/blog\/wp-json\/wp\/v2\/posts\/92\/revisions\/93"}],"wp:attachment":[{"href":"https:\/\/wdlobin.com\/blog\/wp-json\/wp\/v2\/media?parent=92"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wdlobin.com\/blog\/wp-json\/wp\/v2\/categories?post=92"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wdlobin.com\/blog\/wp-json\/wp\/v2\/tags?post=92"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}