[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki7\/archives\/9363#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki7\/archives\/9363","headline":"\u7dda\u7a4d\u5206 &#8211; Wikipedia","name":"\u7dda\u7a4d\u5206 &#8211; Wikipedia","description":"\u91cf\u5b50\u529b\u5b66\u306e\u7406\u8ad6\u624b\u6cd5\u3067\u3042\u308b\u300c\u7d4c\u8def\u7a4d\u5206\u300d\u3068\u306f\u7570\u306a\u308a\u307e\u3059\u3002 \u6570\u5b66\u306b\u304a\u3051\u308b\u7dda\u7a4d\u5206\uff08\u305b\u3093\u305b\u304d\u3076\u3093\u3001\u82f1: line integral; \u7a00\u306b path integral[\u6ce8\u91c8 1], curve integral, curvilinear integral\uff09\u306f\u3001\u66f2\u7dda\u306b\u6cbf\u3063\u3066\u8a55\u4fa1\u3055\u308c\u305f\u51fd\u6570\u306e\u5024\u306b\u3064\u3044\u3066\u306e\u7a4d\u5206\u306e\u7dcf\u79f0\u3002\u30d9\u30af\u30c8\u30eb\u89e3\u6790\u3084\u8907\u7d20\u89e3\u6790\u306b\u304a\u3044\u3066\u91cd\u8981\u306a\u5f79\u5272\u3092\u6f14\u3058\u308b\u3002\u9589\u66f2\u7dda\u306b\u6cbf\u3046\u7dda\u7a4d\u5206\u3092\u7279\u306b\u9589\u8def\u7a4d\u5206\uff08\u3078\u3044\u308d\u305b\u304d\u3076\u3093\uff09\u3042\u308b\u3044\u306f\u5468\u56de\u7a4d\u5206\uff08\u3057\u3085\u3046\u304b\u3044\u305b\u304d\u3076\u3093\uff09\u3068\u547c\u3073\u3001\u5c02\u7528\u306e\u7a4d\u5206\u8a18\u53f7 \u222e{displaystyle oint } \u304c\u4f7f\u308f\u308c\u308b\u3053\u3068\u3082\u3042\u308b\u3002\u5468\u56de\u7a4d\u5206\u6cd5\u306f\u8907\u7d20\u89e3\u6790\u306b\u304a\u3051\u308b\u91cd\u8981\u306a\u624b\u6cd5\u306e\u4e00\u3064\u3067\u3042\u308b\u3002 \u8868\u9762z = f\uff08x\u3001y\uff09\u306b\u6cbf\u3063\u305f\u66f2\u7ddaC\u306e\u4e0b\u306e\u9818\u57df\u3068\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b \u7dda\u7a4d\u5206\u306e\u5bfe\u8c61\u3068\u306a\u308b\u51fd\u6570\u306f\u3001\u30b9\u30ab\u30e9\u30fc\u5834\u3084\u30d9\u30af\u30c8\u30eb\u5834\u306a\u3069\u3068\u3057\u3066\u4e0e\u3048\u308b\u3002\u7dda\u7a4d\u5206\u306e\u5024\u306f\u5834\u306e\u8003\u3048\u3066\u3044\u308b\u66f2\u7dda\u4e0a\u3067\u306e\u5024\u306b\u66f2\u7dda\u4e0a\u306e\u3042\u308b\u30b9\u30ab\u30e9\u30fc\u51fd\u6570\uff08\u5f27\u9577\u3001\u3042\u308b\u3044\u306f\u30d9\u30af\u30c8\u30eb\u5834\u306b\u3064\u3044\u3066\u306f\u66f2\u7dda\u4e0a\u306e\u5fae\u5206\u30d9\u30af\u30c8\u30eb\u3068\u306e\u70b9\u4e57\u7a4d\uff09\u306b\u3088\u308b\u91cd\u307f\u4ed8\u3051\u3092\u3057\u305f\u3082\u306e\u3092\u300c\u8db3\u3057\u5408\u308f\u305b\u305f\u300d\u3082\u306e\u3068\u306a\u308b\u3002\u3053\u306e\u91cd\u307f\u4ed8\u3051\u304c\u3001\u533a\u9593\u4e0a\u3067\u5b9a\u7fa9\u3059\u308b\u7a4d\u5206\u3068\u7dda\u7a4d\u5206\u3068\u3092\u5206\u3051\u308b\u70b9\u3067\u3042\u308b\u3002","datePublished":"2021-09-03","dateModified":"2021-09-03","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki7\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki7\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/f\/fb\/Confusion_grey.svg\/25px-Confusion_grey.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/f\/fb\/Confusion_grey.svg\/25px-Confusion_grey.svg.png","height":"19","width":"25"},"url":"https:\/\/wiki.edu.vn\/jp\/wiki7\/archives\/9363","about":["Wiki"],"wordCount":10964,"articleBody":"\u91cf\u5b50\u529b\u5b66\u306e\u7406\u8ad6\u624b\u6cd5\u3067\u3042\u308b\u300c\u7d4c\u8def\u7a4d\u5206\u300d\u3068\u306f\u7570\u306a\u308a\u307e\u3059\u3002\u6570\u5b66\u306b\u304a\u3051\u308b\u7dda\u7a4d\u5206\uff08\u305b\u3093\u305b\u304d\u3076\u3093\u3001\u82f1: line integral; \u7a00\u306b path integral[\u6ce8\u91c8 1], curve integral, curvilinear integral\uff09\u306f\u3001\u66f2\u7dda\u306b\u6cbf\u3063\u3066\u8a55\u4fa1\u3055\u308c\u305f\u51fd\u6570\u306e\u5024\u306b\u3064\u3044\u3066\u306e\u7a4d\u5206\u306e\u7dcf\u79f0\u3002\u30d9\u30af\u30c8\u30eb\u89e3\u6790\u3084\u8907\u7d20\u89e3\u6790\u306b\u304a\u3044\u3066\u91cd\u8981\u306a\u5f79\u5272\u3092\u6f14\u3058\u308b\u3002\u9589\u66f2\u7dda\u306b\u6cbf\u3046\u7dda\u7a4d\u5206\u3092\u7279\u306b\u9589\u8def\u7a4d\u5206\uff08\u3078\u3044\u308d\u305b\u304d\u3076\u3093\uff09\u3042\u308b\u3044\u306f\u5468\u56de\u7a4d\u5206\uff08\u3057\u3085\u3046\u304b\u3044\u305b\u304d\u3076\u3093\uff09\u3068\u547c\u3073\u3001\u5c02\u7528\u306e\u7a4d\u5206\u8a18\u53f7 \u222e{displaystyle oint }   \u304c\u4f7f\u308f\u308c\u308b\u3053\u3068\u3082\u3042\u308b\u3002\u5468\u56de\u7a4d\u5206\u6cd5\u306f\u8907\u7d20\u89e3\u6790\u306b\u304a\u3051\u308b\u91cd\u8981\u306a\u624b\u6cd5\u306e\u4e00\u3064\u3067\u3042\u308b\u3002  \u8868\u9762z = f\uff08x\u3001y\uff09\u306b\u6cbf\u3063\u305f\u66f2\u7ddaC\u306e\u4e0b\u306e\u9818\u57df\u3068\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u7dda\u7a4d\u5206\u306e\u5bfe\u8c61\u3068\u306a\u308b\u51fd\u6570\u306f\u3001\u30b9\u30ab\u30e9\u30fc\u5834\u3084\u30d9\u30af\u30c8\u30eb\u5834\u306a\u3069\u3068\u3057\u3066\u4e0e\u3048\u308b\u3002\u7dda\u7a4d\u5206\u306e\u5024\u306f\u5834\u306e\u8003\u3048\u3066\u3044\u308b\u66f2\u7dda\u4e0a\u3067\u306e\u5024\u306b\u66f2\u7dda\u4e0a\u306e\u3042\u308b\u30b9\u30ab\u30e9\u30fc\u51fd\u6570\uff08\u5f27\u9577\u3001\u3042\u308b\u3044\u306f\u30d9\u30af\u30c8\u30eb\u5834\u306b\u3064\u3044\u3066\u306f\u66f2\u7dda\u4e0a\u306e\u5fae\u5206\u30d9\u30af\u30c8\u30eb\u3068\u306e\u70b9\u4e57\u7a4d\uff09\u306b\u3088\u308b\u91cd\u307f\u4ed8\u3051\u3092\u3057\u305f\u3082\u306e\u3092\u300c\u8db3\u3057\u5408\u308f\u305b\u305f\u300d\u3082\u306e\u3068\u306a\u308b\u3002\u3053\u306e\u91cd\u307f\u4ed8\u3051\u304c\u3001\u533a\u9593\u4e0a\u3067\u5b9a\u7fa9\u3059\u308b\u7a4d\u5206\u3068\u7dda\u7a4d\u5206\u3068\u3092\u5206\u3051\u308b\u70b9\u3067\u3042\u308b\u3002  \u7269\u7406\u5b66\u306b\u304a\u3051\u308b\u591a\u304f\u306e\u5358\u7d14\u306a\u516c\u5f0f\u304c\u3001\u7dda\u7a4d\u5206\u3067\u66f8\u304f\u3053\u3068\u306b\u3088\u3063\u3066\u81ea\u7136\u306b\u3001\u9023\u7d9a\u7684\u306b\u5909\u5316\u3055\u305b\u305f\u5834\u5408\u306b\u3064\u3044\u3066\u3082\u4e00\u822c\u5316\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308b\u3002\u4f8b\u3048\u3070\u3001\u529b\u5b66\u7684\u306a\u4ed5\u4e8b\u3092\u8868\u3059\u5f0f W = F\u22c5s \u304b\u3089\u66f2\u7dda C \u306b\u6cbf\u3063\u3066\u306e\u4ed5\u4e8b\u3092\u8868\u3059\u5f0f W = \u222bCF\u22c5ds \u3092\u5f97\u308b\u3002\u4f8b\u3048\u3070\u96fb\u5834\u3084\u91cd\u529b\u5834\u306b\u304a\u3044\u3066\u904b\u52d5\u3059\u308b\u7269\u4f53\u306e\u6210\u3059\u4ed5\u4e8b\u304c\u8a08\u7b97\u3067\u304d\u308b\u3002Table of Contents\u5f27\u9577\u5909\u6570\u3068\u7dda\u7d20[\u7de8\u96c6]\u5834\u306e\u7dda\u7a4d\u5206[\u7de8\u96c6]\u30b9\u30ab\u30e9\u30fc\u5834\u306b\u5bfe\u3059\u308b\u7dda\u7a4d\u5206[\u7de8\u96c6]\u504f\u7dda\u7a4d\u5206[\u7de8\u96c6]\u7dda\u7d20\u306b\u95a2\u3059\u308b\u7dda\u7a4d\u5206[\u7de8\u96c6]\u7dda\u7d20\u306b\u95a2\u3059\u308b\u7dda\u7a4d\u5206\u306e\u5c0e\u51fa[\u7de8\u96c6]\u30d9\u30af\u30c8\u30eb\u5834\u306b\u5bfe\u3059\u308b\u7dda\u7a4d\u5206[\u7de8\u96c6]\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7dda\u7a4d\u5206\u306e\u5b9a\u7fa9[\u7de8\u96c6]\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7dda\u7a4d\u5206\u306e\u5c0e\u51fa[\u7de8\u96c6]\u7d4c\u8def\u72ec\u7acb\u306a\u7dda\u7a4d\u5206[\u7de8\u96c6]\u5fdc\u7528[\u7de8\u96c6]\u8907\u7d20\u7dda\u7a4d\u5206[\u7de8\u96c6]\u8907\u7d20\u7dda\u7a4d\u5206\u306e\u4f8b[\u7de8\u96c6]\u8907\u7d20\u7dda\u7a4d\u5206\u3068\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7a4d\u5206\u3068\u306e\u95a2\u4fc2[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u6ce8\u91c8[\u7de8\u96c6]\u51fa\u5178[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]\u5f27\u9577\u5909\u6570\u3068\u7dda\u7d20[\u7de8\u96c6]n \u6b21\u5143\u5b9f\u591a\u69d8\u4f53 M \u306e\u9818\u57df \u03a9 \u3092\u8003\u3048\u308b\u3002\u5c40\u6240\u7684\u306b\u306f \u03a9 \u2282 Rn \u3068\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u03a9 \u5185\u306e\u6ed1\u3089\u304b\u306a\u66f2\u7dda \u03b3: I \u2192 \u03a9 \u304c r = \u03b3(t) = (\u03b31(t), \u03b32(t), \u2026, \u03b3n(t)) \u3067\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3068\u304d\u3001s = s(t) \u304c \u03b3 \u306e\u5f27\u9577\u5909\u6570\u3067\u3042\u308b\u3068\u306f\u3001\u305d\u308c\u304c\u7dda\u5206 \u03b3 \u306b\u6cbf\u3063\u3066\u7aef\u70b9\u304b\u3089\u6e2c\u3063\u305f \u03b3 \u306e\u5f27\u9577\u3092\u4e0e\u3048\u308b\u3082\u306e\u3067\u3042\u308b\u3053\u3068\u3092\u8a00\u3046\u3002\u3044\u307e \u03b3 \u306f\u306a\u3081\u3089\u304b\u3067\u3042\u308b\u304b\u3089\u3001\u305d\u306e\u5f27\u9577\u306f\u533a\u9593 I = [a, b] \u4e0a\u306e\u5404\u70b9 t0 \u306b\u5bfe\u3057\u3066  s(t0)=\u222baa+t0|d\u03b3dt|dt=\u222baa+t0(d\u03b31dt)2+(d\u03b32dt)2+\u22ef+(d\u03b3ndt)2dt{displaystyle s(t_{0})=int _{a}^{a+t_{0}}left|{frac {dgamma }{dt}}right|dt=int _{a}^{a+t_{0}}{sqrt {({tfrac {dgamma _{1}}{dt}})^{2}+({tfrac {dgamma _{2}}{dt}})^{2}+cdots +({tfrac {dgamma _{n}}{dt}})^{2}}};{mathit {dt}}}\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u7279\u306b s \u306fds=|d\u03b3dt|dt=|d\u03b3|{displaystyle ds=left|{frac {dgamma }{dt}}right|{mathit {dt}}=|dgamma |}\u3092\u6e80\u305f\u3059\u304c\u3001\u3053\u308c\u306f\u30d1\u30e9\u30e1\u30fc\u30bf t \u306e\u53d6\u308a\u65b9\u306b\u4f9d\u3089\u305a\u5b9a\u307e\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u3079\u304d\u3067\u3042\u308b\u3002\u8a18\u53f7\u7684\u306b\u306f|dr|2=dx12+dx22+\u22ef+dxn2{displaystyle |dmathbf {r} |^{2}=dx_{1}^{2}+dx_{2}^{2}+dotsb +dx_{n}^{2}}\u306b r = \u03b3(t) \u3092\u4ee3\u5165\u3059\u308b\u3053\u3068\u3067\u5f97\u3089\u308c\u308b\u3002\u3053\u306e ds \u3092 \u03b3 \u306e\u7dda\u7d20\uff08\u305b\u3093\u305d\u3001line element\uff09\u3068\u547c\u3076\u3002\u66f2\u7dda\u304c\u533a\u5206\u7684\u306b\u6ed1\u3089\u304b\u306a\u3089\u3001\u5fae\u5206\u53ef\u80fd\u306a\u533a\u9593\u306e\u548c\u306b\u308f\u3051\u3066\u540c\u3058\u304f\u5f27\u9577\u3092\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u5834\u306e\u7dda\u7a4d\u5206[\u7de8\u96c6]\u5b9a\u6027\u7684\u306b\u306f\u3001\u30d9\u30af\u30c8\u30eb\u89e3\u6790\u306b\u304a\u3051\u308b\u7dda\u7a4d\u5206\u306f\u3001\u4e0e\u3048\u3089\u308c\u305f\u5834\u306e\u4e0e\u3048\u3089\u308c\u305f\u66f2\u7dda\u306b\u6cbf\u3063\u3066\u306e\u5168\u4f53\u7684\u306a\u52b9\u679c\u3092\u8a08\u308b\u3082\u306e\u3068\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3088\u308a\u53b3\u5bc6\u306b\u8a00\u3048\u3070\u3001\u30b9\u30ab\u30e9\u30fc\u5834\u4e0a\u306e\u7dda\u7a4d\u5206\u306f\u3001\u7279\u5b9a\u306e\u66f2\u7dda\u306b\u3088\u3063\u3066\u66f2\u3052\u3089\u308c\u305f\u5834\u306e\u4e0b\u306b\u3042\u308b\u9818\u57df\u306e\u9762\u7a4d\u3068\u89e3\u91c8\u3067\u304d\u308b\u3002\u3053\u308c\u306f z = f(x, y) \u3067\u5b9a\u7fa9\u3059\u308b\u66f2\u9762\u3068 xy-\u5e73\u9762\u4e0a\u306e\u66f2\u7dda C \u3092\u4f7f\u3063\u3066\u8996\u899a\u7684\u306b\u898b\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001f \u306e\u7dda\u7a4d\u5206\u306f\u66f2\u7dda C \u306e\u771f\u4e0a\u306b\u3042\u308b\u66f2\u9762\u4e0a\u306e\u70b9\u3067\u5207\u308a\u53d6\u308b\u3068\u304d\u306b\u3067\u304d\u308b\u300c\u30ab\u30fc\u30c6\u30f3\u300d\u306e\u9762\u7a4d\u306b\u306a\u308b[2]\u3002\u30b9\u30ab\u30e9\u30fc\u5834\u306b\u5bfe\u3059\u308b\u7dda\u7a4d\u5206[\u7de8\u96c6]\u504f\u7dda\u7a4d\u5206[\u7de8\u96c6]\u30b9\u30ab\u30e9\u30fc\u5834 f\u00a0: U \u2286 Rn \u2192 R \u306e\u6ed1\u3089\u304b\u306a\u66f2\u7dda [a, b] \u220b t \u21a6 \u03b3(t) = (\u03b31(t), \u03b32(t), \u2026, \u03b3n(t)) \u306b\u6cbf\u3063\u305f\u5404\u8ef8\u65b9\u5411\u306e\u7dda\u7a4d\u5206\u306f\u222bCfdxi=\u222babf(r(t))d\u03b3i(t)dtdt{displaystyle int _{C}f,dx_{i}=int _{a}^{b}f(mathbf {r} (t)){frac {dgamma _{i}(t)}{dt}}dt}\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u3053\u306e\u3068\u304d\u3001\u51fd\u6570 f \u3092\u88ab\u7a4d\u5206\u51fd\u6570 (integrand)\u3001\u66f2\u7dda C \u3092\u7a4d\u5206\u9818\u57df (domain of integration) \u3042\u308b\u3044\u306f\u7a4d\u5206\u8def (path) \u3068\u547c\u3076\u3002\u7dda\u7d20\u306b\u95a2\u3059\u308b\u7dda\u7a4d\u5206[\u7de8\u96c6]\u30b9\u30ab\u30e9\u30fc\u5834 f\u00a0: U \u2286 Rn \u2192 R \u306e\u6ed1\u3089\u304b\u306a\u66f2\u7dda C \u2282 U \u306b\u6cbf\u3063\u305f\u7dda\u7d20\u306b\u95a2\u3059\u308b\u7dda\u7a4d\u5206\u306f\u222bCfds=\u222babf(r(t))|r\u2032(t)|dt{displaystyle int _{C}f,ds=int _{a}^{b}f(mathbf {r} (t))|mathbf {r} &#8216;(t)|dt}\u3068\u5b9a\u7fa9\u3059\u308b\uff08\u533a\u5206\u7684\u306b\u6ed1\u3089\u304b\u306e\u5834\u5408\u306f\u3001\u6ed1\u3089\u304b\u306a\u533a\u9593\u3054\u3068\u306e\u7a4d\u5206\u306e\u548c\u3068\u5b9a\u3081\u308b\uff09\u3002\u305f\u3060\u3057\u3001r: [a, b] \u2192 C \u306f\u3001r(a) \u3068 r(b) \u304c\u4e0e\u3048\u305f\u66f2\u7dda C \u306e\u4e21\u7aef\u70b9\u3068\u306a\u308b\u3088\u3046\u306a\u3001C \u306e\u52dd\u624b\u306a\u5168\u5358\u5c04\u5a92\u4ecb\u8868\u793a\u3068\u3059\u308b\u3002\u8a18\u53f7 ds \u306f\u76f4\u89b3\u7684\u306b\u306f\u5f27\u9577\u306e\u7121\u9650\u5c0f\u6210\u5206\u3068\u3057\u3066\u306e\u7dda\u7d20\u3068\u89e3\u91c8\u3067\u304d\u308b\u3002\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u66f2\u7dda C \u306b\u6cbf\u3063\u305f\u7dda\u7a4d\u5206\u306f\u3001C \u306e\u5a92\u4ecb\u8868\u793a r \u306e\u53d6\u308a\u65b9\u306b\u4f9d\u3089\u306a\u3044\u3002\u7dda\u7d20\u306b\u95a2\u3059\u308b\u7dda\u7a4d\u5206\u306e\u5c0e\u51fa[\u7de8\u96c6]\u4e0a\u8a18\u306e\u5982\u304f f, C \u3092\u5b9a\u3081\u3001C \u306e\u5a92\u4ecb\u8868\u793a r \u3092\u53d6\u308c\u3070\u3001\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u7dda\u7a4d\u5206\u306f\u30ea\u30fc\u30de\u30f3\u548c\u3068\u3057\u3066\u69cb\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u533a\u9593 [a, b] \u3092\u9577\u3055 \u0394t = (b \u2212 a)\/n \u306e n-\u500b\u306e\u5c0f\u533a\u9593 [ti\u22121,\u2009ti] \u306b\u5206\u5272\u3057\u3001\u66f2\u7dda C \u4e0a\u306b\u5404\u5c0f\u533a\u9593\u306b\u5bfe\u5fdc\u3059\u308b\u6a19\u672c\u70b9 r(ti) \u3092\u3068\u308b\u3002\u6a19\u672c\u70b9\u306e\u96c6\u5408 {r(ti) | 1 \u2264 i \u2264 n} \u306b\u5bfe\u3057\u3066\u3001\u6a19\u672c\u70b9 r(ti\u22121) \u3068 r(ti) \u3092\u7d50\u3093\u3067\u3067\u304d\u308b\u7dda\u5206\u306e\u96c6\u307e\u308a\u306b\u3088\u3063\u3066\u66f2\u7dda C \u3092\u8fd1\u4f3c\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u5404\u6a19\u672c\u70b9\u306e\u9593\u3092\u7d50\u3076\u7dda\u5206\u306e\u9577\u3055\u3092 \u0394si \u3068\u66f8\u304f\u3053\u3068\u306b\u3059\u308c\u3070\u3001\u7a4d f\u2009(r(ti))\u0394si \u306f\u3001\u9ad8\u3055\u3068\u5e45\u304c f&#038;(r(ti)) \u3068 \u0394si \u3067\u4e0e\u3048\u3089\u308c\u308b\u77e9\u5f62\u306e\u7b26\u53f7\u4ed8\u9762\u7a4d\u306b\u5bfe\u5fdc\u3059\u308b\u3002\u305d\u308c\u3089\u306e\u7dcf\u548c\u3092\u53d6\u3063\u3066\u3001\u5206\u5272\u306e\u5404\u5c0f\u533a\u9593\u306e\u9577\u3055\u3092 0 \u306b\u8fd1\u3065\u3051\u308b\u6975\u9650\u3092I=lim\u0394t\u21920\u2211i=1nf(r(ti))\u0394si{displaystyle I=lim _{Delta tto 0}sum _{i=1}^{n}f(mathbf {r} (t_{i}))Delta s_{i}}\u3068\u8003\u3048\u308b\u3068\u304d\u3001\u66f2\u7dda\u4e0a\u306e\u5206\u70b9\u9593\u306e\u8ddd\u96e2\u306f\u0394si=|r(ti+\u0394t)\u2212r(ti)|=|r\u2032(ti)|\u0394t{displaystyle Delta s_{i}=|mathbf {r} (t_{i}+Delta t)-mathbf {r} (t_{i})|=|mathbf {r} &#8216;(t_{i})|Delta t}\u3068\u66f8\u3051\u308b\u304b\u3089\u3001\u3053\u308c\u3092\u4ee3\u5165\u3057\u3066\u5f97\u308bI=lim\u0394t\u21920\u2211i=1nf(r(ti))|r\u2032(ti)|\u0394t{displaystyle I=lim _{Delta tto 0}sum _{i=1}^{n}f(mathbf {r} (t_{i}))|mathbf {r} &#8216;(t_{i})|Delta t}\u306f\u3001\u7a4d\u5206I=\u222babf(r(t))|r\u2032(t)|dt{displaystyle I=int _{a}^{b}f(mathbf {r} (t))|mathbf {r} &#8216;(t)|{mathit {dt}}}\u306b\u5bfe\u5fdc\u3059\u308b\u30ea\u30fc\u30de\u30f3\u548c\u3067\u3042\u308b\u3002\u57fa\u672c\u7684\u306b\u3053\u306e\u7a4d\u5206\u306f\u3001x = u(t) \u304a\u3088\u3073 y = v(t) \u3068\u306a\u308b\u5236\u7d04\u6761\u4ef6\u4e0b\u3067\u30b9\u30ab\u30e9\u30fc\u51fd\u6570 z = f(x, y) \u306e\u4e0b\u306b\u3042\u308b\u9818\u57df\u306e\u9762\u7a4d\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u30d9\u30af\u30c8\u30eb\u5834\u306b\u5bfe\u3059\u308b\u7dda\u7a4d\u5206[\u7de8\u96c6]\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7dda\u7a4d\u5206\u306e\u5b9a\u7fa9[\u7de8\u96c6]\u30d9\u30af\u30c8\u30eb\u5834 F: U \u2286 Rn \u2192 Rn \u306e r \u306e\u5411\u304d\u3078\u306e\u533a\u5206\u7684\u306b\u6ed1\u3089\u304b\u306a\u66f2\u7dda C \u2282 U \u306b\u6cbf\u3063\u305f\u7dda\u7a4d\u5206\u306f\u222bCF(r)\u22c5dr=\u222babF(r(t))\u22c5r\u2032(t)dt{displaystyle int _{C}mathbf {F} (mathbf {r} )cdot dmathbf {r} =int _{a}^{b}mathbf {F} (mathbf {r} (t))cdot mathbf {r} &#8216;(t),dt}\u3068\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u305f\u3060\u3057\u3001&#8221;\u22c5&#8221; \u306f\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3067\u3042\u308a\u3001r: [a, b] \u2192 C \u306f\u3001r(a) \u3068 r(b) \u304c\u66f2\u7dda C \u306e\u4e21\u7aef\u70b9\u3068\u306a\u308b C \u306e\u5168\u5358\u5c04\u5a92\u4ecb\u8868\u793a\u3068\u3059\u308b\u3002\u5f93\u3063\u3066\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u7dda\u7a4d\u5206\u306f\u3001\u5404\u30d9\u30af\u30c8\u30eb\u304c\u5e38\u306b\u7a4d\u5206\u8def\u306b\u63a5\u3059\u308b\u3088\u3046\u306a\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7dda\u7a4d\u5206\u306b\u4e00\u81f4\u3059\u308b\u3002\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7dda\u7a4d\u5206\u306f\u3001\u7d76\u5bfe\u5024\u306b\u95a2\u3057\u3066\u306f\u5a92\u4ecb\u5909\u6570 r \u306e\u53d6\u308a\u65b9\u306b\u4f9d\u3089\u306a\u3044\u304c\u3001\u5411\u304d\u306b\u95a2\u3057\u3066\u306f\u4f9d\u5b58\u3059\u308b\u3002\u7279\u306b\u3001\u5a92\u4ecb\u5909\u6570\u306e\u5411\u304d\u3092\u9006\u306b\u3059\u308c\u3070\u3001\u7dda\u7a4d\u5206\u306e\u7b26\u53f7\u304c\u5909\u308f\u308b\u3002\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7dda\u7a4d\u5206\u306e\u5c0e\u51fa[\u7de8\u96c6]  \u30d9\u30af\u30c8\u30eb\u5834\u5185\u306e\u66f2\u7dda\u306b\u6cbf\u3063\u305f\u7c92\u5b50\u306e\u8ecc\u8de1\u3002\u4e0b\u306b\u8868\u793a\u3055\u308c\u3066\u3044\u308b\u306e\u306f\u3001\u66f2\u7dda\u306b\u6cbf\u3063\u3066\u7c92\u5b50\u304c\u52d5\u3044\u305f\u3068\u304d\u306b\u7c92\u5b50\u304c\u51fa\u4f1a\u3046\u5834\u306e\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3002\u305d\u308c\u3089\u306e\u30d9\u30af\u30c8\u30eb\u3068\u8ecc\u8de1\u306e\u5404\u70b9\u306b\u304a\u3051\u308b\u66f2\u7dda\u306e\u63a5\u30d9\u30af\u30c8\u30eb\u3068\u306e\u70b9\u4e57\u7a4d\u306e\u548c\u3092\u53d6\u3063\u305f\u3082\u306e\u304c\u3001\u6c42\u3081\u308b\u7dda\u7a4d\u5206\u306b\u306a\u308b\u3002\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7dda\u7a4d\u5206\u3082\u3001\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u7dda\u7a4d\u5206\u306e\u5834\u5408\u3068\u3088\u304f\u4f3c\u305f\u65b9\u6cd5\u3067\u5c0e\u3051\u308b\u3002\u30d9\u30af\u30c8\u30eb\u5834 F\u3001\u66f2\u7dda C\u3001\u5a92\u4ecb\u8868\u793a r(t) \u306f\u4e0a\u8a18\u306e\u5982\u304f\u3068\u3057\u3066\u3001\u30ea\u30fc\u30de\u30f3\u548c\u3092\u69cb\u6210\u3057\u3088\u3046\u3002\u533a\u9593 [a,\u2009b] \u3092\u9577\u3055 \u0394t = (b \u2212 a)\/n \u306e n-\u500b\u306e\u5c0f\u533a\u9593\u306b\u5206\u5272\u3057\u3001i-\u756a\u76ee\u306e\u5c0f\u533a\u9593\u304b\u3089\u6a19\u672c\u70b9 ti \u3092\u53d6\u3063\u3066\u3001\u66f2\u7dda\u4e0a\u306e\u5206\u70b9 r(ti) \u3092\u8003\u3048\u308b\u3002\u3053\u3053\u3067\u306f\u5206\u70b9\u9593\u306e\u8ddd\u96e2\u3092\u8db3\u3057\u5408\u308f\u305b\u308b\u306e\u3067\u306f\u306a\u304f\u3066\u3001\u5206\u70b9\u9593\u306e\u5909\u4f4d\u30d9\u30af\u30c8\u30eb \u0394si \u3092\u8db3\u3057\u5408\u308f\u305b\u308b\u3002\u524d\u3068\u540c\u3058\u304f\u3001F \u3092\u653e\u5c04\u66f2\u7dda\u4e0a\u306e\u5404\u70b9\u3067\u8a55\u4fa1\u3057\u3066\u3001\u305d\u308c\u3068\u66f2\u7dda C \u306e\u5404\u5c0f\u7247\u3067\u306e F \u306e\u7121\u9650\u5c0f\u5bc4\u4e0e\u3092\u4e0e\u3048\u308b\u5909\u4f4d\u30d9\u30af\u30c8\u30eb\u3068\u306e\u70b9\u4e57\u7a4d\u3092\u3068\u3063\u305f\u3082\u306e\u5168\u3066\u548c\u306e\u3001\u5206\u5272\u306e\u30b5\u30a4\u30ba\u3092 0 \u306b\u3059\u308b\u6975\u9650I=lim\u0394t\u21920\u2211i=1nF(r(ti))\u22c5\u0394si{displaystyle I=lim _{Delta tto 0}sum _{i=1}^{n}mathbf {F} (mathbf {r} (t_{i}))cdot Delta mathbf {s} _{i}}\u3092\u8003\u3048\u308b\u3002\u66f2\u7dda\u4e0a\u306e\u96a3\u308a\u5408\u3046\u5206\u70b9\u306e\u9593\u306e\u5909\u4f4d\u30d9\u30af\u30c8\u30eb\u306f\u0394si=r(ti+\u0394t)\u2212r(ti)=r\u2032(ti)\u0394t{displaystyle Delta mathbf {s} _{i}=mathbf {r} (t_{i}+Delta t)-mathbf {r} (t_{i})=mathbf {r} &#8216;(t_{i})Delta t}\u3068\u66f8\u3051\u308b\u304b\u3089\u3001\u4ee3\u5165\u3057\u3066\u30ea\u30fc\u30de\u30f3\u548cI=lim\u0394t\u21920\u2211i=1nF(r(ti))\u22c5r\u2032(ti)\u0394t{displaystyle I=lim _{Delta trightarrow 0}sum _{i=1}^{n}mathbf {F} (mathbf {r} (t_{i}))cdot mathbf {r} &#8216;(t_{i})Delta t}\u3092\u5f97\u3001\u3053\u308c\u306b\u3088\u308a\u4e0a\u8a18\u306e\u7dda\u7a4d\u5206\u304c\u5b9a\u307e\u308b\u3002\u7d4c\u8def\u72ec\u7acb\u306a\u7dda\u7a4d\u5206[\u7de8\u96c6]\u30d9\u30af\u30c8\u30eb\u5834 F \u304c\u4f55\u3089\u304b\u306e\u30b9\u30ab\u30e9\u30fc\u5834 G \u306e\u52fe\u914d\u3068\u3057\u3066\u2207G=F{displaystyle nabla G=mathbf {F} }\u3068\u66f8\u3051\u308b\u3068\u304d\u3001G \u3068 r(t) \u3068\u306e\u5408\u6210\u306e\u5c0e\u51fd\u6570dG(r(t))dt=\u2207G(r(t))\u22c5r\u2032(t)=F(r(t))\u22c5r\u2032(t){displaystyle {frac {dG(mathbf {r} (t))}{dt}}=nabla G(mathbf {r} (t))cdot mathbf {r} &#8216;(t)=mathbf {F} (mathbf {r} (t))cdot mathbf {r} &#8216;(t)}\u306f\u3001F \u306e r(t) \u4e0a\u306e\u7dda\u7a4d\u5206\u306e\u88ab\u7a4d\u5206\u51fd\u6570\u3067\u3042\u308b\u3002\u5f93\u3063\u3066\u3001\u7a4d\u5206\u8def C \u3092\u4e0e\u3048\u308c\u3070\u222bCF(r)\u22c5dr=\u222babF(r(t))\u22c5r\u2032(t)dt=\u222babdG(r(t))dtdt=G(r(b))\u2212G(r(a)){displaystyle int _{C}mathbf {F} (mathbf {r} )cdot ,dmathbf {r} =int _{a}^{b}mathbf {F} (mathbf {r} (t))cdot mathbf {r} &#8216;(t),dt=int _{a}^{b}{frac {dG(mathbf {r} (t))}{dt}},dt=G(mathbf {r} (b))-G(mathbf {r} (a))}\u304c\u6210\u308a\u7acb\u3064\u3002\u8a00\u3044\u63db\u3048\u308c\u3070\u3001F \u306e C \u4e0a\u306e\u7a4d\u5206\u306f\u3001\u70b9 r(b) \u304a\u3088\u3073 r(a) \u4e0a\u306e G \u306e\u5024\u306e\u307f\u306b\u4f9d\u5b58\u3057\u3001\u305d\u308c\u3089\u3092\u7d50\u3076\u7a4d\u5206\u8def\u306e\u53d6\u308a\u65b9\u306b\u4f9d\u3089\u306a\u3044\u3002\u7279\u306b\u7a4d\u5206\u8def C \u304c\u9589\u7d4c\u8def\u3067\u3042\u308b\u306a\u3089\u3070\u3001\u7a4d\u5206\u306f\u5fc5\u305a 0 \u306b\u306a\u308b\u305f\u3081\u3001\u30d9\u30af\u30c8\u30eb\u5834 F \u306f\u4fdd\u5b58\u30d9\u30af\u30c8\u30eb\u5834\uff08\u82f1\u8a9e\u7248\uff09\u3068\u547c\u3070\u308c\u308b\u3002\u307e\u305f\u3001\u7269\u7406\u5b66\u306b\u304a\u3044\u3066\u3001\u3053\u306e\u3088\u3046\u306a\u6027\u8cea\u3092\u6301\u3064\u529b\u306e\u5834\u3092\u4fdd\u5b58\u529b\u3068\u547c\u3076\u3002\u3053\u306e\u3053\u3068\u304b\u3089\u3001\u4fdd\u5b58\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7dda\u7a4d\u5206\u306f\u7d4c\u8def\u72ec\u7acb (path independent) \u3042\u308b\u3044\u306f\u300c\u7a4d\u5206\u7d4c\u8def\u306b\u4f9d\u3089\u306a\u3044\u300d\u3068\u8a00\u3046\u3002\u5fdc\u7528[\u7de8\u96c6]\u3053\u306e\u7dda\u7a4d\u5206\u306f\u7269\u7406\u5b66\u3067\u3088\u304f\u7528\u3044\u308b\u3002\u305f\u3068\u3048\u3070\u3001\u30d9\u30af\u30c8\u30eb\u5834 F \u3067\u8868\u3059\u529b\u5834\u306e\u5185\u5074\u3067\u66f2\u7dda C \u306b\u6cbf\u3063\u3066\u904b\u52d5\u3059\u308b\u7c92\u5b50\u306e\u6210\u3059\u4ed5\u4e8b\u3092 F \u306e C \u4e0a\u306e\u7dda\u7a4d\u5206\u3067\u8868\u3059\u3002W(t0;t1)=\u222bCF(r(t),t)\u22c5dr(t)=\u222bt0t1F(r(t),t)\u22c5drdt(t)dt.{displaystyle W(t_{0};t_{1})=int _{C}mathbf {F} (mathbf {r} (t),t)cdot dmathbf {r} (t)=int _{t_{0}}^{t_{1}}mathbf {F} (mathbf {r} (t),t)cdot {frac {dmathbf {r} }{dt}}!(t),dt.}\u8907\u7d20\u7dda\u7a4d\u5206[\u7de8\u96c6]\u7dda\u7a4d\u5206\u306f\u8907\u7d20\u89e3\u6790\u306b\u304a\u3051\u308b\u57fa\u672c\u7684\u306a\u9053\u5177\u3067\u3042\u308b\u3002U \u3092\u8907\u7d20\u6570\u5e73\u9762 C \u306e\u958b\u96c6\u5408\u3001\u03b3: [a, b] \u2192 U \u3092\u6709\u9650\u9577\u66f2\u7dda\u3068\u3059\u308b\u3068\u3001\u51fd\u6570 f: U \u2192 C \u306e\u7dda\u7a4d\u5206\u222b\u03b3f(z)dz{displaystyle int _{gamma }f(z),dz}\u306f\u3001\u533a\u9593 [a, b] \u306e a = t0 < t1 < \u22ef <  tn = b \u3078\u306e\u7d30\u5206\u3092\u8003\u3048\u3066\u5f97\u308b\u30ea\u30fc\u30de\u30f3\u548c\u22111\u2264k\u2264nf(\u03b3(tk))(\u03b3(tk)\u2212\u03b3(tk\u22121)){displaystyle sum _{1leq kleq n}f(gamma (t_{k}))(gamma (t_{k})-gamma (t_{k-1}))}\u306e\u3001\u5c0f\u533a\u9593\u306e\u5e45\u3092 0 \u306b\u8fd1\u3065\u3051\u308b\u6975\u9650\u3068\u3057\u3066\u5b9a\u7fa9\u3059\u308b\u3002\u03b3 \u304c\u9023\u7d9a\u7684\u5fae\u5206\u53ef\u80fd\u306a\u66f2\u7dda\u306a\u3089\u3070\u3001\u3053\u306e\u7dda\u7a4d\u5206\u306e\u5024\u306f\u5b9f\u5909\u6570\u51fd\u6570\u306e\u7a4d\u5206\u222b\u03b3f(z)dz=\u222babf(\u03b3(t))\u03b3\u2032(t)dt{displaystyle int _{gamma }f(z),dz=int _{a}^{b}f(gamma (t))gamma &#8216;(t),dt}\u3068\u3057\u3066\u8a55\u4fa1\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u5f27\u9577\u306b\u95a2\u3059\u308b\u7dda\u7a4d\u5206\u3082\u540c\u69d8\u306b\u222b\u03b3f(z)|dz|=\u222bf(\u03b3(t))|d\u03b3dt|dt{displaystyle int _{gamma }f(z)|dz|=int f(gamma (t))left|{frac {dgamma }{dt}}right|{mathit {dt}}}\u3068\u5b9a\u7fa9\u3067\u304d\u308b\u3002\u3053\u308c\u3089\u4e8c\u7a2e\u985e\u306e\u7dda\u7a4d\u5206\u306b\u3064\u3044\u3066\u3001\u7279\u306b|\u222b\u03b3f(z)dx|\u2264\u222b\u03b3|f(z)||dz|{displaystyle left|int _{gamma }f(z)dxright|leq int _{gamma }|f(z)||dz|}\u304c\u6210\u308a\u7acb\u3064\u3002\u8907\u7d20\u51fd\u6570\u306e\u7dda\u7a4d\u5206\u3092\u8a08\u7b97\u3059\u308b\u65b9\u6cd5\u306f\u3044\u308d\u3044\u308d\u3042\u308b\u3002\u4f8b\u3048\u3070\u3001\u8907\u7d20\u51fd\u6570\u3092\u5b9f\u90e8\u3068\u865a\u90e8\u306b\u5206\u3051\u3066\u8003\u3048\u308c\u3070\u30012 \u3064\u306e\u5b9f\u6570\u5024\u7dda\u7a4d\u5206\u3092\u8a08\u7b97\u3059\u308b\u554f\u984c\u306b\u5e30\u7740\u3067\u304d\u308b\u3002\u30b3\u30fc\u30b7\u30fc\u306e\u7a4d\u5206\u516c\u5f0f\u3092\u7528\u3044\u3066\u8a08\u7b97\u3059\u308b\u65b9\u6cd5\u3082\u3042\u308b\u3002\u5f8c\u8005\u306f\u8907\u7d20\u7dda\u7a4d\u5206\u306e\u88ab\u7a4d\u5206\u51fd\u6570\u304c\u3001\u305d\u306e\u7a4d\u5206\u8def\u3092\u542b\u3080\u9818\u57df\u5185\u3067\u89e3\u6790\u7684\u304b\u3064\u7279\u7570\u70b9\u3092\u542b\u307e\u306a\u3044\u306a\u3089\u3070\u3001\u305d\u306e\u7dda\u7a4d\u5206\u306e\u5024\u306f\u5358\u306b 0 \u306b\u306a\u308b\u3068\u3044\u3046\u30b3\u30fc\u30b7\u30fc\u306e\u7a4d\u5206\u5b9a\u7406\u304b\u3089\u306e\u5e30\u7d50\u3067\u3042\u308b\u3002\u7559\u6570\u5b9a\u7406\u306f\u30b3\u30fc\u30b7\u30fc\u306e\u7a4d\u5206\u5b9a\u7406\u306e\u4e00\u822c\u5316\u3067\u3042\u308b\u3002\u3053\u306e\u5b9a\u7406\u306f\u8907\u7d20\u5e73\u9762\u5185\u306e\u5468\u56de\u7a4d\u5206\u306b\u3088\u3063\u3066\u5b9f\u51fd\u6570\uff08\u5b9f\u5909\u6570\u5b9f\u6570\u5024\u51fd\u6570\uff09\u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u3001\u3057\u3070\u3057\u3070\u7528\u3044\u308b\u3002\u8907\u7d20\u7dda\u7a4d\u5206\u306e\u4f8b[\u7de8\u96c6]\u8907\u7d20\u51fd\u6570 f(z) = 1\/z \u3068\u9589\u8def C \u3068\u3057\u3066 0 \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u5358\u4f4d\u5186\u3092 1 \u304b\u3089\u53cd\u6642\u8a08\u56de\u308a\u306b\u4e00\u5468\u3059\u308b\u3082\u306e\u8003\u3048\u308b\u3002C \u306f eit (t \u2208 [0, 2\u03c0]) \u3068\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3067\u304d\u308b\u304b\u3089\u3001\u4ee3\u5165\u3057\u3066\u222eCf(z)dz=\u222b02\u03c01eitieitdt=i\u222b02\u03c0dt=2\u03c0i{displaystyle oint _{C}f(z),dz=int _{0}^{2pi }{1 over e^{it}}ie^{it},dt=iint _{0}^{2pi },dt=2pi i}\u3092\u5f97\u308b\u3002\u4e0a\u8a18\u306e\u7a4d\u5206\u306f\u30b3\u30fc\u30b7\u30fc\u306e\u7a4d\u5206\u516c\u5f0f\u3092\u7528\u3044\u3066\u3082\u540c\u3058\u8a08\u7b97\u7d50\u679c\u304c\u5f97\u3089\u308c\u308b\u3002\u8907\u7d20\u7dda\u7a4d\u5206\u3068\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7a4d\u5206\u3068\u306e\u95a2\u4fc2[\u7de8\u96c6]\u8907\u7d20\u5e73\u9762 C \u3092\u5b9f 2 \u6b21\u306e\u7a7a\u9593 R2 \u3068\u898b\u306a\u305b\u3070\u3001\u4e8c\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u5834\u306e\u7dda\u7a4d\u5206\u306f\u3001\u5bfe\u5fdc\u3059\u308b\u8907\u7d20\u51fd\u6570\u306e\u5171\u8edb\u306e\u7dda\u7a4d\u5206\u306e\u5b9f\u90e8\u306b\u5bfe\u5fdc\u3059\u308b\u3002\u3059\u306a\u308f\u3061\u3001x, y \u8ef8\u65b9\u5411\u306e\u5358\u4f4d\u30d9\u30af\u30c8\u30eb j, k \u3092\u7528\u3044\u3066\u3001r(t) = x(t)j + y(t)k \u304a\u3088\u3073 f(z) = u(z) + iv(z) \u3068\u7f6e\u304f\u3068\u222bCf(z)\u00afdz=\u222bC(u\u2212iv)dz=\u222bC(uj+vk)\u22c5dr\u2212i\u222bC(vj\u2212uk)\u22c5dr{displaystyle int _{C}{overline {f(z)}},dz=int _{C}(u-iv),dz=int _{C}(umathbf {j} +vmathbf {k} )cdot dmathbf {r} -iint _{C}(vmathbf {j} -umathbf {k} )cdot dmathbf {r} }\u306a\u308b\u95a2\u4fc2\u5f0f\u304c\u3001\u53f3\u8fba\u306e 2 \u3064\u306e\u7a4d\u5206\u304c\u3068\u3082\u306b\u5b58\u5728\u3059\u308b\u3053\u3068\u304b\u3089\u8a00\u3048\u308b\u3002\u305f\u3060\u3057 C \u306e\u5a92\u4ecb\u5909\u6570\u8868\u793a z(t) \u306f r(t) \u3068\u540c\u3058\u5411\u304d\u3092\u6301\u3064\u3088\u3046\u306b\u3068\u308b\u3002\u540c\u3058\u3053\u3068\u3060\u304c\u3001\u5fae\u5206\u5f62\u5f0f\u3068\u3057\u3066\u898b\u308c\u3070 f(z)dz \u306ff(z)dz=(u(x,y)dx\u2212v(x,y)dy)+i(v(x,y)dx+u(x,y)dy){displaystyle f(z),dz=(u(x,y)dx-v(x,y)dy)+i(v(x,y)dx+u(x,y)dy)}\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u3066\u3001\u3053\u308c\u3068\u5171\u8edb\u8907\u7d20\u7a4d\u5206[6]f(z)dz\u00af(=f(z)\u00afdz)=(u(x,y)dx+v(x,y)dy)+i(v(x,y)dx\u2212u(x,y)dy){displaystyle f(z),d{bar {z}}(={overline {f(z)}},dz)=(u(x,y)dx+v(x,y)dy)+i(v(x,y)dx-u(x,y)dy)}\u3092\u3042\u308f\u305b\u3066\u8003\u3048\u308c\u3070\u3001\u30d9\u30af\u30c8\u30eb\u5834\u3068\u3057\u3066\u306e\u7dda\u7a4d\u5206\u3068\u9762\u7a4d\u5206\u3092\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u8907\u7d20\u6b63\u5247\u51fd\u6570\u304c\u30b3\u30fc\u30b7\u30fc\uff1d\u30ea\u30fc\u30de\u30f3\u306e\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3059\u3053\u3068\u304b\u3089\u3001\u6b63\u5247\u51fd\u6570\u306e\u5171\u8edb\u306b\u5bfe\u5fdc\u3059\u308b\u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2\u306f 0 \u306b\u306a\u308b\u3002\u3053\u308c\u306f\u3069\u3061\u3089\u306e\u7a2e\u985e\u306e\u7dda\u7a4d\u5206\u3067\u3082\u305d\u308c\u304c 0 \u306b\u306a\u308b\u3068\u304d\u306e\u30b9\u30c8\u30fc\u30af\u30b9\u306e\u5b9a\u7406\u3068\u95a2\u9023\u304c\u3042\u308b\u3002\u3059\u306a\u308f\u3061\u3001\u30ac\u30a6\u30b9\uff1d\u30b0\u30ea\u30fc\u30f3\u306e\u5b9a\u7406\u3092\u9069\u7528\u3059\u308c\u3070\u8907\u7d20\u95a2\u6570\u306e\u9762\u7a4d\u5206\u306f\u3001\u305d\u306e\u9818\u57df\u306e\u5883\u754c\u4e0a\u306e\u7dda\u7a4d\u5206\u306b\u5e30\u7740\u3055\u308c\u308b\u305f\u3081\u3001\u8907\u7d20\u95a2\u6570\u306e\u7a4d\u5206\u3067\u306f\u7dda\u7a4d\u5206\u304c\u672c\u8cea\u7684\u3067\u3042\u308b\u3002\u7279\u306b\u6b63\u5247\u95a2\u6570 f \u306e\u5358\u7d14\u9589\u66f2\u7dda \u03b3 \u4e0a\u306e\u9589\u8def\u7a4d\u5206\u306b\u95a2\u3059\u308b\u30b3\u30fc\u30b7\u30fc\u306e\u5b9a\u7406\u222e\u03b3f(z)dz=0{displaystyle oint _{gamma }f(z)dz=0}\u306f\u3001\u03b3 \u3092\u5883\u754c \u2202D \u3068\u3059\u308b\u9818\u57df D \u3067\u306e\u30b0\u30ea\u30fc\u30f3\u306e\u5b9a\u7406\u306b\u30b3\u30fc\u30b7\u30fc\u30fb\u30ea\u30fc\u30de\u30f3\u306e\u95a2\u4fc2\u5f0f\u3092\u4ee3\u5165\u3059\u308b\u3053\u3068\u306b\u5bfe\u5fdc\u3059\u308b\u3002\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u6ce8\u91c8[\u7de8\u96c6]^ path integral\u306f\u91cf\u5b50\u529b\u5b66\u306e\u7d4c\u8def\u7a4d\u5206\u3092\u6307\u3059\u8a00\u8449\u3068\u3057\u3066\u5b9a\u7740\u3057\u3066\u3044\u308b\u3002\u7dda\u7a4d\u5206\u306e\u610f\u5473\u3067\u306f\u3042\u307e\u308a\u7528\u3044\u3089\u308c\u306a\u3044[\u8981\u51fa\u5178]\u3002\u51fa\u5178[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u9ad8\u6728\u8c9e\u6cbb \u300e\u89e3\u6790\u6982\u8ad6\u300f \u5ca9\u6ce2\u66f8\u5e97\u30011983\u5e74\u3002\u00a0\u9577\u6cbc\u4f38\u4e00\u90ce \u300e\u7269\u7406\u6570\u5b66\u306e\u76f4\u611f\u7684\u65b9\u6cd5\u300f \u8b1b\u8ac7\u793e\u3008\u30d6\u30eb\u30fc\u30d0\u30c3\u30af\u30b9\u3009\u30012011\u5e74\u3002ISBN\u00a0978-4062577380\u3002\u00a0\u6728\u6751\u4fca\u623f; \u9ad8\u91ce\u606d\u4e00 \u300e\u95a2\u6570\u8ad6\u300f 7\u5dfb \u671d\u5009\u66f8\u5e97\u3008\u65b0\u6570\u5b66\u8b1b\u5ea7\u3009\u30011991\u5e74\u3002\u00a0\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki7\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki7\/archives\/9363#breadcrumbitem","name":"\u7dda\u7a4d\u5206 &#8211; Wikipedia"}}]}]