重積分

04-20

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預備知識　定積分

面積分

面積分 也叫二重積分，可以看做一元函數定積分的一種拓展．從幾何上來理解，如果後者是計算一定區間內被積函數曲線與坐標軸之間的面積，那麼前者就是計算一定二維區域內被積函數曲面與坐標軸平面之間的體積．

例1　曲面下的體積

　　二元函數 $f(x,y) = x^2 + y^2$ ，可以在直角坐標系 $xyz$ 代表一個拋物面．現在我們指定一個二維區域，是由 $y_1(x) = -x,y_2(x) = x$ 以及 $x=1$ 所圍成的三角形．現在我們以這個三角形為底取一個無限高的三稜柱，求三稜柱被 $f(x,y)$ 截去的有限部分的體積 $V$ ．

　　與定積分的思想一樣，我們可以把三角形區域劃分成許多更小的區域，每一個區域都對應一個被曲面截取的小柱體，由於我們使用直角坐標系，我們不妨沿 $x$ 和 $y$ 方向劃分出許多小矩形，用 $(i,j)$ 給它們編號，每個矩形的面積分別為 $Delta x_i Delta y_j$ ，小柱體的體積近似用底乘以高計算得 $Delta V_{ij} = f(x_i,y_i) Delta x_i Delta y_j$ ，其中 xi,yixi,yi 分別為區間 $x_i, y_i$ 內任意一點．所以總體積就可以用極限表示為

$\egin{equation} V = lim_{substack{Delta x_i o 0\ Delta y_j o 0}} sum_{i, j} f(x_i,y_j) Delta x_i Delta y_j end{equation}$ (1)

這是一個雙重極限，類比定積分的定義，用面積分簡寫上式記為

$ewcommand{I}{mathrm{i}} ewcommand{E}{mathrm{e}} enewcommand{vec}[1]{oldsymbol{mathbf{#1}}} ewcommand{mat}[1]{oldsymbol{mathbf{#1}}} ewcommand{ en}[1]{oldsymbol{mathbf{#1}}} ewcommand{Nabla}{vec abla} enewcommand{Tr}{^{mathrm{T}}} ewcommand{uvec}[1]{hat{oldsymbol{mathbf{#1}}}} enewcommand{pmat}[1]{egin{pmatrix}#1end{pmatrix}} ewcommand{ali}[1]{egin{aligned}#1end{aligned}} ewcommand{leftgroup}[1]{left{egin{aligned}#1end{aligned} ight.} ewcommand{vmat}[1]{egin{vmatrix}#1end{vmatrix}} ewcommand{Cj}{^*} ewcommand{Her}{^dagger} ewcommand{Q}[1]{hat #1} ewcommand{sinc}{mathop{mathrm{sinc}}} ewcommand{Si}[1]{,mathrm{#1}} ewcommand{les}{leqslant} ewcommand{ges}{geqslant} ewcommand{qty}[1]{left{{#1} ight}} ewcommand{qtyRound}[1]{left({#1} ight)} ewcommand{qtySquare}[1]{left[{#1} ight]} ewcommand{abs}[1]{leftlvert{#1} ight vert} ewcommand{eval}[1]{left.{#1} ight vert} ewcommand{comm}[2]{left[{#1},{#2} ight]} ewcommand{commStar}[2]{[{#1},{#2}]} ewcommand{pb}[2]{left{{#1},{#2} ight}} ewcommand{pbStar}[2]{{{#1},{#2}}} ewcommand{vdot}{oldsymbolcdot} ewcommand{cross}{oldsymbol imes} ewcommand{Nabla}{vec abla} ewcommand{grad}{oldsymbol abla} ewcommand{div}{vec ablaoldsymbolcdot} ewcommand{curl}{vec ablaoldsymbol imes} ewcommand{laplacian}{ abla^2} ewcommand{sinRound}[2][{}]{sin^{#1}left(#2 ight)} ewcommand{cosRound}[2][{}]{cos^{#1}left(#2 ight)} ewcommand{ anRound}[2][{}]{ an^{#1}left(#2 ight)} ewcommand{cscRound}[2][{}]{csc^{#1}left(#2 ight)} ewcommand{secRound}[2][{}]{sec^{#1}left(#2 ight)} ewcommand{cotRound}[2][{}]{cot^{#1}left(#2 ight)} ewcommand{sinhRound}[2][{}]{sinh^{#1}left(#2 ight)} ewcommand{coshRound}[2][{}]{cosh^{#1}left(#2 ight)} ewcommand{ anhRound}[2][{}]{ anh^{#1}left(#2 ight)} ewcommand{arcsinRound}[2][{}]{arcsin^{#1}left(#2 ight)} ewcommand{arccosRound}[2][{}]{arccos^{#1}left(#2 ight)} ewcommand{arctanRound}[2][{}]{arctan^{#1}left(#2 ight)} ewcommand{expRound}[1]{expleft(#1 ight)} ewcommand{logRound}[2][{}]{log^{#1}left(#2 ight)} ewcommand{lnRound}[2][{}]{ln^{#1}left(#2 ight)} enewcommand{Re}{mathrm{Re}} enewcommand{Im}{mathrm{Im}} ewcommand{dd}[1][]{,mathrm{d}^{#1}} ewcommand{dv}[2][{}]{frac{mathrm{d}^{#1}}{mathrm{d}{#2}^{#1}}} ewcommand{dvStar}[2][{}]{mathrm{d}^{#1}/mathrm{d}{#2}^{#1}} ewcommand{dvTwo}[3][{}]{frac{mathrm{d}^{#1}{#2}}{mathrm{d}{#3}^{#1}}} ewcommand{dvStarTwo}[3][{}]{mathrm{d}^{#1}{#2}/mathrm{d}{#3}^{#1}} ewcommand{pdv}[2][{}]{frac{partial^{#1}}{partial{#2}^{#1}}} ewcommand{pdvStar}[2][{}]{partial^{#1}/partial{#2}^{#1}} ewcommand{pdvTwo}[3][{}]{frac{partial^{#1}{#2}}{partial{#3}^{#1}}} ewcommand{pdvStarTwo}[3][{}]{partial^{#1}{#2}/partial{#3}^{#1}} ewcommand{pdvThree}[3]{frac{partial^2{#1}}{partial{#2}partial{#3}}} ewcommand{pdvStarThree}[3]{partial^2{#1}/partial{#2}partial{#3}} ewcommand{ra}[1]{leftlangle{#1} ight vert} ewcommand{raStar}[1]{langle{#1} vert} ewcommand{ket}[1]{leftlvert{#1} ight angle} ewcommand{ketStar}[1]{lvert{#1} angle} ewcommand{raket}[2]{leftlangle{#1}middle|{#2} ight angle} ewcommand{raketStar}[2]{langle{#1}|{#2} angle} ewcommand{ev}[1]{leftlangle{#1} ight angle} ewcommand{evStar}[1]{langle{#1} angle} ewcommand{evTwo}[2]{leftlangle{#2}middle|{#1}middle|{#2} ight angle} ewcommand{evStarTwo}[2]{langle{#2}|{#1}|{#2} angle} ewcommand{mel}[3]{leftlangle{#1}middle|{#2}middle|{#3} ight angle} ewcommand{melStar}[3]{langle{#1}|{#2}|{#3} angle} ewcommand{order}[1]{mathcal{O}left(#1 ight)} \ egin{equation} V = iint_{mathcal{S}} f(x,y) dd{x}dd{y} end{equation}$ (2)

其中 $S$ 代表以上定義的三角形積分區域．

（剩下部分見頂部的「閱讀原文」）