Green’s Theorem and Its Applications | 格林定理及其应用

📚 Green’s Theorem and Its Applications | 格林定理及其应用

Green’s theorem is a powerful bridge between line integrals and double integrals. It allows us to compute a complicated boundary integral by a simpler area integral, or vice versa. This makes it an essential tool in physics, engineering, and advanced calculus.

格林定理是连接线积分与二重积分的一座强大桥梁。它让我们可以通过更简单的面积分来计算复杂的边界积分，反之亦然。因此，它是物理、工程和高等微积分中的重要工具。

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1. The Statement of Green’s Theorem | 格林定理的表述

Let C be a positively oriented, piecewise smooth, simple closed curve in the xy-plane, and let D be the region enclosed by C. If the vector field F(x,y) = (P(x,y), Q(x,y)) has continuous first partial derivatives on an open region that contains D, then

设C为xy平面中一条正向定向、分段光滑的简单闭曲线，D为由C围成的区域。若向量场F(x,y)=(P(x,y), Q(x,y))在包含D的开区域上有连续一阶偏导数，则

∮_C (P dx + Q dy) = ∬_D (∂Q/∂x − ∂P/∂y) dA

The left side is the line integral of F along C, also written ∮_C F·dr. The right side is the double integral of the scalar curl of F over D.

左边是F沿C的线积分，也写作∮_C F·dr；右边是F的标量旋度在D上的二重积分。

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2. Orientation of the Curve | 曲线定向

A simple closed curve is said to be positively oriented when the region D is always on the left as one traverses the curve. For a standard region, this means counterclockwise traversal.

简单闭曲线称为正向定向，是指当人沿曲线行进时区域D始终在左侧。对常规区域而言，这意味着逆时针方向。

If C is traversed in the opposite direction, both the line integral and the double integral in Green’s theorem change sign. Therefore, it is crucial to check orientation before applying the theorem.

若C沿相反方向行进，则格林定理中的线积分与二重积分都会改变符号。因此在应用定理前，必须检查定向。

For example, if the curve is traversed clockwise, then ∮_C (P dx + Q dy) becomes −∬_D (∂Q/∂x − ∂P/∂y) dA.

例如，若曲线按顺时针方向行进，则∮_C (P dx + Q dy)将等于−∬_D (∂Q/∂x − ∂P/∂y) dA。

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3. The Idea Behind the Proof | 证明背后的思想

To see why Green’s theorem is true, first consider a simple region that is both type I and type II. The double integral can be split into two iterated integrals, and each inner integral is evaluated using the Fundamental Theorem of Calculus.

为理解格林定理为何成立，先考虑一个既是I型又是II型的简单区域。二重积分可拆成两个累次积分，每一个内层积分都可用微积分基本定理求值。

For example, ∬_D ∂Q/∂x dA = ∫_a^b [Q(x, g₂(x)) − Q(x, g₁(x))] dx, which exactly matches the contribution from the right and left sides of the boundary. Similar cancellation happens for ∂P/∂y, and the sum of boundary terms reconstructs the full line integral.

例如，∬_D ∂Q/∂x dA = ∫_a^b [Q(x, g₂(x)) − Q(x, g₁(x))] dx，正好对应边界左右两侧的贡献；对∂P/∂y也有类似抵消，各边界项之和重新构成完整的线积分。

The key idea is that internal boundaries cancel out when we cut a large region into small simpler pieces. Only the outer boundary remains.

核心思想是：将一个大的区域切割成若干简单小块时，内部边界会两两抵消，最终只剩下外边界。

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4. Conditions of Validity | 成立条件

Green’s theorem requires C to be a simple closed curve with no self-intersections and only finitely many corners. The functions P and Q must have continuous first partial derivatives on an open region containing D.

格林定理要求C是简单闭曲线，即没有自交点，且只含有限多个角点；函数P与Q在包含D的开区域上需要有连续一阶偏导数。

If D has holes, the theorem still applies if we orient all inner boundaries in the opposite direction to the outer boundary. In this way, the region D is always on the left as each boundary is traversed.

若D有空洞，只要将所有内边界与外边界取相反方向，定理仍然适用。这样，沿每条边界行进时，D始终位于左侧。

These conditions ensure there are no singularities inside D. If a singular point exists, one must cut out a small circle around it and apply Green’s theorem to the resulting region.

这些条件确保D内没有奇点。若存在奇异点，需要在该点周围挖去一个小圆，再对剩余区域应用格林定理。

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5. Application 1: Computing Work Done in a Force Field | 应用一：力场做功

In mechanics, the work done by a force field F along a closed path C is W = ∮_C F·dr = ∮_C (P dx + Q dy). Green’s theorem converts this to a double integral of the scalar curl ∂Q/∂x − ∂P/∂y.

在力学中，力场F沿闭路径C所做的功为W = ∮_C F·dr = ∮_C (P dx + Q dy)。格林定理将其转化为标量旋度∂Q/∂x − ∂P/∂y的二重积分。

If the vector field is conservative, this curl is zero, so the work around any closed loop is zero. This is a quick way to test whether a field is conservative on a simply connected region.

若向量场是保守场，该旋度为零，因此沿任意闭合回路的功为零。这是在单连通区域上检验保守场的快捷方法。

For example, if F = (y, x), then ∂Q/∂x − ∂P/∂y = 1 − 1 = 0, so ∮_C F·dr = 0 for every simple closed curve C.

例如，若F = (y, x)，则∂Q/∂x − ∂P/∂y = 1 − 1 = 0，因此对任意简单闭曲线C都有∮_CPublished by TutorHao | IB Mathematics Revision Series | aleveler.com

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