6. Green's Theorem | 格林定理

📚 6. Green’s Theorem | 格林定理

Green’s Theorem is a fundamental result in vector calculus that reveals a deep relationship between a line integral around a simple closed curve and a double integral over the plane region bounded by that curve. Named after the English mathematician George Green, this theorem serves as a two‑dimensional analogue of the Fundamental Theorem of Calculus and forms the cornerstone for more advanced theorems such as Stokes’ Theorem and the Divergence Theorem.

格林定理是矢量微积分中的一个基本结果，它揭示了绕简单闭合曲线的线积分与该曲线所围平面区域上的二重积分之间的深刻联系。该定理以英国数学家乔治·格林的名字命名，是微积分基本定理在二维空间的对应物，也为斯托克斯定理和高斯散度定理等更高级的定理奠定了基础。

1. Statement of Green’s Theorem | 格林定理的陈述

Let C be a positively oriented, piecewise‑smooth, simple closed curve in the plane, and let D be the region bounded by C. If P(x, y) and Q(x, y) have continuous first partial derivatives on an open region that contains D, then Green’s Theorem states:

设 C 为平面中一条正向分段光滑的简单闭合曲线，D 是由 C 所围成的区域。若 P(x, y) 和 Q(x, y) 在包含 D 的某个开区域上有一阶连续偏导数，则格林定理表述为：

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

In the single‑variable Fundamental Theorem of Calculus, the integral of a derivative over an interval equals the difference of the antiderivative at the endpoints. Green’s Theorem generalises this idea: a double integral of a “derivative” (the curl in 2D) over a region equals the line integral of the original vector field along the boundary.

在单变量微积分基本定理中，导数的区间积分等于原函数在端点的差值。格林定理推广了这一思想：区域内某“导数”（二维旋度）的二重积分等于原矢量场沿边界的线积分。

2. Orientation of the Curve | 曲线的定向

Positive orientation means the curve C is traversed counter‑clockwise. As you walk along C in this direction, the region D always stays on your left. If no orientation is specified, Green’s Theorem assumes positive orientation. For a clockwise traversal, the theorem introduces a minus sign:

正向是指曲线 C 沿逆时针方向行进。你沿该方向行走时，区域 D 始终在你的左侧。若无特别说明，格林定理均默认采用正向。对于顺时针方向，定理会多出一个负号：

∮_–C (P dx + Q dy) = –∮_C (P dx + Q dy)

This convention is essential when calculating work or circulation, because reversing the path flips the sign of the line integral. In IB exam problems, always check the given orientation; if the curve is not positively oriented, apply the negative sign before using Green’s Theorem.

在计算功或环量时，这一约定至关重要，因为反转路径会改变线积分的符号。在IB考试题目中，务必检查所给出的定向；若曲线不是正向，须在使用格林定理前添加负号。

3. Simple, Closed, and Piecewise‑Smooth Curves | 简单、闭合与分段光滑曲线

Green’s Theorem requires C to be a simple closed curve – it does not intersect itself, and it completely encloses a region. Moreover, the curve must be piecewise‑smooth, meaning it can be divided into a finite number of smooth segments (each with a continuously turning tangent). Common shapes like circles, ellipses, squares, and triangles all satisfy these requirements.

格林定理要求 C 是一条简单闭合曲线——它不自交，并且完全包围一个区域。此外，曲线必须分段光滑，即它可以拆分为有限个光滑弧段（每段有连续转动的切线）。圆、椭圆、正方形、三角形等常见形状都满足这些条件。

If the curve has a “hole” (the region D is not simply connected), Green’s Theorem still applies if the region is treated as a union of simply connected sub‑regions or if the inner boundaries are oriented appropriately. This extension can handle annuli and other multiply connected domains, though the IB syllabus typically sticks to simply connected regions.

若曲线带“洞”（区域 D 非单连通），只要将区域视为若干单连通子区域的并集，或为内边界选定适当定向，格林定理依然适用。这一推广可处理圆环等多连通区域，但IB大纲通常仅涉及单连通区域。

4. Proof Outline | 证明概要

Green’s Theorem can be proved by decomposing a general region into elementary ones (Type I and Type II regions) where the double integral can be evaluated as an iterated integral. For a Type I region given by a ≤ x ≤ b, g₁(x) ≤ y ≤ g₂(x), one shows that ∮_C P dx = –∬_D ∂P/∂y dA by direct computation of the line integral along the four boundary pieces. Similarly, for a Type II region, ∮_C Q dy = ∬_D ∂Q/∂x dA. Adding these two equalities yields the full theorem.

格林定理的证明思路是将一般区域分解为基本区域（I型和II型区域），在这些区域上二重积分可化为累次积分。对于由 a ≤ x ≤ b，g₁(x) ≤ y ≤ g₂(x) 给出的I型区域，可沿四条边界直接计算线积分，验证 ∮_C P dx = –∬_D ∂P/∂y dA。类似地，对II型区域有 ∮_C Q dy = ∬_D ∂Q/∂x dA。两式相加便得完整定理。

The key step uses the Fundamental Theorem of Calculus to link the difference of values of P and Q on the upper and lower boundaries to the partial derivatives inside the region. This proof is a standard topic in IB HL Analysis & Approaches and Further Mathematics.

关键步骤是利用微积分基本定理，将 P 和 Q 在上、下边界上的值之差与区域内的偏导数联系起来。该证明是IB HL分析与方法和进阶数学中的标准内容。

5. Special Case: Area Calculation | 特殊情况：面积计算

One elegant consequence of Green’s Theorem is a line‑integral formula for the area of a plane region. By choosing particular functions P and Q such that (∂Q/∂x – ∂P/∂y) = 1, we obtain:

格林定理的一个优美推论是用线积分计算平面区域面积的公式。选取特定的 P 和 Q 使 (∂Q/∂x – ∂P/∂y) = 1，可得：

A = ∬_D dA = ½ ∮_C (x dy – y dx)

Other valid choices include P = 0, Q = x (giving A = ∮_C x dy) or P = –y, Q = 0 (A = –∮_C y dx). These formulas are particularly useful when the boundary C is described parametrically, such as an ellipse or a parametric closed loop. In IB exams, you may be asked to derive these area formulas or to apply them to find the area of an ellipse.

其他可行选择有 P = 0, Q = x（得 A = ∮_C x dy）或 P = –y, Q = 0（A = –∮_C y dx）。当边界 C 以参数形式给出（如椭圆或参数闭合环路）时，这些公式尤为实用。IB考试中可能要求推导这些面积公式，或将其用于求椭圆面积。

6. Worked Example 1: From Line Integral to Double Integral | 例1：线积分转化为二重积分

Evaluate the line integral ∮_C (xy² dx + x² dy) where C is the positively oriented square with vertices (0,0), (2,0), (2,2), (0,2).

计算线积分 ∮_C (xy² dx + x² dy)，其中 C 为以 (0,0), (2,0), (2,2), (0,2) 为顶点的正向正方形。

Here P = xy² and Q = x². Their partial derivatives are:

此处 P = xy², Q = x²。它们的偏导数为：

∂Q/∂x = 2x, ∂P/∂y = 2xy

Applying Green’s Theorem:

应用格林定理：

∮_C (xy² dx + x² dy) = ∬_D (2x – 2xy) dA

The region D is the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 2. The double integral becomes:

区域 D 为 0 ≤ x ≤ 2, 0 ≤ y ≤ 2 的正方形。二重积分为：

∫₀² ∫₀² (2x – 2xy) dy dx = ∫₀² [2xy – xy²]₀² dx = ∫₀² (4x – 4x) dx = 0

Thus, the line integral evaluates to 0. A direct parameterisation of the four line segments would have been much lengthier; Green’s Theorem instantly reveals the cancellation.

因此该线积分的值为0。若直接对四段线段进行参数化，计算量将大得多；格林定理立刻揭示出抵消关系。

7. Worked Example 2: Area of an Ellipse | 例2：椭圆的面积

Use the area formula A = ½ ∮_C (x dy – y dx) to find the area enclosed by an ellipse with parametric equations x = a cos t, y = b sin t, 0 ≤ t ≤ 2π.

利用面积公式 A = ½ ∮_C (x dy – y dx) 求由参数方程 x = a cos t, y = b sin t, 0 ≤ t ≤ 2π 给出的椭圆面积。

Compute dx = –a sin t dt, dy = b cos t dt. Then:

计算 dx = –a sin t dt, dy = b cos t dt。于是：

x dy – y dx = (a cos t)(b cos t dt) – (b sin t)(–a sin t dt) = ab (cos² t + sin² t) dt = ab dt

Integrating from 0 to 2π yields:

对 t 从 0 到 2π 积分：

A = ½ ∫₀²π ab dt = ½ ab (2π) = πab

This matches the well‑known formula. Green’s Theorem thus provides an alternative derivation that elegantly avoids the double integral over an elliptical region.

这正与熟知的公式一致。格林定理由此提供了一种无需在椭圆区域上进行二重积分的优雅推导。

8. Circulation and Flux Forms | 环量与通量形式

Green’s Theorem can be expressed in two distinct physical contexts. The circulation (or tangential) form is the original statement:

格林定理可用两种不同的物理解释来表述。环量（切向）形式就是原始陈述：

∮_C F · T ds = ∮_C (P dx + Q dy) = ∬_D (∂Q/∂x – ∂P/∂y) dA

This calculates the circulation of a vector field F = P i + Q j around C. The flux (normal) form uses an outward‑pointing normal vector and states:

这计算了矢量场 F = P i + Q j 绕 C 的环量。通量（法向）形式则采用外向法向量，表述为：

∮_C F · n ds = ∬_D (∂P/∂x + ∂Q/∂y) dA

In an IB context, the circulation form is the standard one taught. Understanding the flux form provides deeper insight into the Divergence Theorem in 3D and helps with interpreting conservation laws in physics.

IB课程中，环量形式是标准教学内容。理解通量形式能深入洞察三维散度定理，并有助于解释物理学中的守恒定律。

9. Applications in Physics and Engineering | 物理与工程应用

Green’s Theorem has numerous practical applications. In fluid dynamics, the circulation form quantifies the “swirl” of a fluid around a closed loop; if the integrand ∂Q/∂x – ∂P/∂y (the curl in 2D) vanishes throughout, the flow is irrotational. In electrostatics, the theorem relates the electric field along a closed path to the charge enclosed.

格林定理有众多实际应用。在流体动力学中，环量形式量化了流体绕闭合环路的“旋转”程度；若被积函数 ∂Q/∂x – ∂P/∂y（二维旋度）处处为零，则流动是无旋的。在静电学中，该定理将沿闭合路径的电场与所包围的电荷联系起来。

Engineers use Green’s Theorem to compute areas of irregular polygons (via the surveyor’s formula), to determine the centre of mass, and to evaluate moments of inertia through boundary integrals rather than area integrals. These computations become extremely efficient when the boundary is made up of simple line segments.

工程师利用格林定理计算不规则多边形的面积（通过测量员公式）、确定质心，并通过边界积分而非面积积分来评估转动惯量。当边界由简单线段组成时，这些计算会变得极为高效。

10. Green’s Theorem and Stokes’ Theorem | 格林定理与斯托克斯定理的关系

Green’s Theorem is a planar special case of the more general Stokes’ Theorem, which applies to surfaces in three‑dimensional space. If a surface lies flat in the xy‑plane with unit normal k, Stokes’ Theorem reduces exactly to Green’s Theorem:

格林定理是更一般的斯托克斯定理在平面上的特例，后者适用于三维空间中的曲面。若曲面平躺于 xy 平面上，单位法向量为 k，则斯托克斯定理就退化为格林定理：

∮_C F · dr = ∬_D (∇ × F) · k dA

Recognising this connection helps students transition seamlessly from two‑dimensional line integrals to surface integrals in three dimensions—a theme often explored in IB HL Options and university‑level calculus.

认识这一联系有助于学生从二维线积分顺利过渡到三维曲面积分，这是IB HL选修课程和大学微积分中常见的内容。

11. Common Pitfalls and Tips | 常见错误与技巧

A frequent mistake is forgetting to verify that the curve is closed and that the vector field components are defined and have continuous partial derivatives over the region. If the curve has a discontinuity or a hole where the derivatives fail, Green’s Theorem cannot be applied directly without excluding the problematic points.

一个常见错误是忘记验证曲线是否闭合，以及矢量场分量是否在区域上有定义且具有连续偏导数。若曲线有间断点或存在使导数失效的“洞”，则不能直接应用格林定理，除非将问题点排除。

Another pitfall is misidentifying the orientation. If the curve is traversed clockwise, the integral equals the negative of the double integral. In IB problems, always write down the oriented integral first, then assign the correct sign. Finally, when computing the double integral, remember to integrate with respect to the correct limits—it is easy to mix up x and y bounds on non‑rectangular regions.

另一个易错点是误判方向。若曲线是顺时针方向的，则积分等于二重积分的相反数。在IB题目中，务必先写出含定向的积分，再赋予正确符号。最后，计算二重积分时，要记住在非矩形区域上正确使用积分限——很容易混淆 x 与 y 的范围。

12. Summary and Key Takeaways | 总结与要点

Green’s Theorem equates a line integral around a simple closed curve to a double integral over the interior: ∮_C P dx + Q dy = ∬_D (∂Q/∂x – ∂P/∂y) dA.
格林定理 将简单闭合曲线的线积分与内部区域的二重积分相等同：∮_C P dx + Q dy = ∬_D (∂Q/∂x – ∂P/∂y) dA。
It is valid for positively oriented, piecewise‑smooth, simple closed curves and continuously differentiable P and Q.
该定理适用于正向分段光滑简单闭合曲线，且要求 P 和 Q 连续可微。
A powerful special case yields area: A = ½ ∮_C (x dy – y dx).
一个强大的特例用于求面积：A = ½ ∮_C (x dy – y dx)。
Always check orientation, closedness, and the region’s connectivity before applying the theorem.
在应用定理前，一定要检查曲线定向、闭合性和区域的连通性。
The proof relies on the Fundamental Theorem of Calculus and decomposition into Type I and II regions.
其证明依赖于微积分基本定理以及分解为I型和II型区域。
Green’s Theorem is the 2D precursor to Stokes’ Theorem and the Divergence Theorem, providing a bridge to advanced vector calculus.
格林定理是斯托克斯定理和高斯散度定理的二维前身，搭建了通往高等矢量微积分的桥梁。

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6. Green’s Theorem | 格林定理