Name: green’s theorem: curl and boundary circulation
Uploaded: 2026-06-08T09:38:46.205Z
Duration: 1 min 27 s
Channel: JoVE

考虑水流在池塘表面流动的情况。目标是将池塘边界周围的环流与整个表面的局部旋转总量联系起来。在每一点上，流动都有其方向和速度，以矢量场的形式表示。该场包含两个分量：P 代表水平分量，Q 代表垂直分量。格林定理将边界上的环量与区域内部的局部旋度联系起来。环量用于衡量流体沿边界的流动强度，而局部旋度则描述某一点上的旋转情况。在池塘内部，这些微小的旋度相互叠加。相邻区域共享的边界被沿相反方向经过，因此它们的贡献相互抵消。最终只剩下外边界起作用。数学上，这意味着沿闭合边界曲线的线积分等于对所围区域上的二重积分，其中使用了 P 和 Q 的偏导数。应用该定理时，区域必须是单连通的，且边界曲线必须是闭合、光滑并按逆时针方向定向的。格林定理将复杂的基于路径的问题简化为更易于处理的基于面积的问题。

格林定理

Green’s Theorem

格林定理建立了围绕平面闭合曲线的线积分与该曲线所围区域上的二重积分之间的关系。该定理适用于向量场 F(x, y) = 〈P(x, y), Q(x, y)〉，其中 P 和 Q 在包含该区域的开集上具有连续的一阶偏导数。设 C 为一条正向的、简单的、闭合的、分段光滑的曲线，R 为由 C 所围成的平面区域。格林定理指出：\begin{equation*}\oint_C P\,dx+Q\,dy =\iint_R \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA\end{equation*}左侧是沿边界 C 的线积分，而右侧是区域 R 上的二重积分。表达\begin{equation*}\jfrac{\partial Q}{\partial x}-\jfrac{\partial P}{\partial y}\end{equation*}是向量场的二维标量旋度，用于度量向量场在 R 内每一点处的无穷小旋转行为。格林定理指出，边界上的总环流量等于旋度在整个区域上的积分。正向是指沿曲线 C 逆时针方向行进，使得区域 R 始终位于运动方向的左侧。该定理要求 C 为闭合且分段光滑的曲线，R 必须是一个合适的平面区域。P 和 Q 两个函数必须在包含 R 的某个区域内具有连续的偏导数。在这些条件下，格林公式可将边界积分转化为等价的面积分。

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Vector Calculus and Theorems

向量微积分与定理

KEY TERMS AND CONCEPTS

关键术语与概念

green’s theorem: curl and boundary circulation

Green’s Theorem links a line integral around a closed curve to a double integral over the region inside it. It is used with a vector field F(x, y) = ⟨P(x, y), Q(x, y)⟩, where P and Q have continuous first partial derivatives on an open set that contains the region.

The curve C must be simple, closed, piecewise smooth, and positively oriented. Positive orientation means the curve is traced counterclockwise, so the region R stays on the left side as you move along the boundary. Green’s Theorem then states that the line integral around C equals the double integral over R.

The formula is ∮C P dx + Q dy = ∬R (∂Q/∂x − ∂P/∂y) dA. The term ∂Q/∂x − ∂P/∂y is the two-dimensional scalar curl of the field. It describes the local rotational behavior at each point in the region.

In this way, Green’s Theorem says that the total circulation along the boundary matches the integral of the curl across the whole region. The theorem applies only when the curve and region meet the stated conditions. Under those assumptions, it turns a boundary integral into an equivalent area integral.

Green’s Theorem: Curl and Boundary Circulation

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green’s theorem

Green’s Theorem links a line integral around a closed curve to a double integral over the region inside it. It is used with a vector field F(x, y) = ⟨P(x, y), Q(x, y)⟩, where P and Q have continuous first partial derivatives on an open set that contains the region. The curve C must be simple, closed, piecewise smooth, and positively oriented. Positive orientation means the curve is traced counterclockwise, so the region R stays on the left side as you move along the boundary. Green’s Theorem...

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key terms and concepts

Learn how Green’s Theorem connects a counterclockwise boundary line integral to a double integral of 2D curl over the enclosed region.

Green’s Theorem: Curl and Boundary Circulation | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Green’s Theorem: Curl and Boundary Circulation

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