Green's Theorem is a fundamental result in vector calculus that establishes a powerful connection between line integrals around closed curves and double integrals over regions. This theorem allows us to convert between these two types of integrals, often simplifying complex calculations in physics and engineering.
The formal statement of Green's Theorem is as follows: If C is a positively oriented, piecewise smooth, simple closed curve in a plane, and D is the region bounded by C, then the line integral around C equals the double integral over D. The left side represents a line integral of a vector field along the boundary, while the right side represents a double integral of the curl over the entire region.
格林公式是向量微积分中的一个重要定理，由英国数学家乔治·格林在1828年提出。它建立了平面上二重积分与曲线积分之间的联系，为我们提供了一个强有力的工具来计算复杂的积分。
格林公式的数学表述如下：沿正向封闭简单曲线C的曲线积分，等于由C围成区域D上的二重积分。其中P和Q是定义在D上且具有连续偏导数的函数。正向是指逆时针方向。
格林公式从几何角度理解，左边的曲线积分表示向量场沿边界曲线的环流，右边的二重积分表示区域内向量场的旋度。图中绿色箭头表示向量场，红色箭头表示沿边界的正向（逆时针）方向。
格林公式有广泛的应用。首先，它可以用来计算由封闭曲线围成的区域面积。其次，它能够简化复杂的曲线积分计算。第三，可以用来验证向量场是否为保守场。最后，在物理学中用于计算流体的环流和通量。
格林公式具有重要的理论意义和实用价值。它是向量微积分的基石之一，巧妙地连接了微分与积分的概念。格林公式也为后续学习更高维的斯托克斯公式和散度定理奠定了坚实基础，在数学物理、工程技术等众多领域都有广泛应用。
Let's see a practical example of Green's Theorem. We can use it to compute the area enclosed by a simple closed curve. The area formula becomes A equals one half times the line integral of x dy minus y dx around the curve C. Here we show an ellipse being traced out, and we can calculate its area using this formula, which gives us pi times the product of the semi-axes.
Green's Theorem has profound significance in mathematics and physics. It provides a powerful tool for area calculations, simplifies complex line integrals, and helps test whether vector fields are conservative. In fluid dynamics, it relates circulation around a boundary to the curl within the region. Most importantly, it serves as the foundation for higher-dimensional generalizations like Stokes' Theorem, bridging local differential properties with global integral properties.