Vector Calculus: An Introduction

If $f$ is a continuously differentiable function of one variable, defined on a closed bounded interval $[a,b]\text{,}$ then the \textit{fundamental theorem of calculus} asserts that

The above formula expresses the integral of the derivative $f^{\prime }$ over an interval by means of the values of $f$ at the boundary points $a$ and $b$ of the interval $[a,b]\text{.}$

Green’s formula provides a generalisation of this fact to functions of two variables. For a function of two variables, the domain is a region, $D\text{,}$ in the plane ${\mathbb{R}}^2\text{,}$ and the boundary, $\partial D\text{,}$ of $D$ is a curve. The formula will apply to vector fields $\mathit{f}$ and have the following form.

The interval is replaced by the domain $D\text{,}$ and the boundary points $a$ and $b$ by the rim (or boundary) $\partial D$ of $D\text{.}$ In two dimensions the degree of freedom is much higher than in one dimension, and therefore we need more “structure” for a formula like the above to be true. Not surprisingly, the key ingredient in the proof of Green’s Theorem will be the fundamental theorem of calculus.