Theory

Introduction

The following conditions for a vector field $\mathbf{F}(x,y)$ are equivalent:

• $\int \mathbf{F}\cdot d\mathbf{s}$ is path independent,
• $\nabla {\times }_z\mathbf{F}=0$ in the region (which is simply connected, i.e. no holes),
• $\mathbf{F}=\nabla f$ for some scalar field $f(x,y)$.

When these conditions hold, $\mathbf{F}$ is called conservative. Then ${\int }_{C_1}\mathbf{F}\cdot d\mathbf{s}={\int }_{C_2}\mathbf{F}\cdot d\mathbf{s}$ for two paths $C_1$ and $C_2$ from $\mathbf{p}$ to $\mathbf{q}$, and $\mathbf{F}=\nabla f$ for a scalar field $f(x,y)$.

(Note: when the domain is not simply connected, then the curl condition is not strong enough, but the first and last conditions are still equivalent to each other and the word ‘conservative’ applies when these two conditions hold.)

The Fundamental Theorem of Line Integrals gives the value that is common to these line integrals over various paths:

FT of Line Integrals

For $\mathcal{C}$ any path from point $\mathbf{p}$ to point $\mathbf{q}$, we have:

$${\int }_{\mathcal{C}}\nabla f\cdot d\mathbf{s}=f(\mathbf{q})-f(\mathbf{p}).$$

If $f(x,y)$ gives the elevation on a mountain at point $(x,y)$, then $\nabla f$ gives the rate of incline in the direction of steepest ascent. The line integral of $f$ along any hiking path $\mathcal{C}$ then gives the total change in elevation.

Derivation of FT of Line Integrals

Take any parametrization $\mathbf{r}(t)$ of $\mathcal{C}$ given for $t\in [a,b]$. Now $\nabla f\cdot d\mathbf{s}=\nabla f\cdot {\mathbf{r}}^{\prime }(t)dt$. By the chain rule for paths (in reverse), this is equal to $\frac{d}{dt}f(\mathbf{r}(t))$. So can apply the ordinary FTC:

$$\begin{array}{rl}{\int }_{\mathcal{C}}\nabla f\cdot d\mathbf{s}& ={\int }_a^b(\nabla f)(\mathbf{r}(t))\cdot {\mathbf{r}}^{\prime }(t)dt\\ & ={\int }_a^b\frac{d}{dt}f(\mathbf{r}(t))dt\\ & =f(\mathbf{r}(b))-f(\mathbf{r}(a))\\ & =f(\mathbf{q})-f(\mathbf{p}).\end{array}$$