This amazing theorem is also called the Fundamental Theorem of Calculus for Line Integrals. It is quite a powerful theorem that sometimes allows fast computations of line integrals.

## Gradient Theorem (Fundamental Theorem of Calculus for Line Integrals)

Let $C$ be a differentiable curve given by the vector function $\mathbf{r}(t)$, $a\leq t\leq b$.

Let $f$ be a differentiable function of $n$ variables whose gradient vector $\nabla f$ is continuous on $C$. Then $\displaystyle \int_C \nabla f\cdot d\mathbf{r}=f(\mathbf{r}(b))-f(\mathbf{r}(a)).$

## Proof

\begin{aligned} \int_C\nabla f\cdot d\mathbf{r}&=\int_a^b\nabla f(\mathbf{r}(t))\cdot \mathbf{r}'(t)\,dt\ \ \ \text{(Definition of line integral)}\\ &=\int_a^b (\frac{\partial f}{\partial x_1}\frac{dx_1}{dt}+\frac{\partial f}{\partial x_2}\frac{dx_2}{dt}+\dots+\frac{\partial f}{\partial x_n}\frac{dx_n}{dt})\,dt\\ &=\int_a^b \frac{d}{dt}f(\mathbf{r}(t))\,dt\ \ \ \text{(by Multivariate Chain Rule)}\\ &=f(\mathbf{r}(b))-f(\mathbf{r}(a))\ \ \ \text{(by Fundamental Theorem of Calculus)} \end{aligned}