Fundamental Theorem of Line Integrals

Theorem
Let $\mathcal{C}$ be a smooth curve given by the vector function $\vec{r}(t)$, $a \leq t \leq b$. Let $f$ be a differentiable function of two or three variables whose gradient vector $\vec{\nabla}f$ is continuous on $\mathcal{C}$. Then we have

$\displaystyle{\int_{\mathcal{C}} \vec{\nabla}f \cdot d \vec{r} = f(\vec{r}(b))-f(\vec{r}(a))}$.

Proof
We have by definition
 * $\displaystyle{\int_{\mathcal{C}} \vec{\nabla}f \cdot d \vec{r}}$
 * $= \displaystyle{\int_a^b \vec{\nabla}f \cdot \vec{r} \ '(t) \ dt}$


 * $= \displaystyle{ \int_a^b \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt} + \frac{\partial f}{\partial z}\frac{dz}{dt} \ dt}$

which, by Chain Rule, provides
 * $\displaystyle{\int_a^b \frac{d}{dt} f(\vec{r}(t)) \ dt}$

and through the Fundamental Theorem of Calculus, we get
 * $f(\vec{r}(a))-f(\vec{r}(b))$.