Fundamental Theorem of Line Integrals

Theorem
Let $\mathcal C$ be a smooth curve given by the vector function $\mathbf r \left({t}\right)$ for $a \le t \le b$.

Let $f$ be a differentiable function of two or three variables whose gradient vector $\nabla f$ is continuous on $\mathcal C$.

Then:


 * $\displaystyle \int_\mathcal C \nabla f \cdot d \mathbf r = f \left({\mathbf r \left({b}\right)}\right) - f \left({\mathbf r \left({a}\right)}\right)$