Fundamental Theorem of Line Integrals

Theorem
Let $\CC$ be a smooth curve given by the vector function $\map {\mathbf r} t$ 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 $\CC$.

Then:


 * $\ds \int_\CC \nabla f \cdot d \mathbf r = \map f {\map {\mathbf r} b} - \map f {\map {\mathbf r} a}$