Line Integral

Definition

${\int }_C\nabla f\cdot d\mathbf{r}:={\lim }_{n\to \infty }\sum _{k=1}^n\nabla f({\mathbf{r}}_k)\cdot ({\mathbf{r}}_{k+1}-{\mathbf{r}}_k)$ where $f:{\mathbf{R}}^n\to \mathbf{R}$, and $C$ is a piecewise smooth curve

Facts

Gradient Theorem

Definition

${\int }_C\nabla f\cdot d\mathbf{r}=f(\mathbf{q})-f(\mathbf{q})$ where $C$ is a differentiable curve which starts at a point $\mathbf{p}$ and ends at a point $\mathbf{q}$

Link to original