Gradient theorem

The gradient theorem (also the fundamental theorem of line integrals) says that for a curve $C$ beginning at a point $\vec{p}$ and ending at a point $\vec{q}$, then:

$${\int }_C\nabla \vec{f}\cdot d\vec{r}=f(\vec{q})-f(\vec{p})$$

where $\nabla \vec{f}$ is the gradient vector field of $f$. It’s more or less analogous to the fundamental theorem of calculus.

Note the orientation of the curve matters. If the curve is closed ($\vec{P}$ = $\vec{Q}$), then we get 0, i.e., the integral of $\nabla F$ over any closed curve is 0.

The fundamental theorem is not relevant in all cases ${\int }_C\vec{F}\cdot d\vec{r}$. It only holds when $\nabla f=\vec{F}$ for some vector field $\vec{F}$, i.e., the vector field is conservative.

See also

• Fundamental theorem of contour integrals