## FANDOM

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The gradient theorem, also known as the fundamental theorem of line integrals, is a theorem which states that a line integral taken over a vector field which is the gradient of a scalar function can be evaluated only by looking at the endpoints of the scalar function. In mathematical terms,

$\int\limits_{p_1}^{p_2}\nabla f\cdot\vec{dr}=f(p_2)-f(p_1)$

As a corollary, the line integral over $\nabla f$ is path independent, therefore any closed path over $\nabla f$ will be equal to zero. Gradient vector fields are also known as conservative.

## Proof

Let $f$ be a differentiable function and $\nabla f$ be its gradient.

$\int_{\gamma}\nabla f(\mathbf{u})d\mathbf{u}=\int\limits_a^b\nabla f(\mathbf{r}(t)\cdot\mathbf{r}'(t)dt$

Since the derivative of $f$ with respect to $t$ will be

$\frac{d}{dt}(f(\mathbf{r}(t))=\nabla f(\mathbf{r}(t))\cdot\mathbf{r}'(t)$

by the multivariable chain rule, this expression becomes

$\int\limits_b^a\frac{d}{dt}(f(\mathbf{r}(t))dt=f(\mathbf{r}(b))-f(\mathbf{r}(a))$
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