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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))

by the fundamental theorem of calculus.

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