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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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