Gradient theorem

Topics in Calculus

Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus  Derivative Change of variables Implicit differentiation Taylor's theorem Related rates Rules and identities: Power rule, Product rule, Quotient rule, Chain rule Integral calculus  Integral Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order Vector calculus  Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem Multivariable calculus  Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian

The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field (any conservative vector field can be expressed as the gradient of a scalar field) can be evaluated by evaluating the original scalar field at the endpoints of the curve:

$\phi\left(\mathbf{q}\right)-\phi\left(\mathbf{p}\right) = \int_L \nabla\phi\cdot d\mathbf{r}.$

It is a generalization of the fundamental theorem of calculus to any curve in a plane or space (generally n-dimensional) rather than just the real line.

The gradient theorem implies that line integrals through irrotational vector fields are path independent. In physics this theorem is one of the ways of defining a "conservative" force. By placing $\ \phi$ as potential, $\nabla\phi$ is a conservative field. Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows.

See also

• state function