fundamental theorem of calculus for line integrals

the fundamental theorem of calculus for line integrals

1. (calculus) A generalization of the fundamental theorem of calculus to line integrals of vector fields, which states that the line integral of a conservative vector field along a path can be computed by evaluating the potential function at the endpoints of the path. Specifically, given a potential function ${\displaystyle f}$ with continuous first partial derivatives on an open region R containing a curve C parameterized by ${\displaystyle \mathbf {r} (t)}$ for ${\displaystyle a\leq t\leq b}$:

   ${\displaystyle \int _{C}\nabla f\cdot d\mathbf {r} =f\left(\mathbf {r} (b)\right)-f\left(\mathbf {r} (a)\right)}$.