Question on integration of a gradient.

1. Jun 17, 2011

yungman

Why

$$\int_a^b \nabla T\; d\vec l \;=\; T(b)-T(a)$$

Why integration of a gradient is always path independent?

2. Jun 17, 2011

Hootenanny

Staff Emeritus
This is the essence of the gradient theorem, which is a generalisation of the fundamental theorem of calculus. One can prove the gradient theorem with a simple application of Stokes' theorem.