Question on integration of a gradient.

[yungman]

Why

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

Why integration of a gradient is always path independent?

 

[Hootenanny]

Why

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

Why integration of a gradient is always path independent?

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.