The integration of a gradient, represented mathematically as \int_a^b \nabla T\; d\vec l \;=\; T(b)-T(a), is path independent due to the properties of conservative vector fields. This principle is a direct application of the gradient theorem, which generalizes the fundamental theorem of calculus. The proof of this theorem can be effectively demonstrated using Stokes' theorem, establishing a clear connection between line integrals and surface integrals.
Students and professionals in mathematics, physics, and engineering who are looking to deepen their understanding of vector calculus and its applications in real-world scenarios.
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.yungman said:Why
[tex]\int_a^b \nabla T\; d\vec l \;=\; T(b)-T(a)[/tex]
Why integration of a gradient is always path independent?