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?
Integration of a gradient is a mathematical process that involves finding the antiderivative of a gradient, which is a vector field representing the rate of change of a scalar field in different directions.
Integration of a gradient is important because it allows us to find the original scalar field from its gradient, which is useful in many applications such as physics, engineering, and economics.
Integration of a gradient is performed by finding the antiderivative of each component of the gradient vector and combining them to form the original scalar field.
The relationship between integration of a gradient and differentiation is that they are inverse processes. Integration of a gradient is the reverse of differentiation, and vice versa.
Some real-life examples of the use of integration of a gradient include calculating the potential energy of an object in a gravitational field, finding the velocity of a fluid flow from its acceleration, and determining the electric field from the electric potential.