The gradient theorem goes like this:

C f = f(b) - f(a)

where C is a curve, f is a scalar-valued function defined along C, and a and b are the endpoints of C.

Things to notice:

If a force field is described by the gradient of some function, then the energy that an object gains or loses by following a given path through the force field is just the difference between the function's values at the beginning and at the end of the path. The function is then called the potential energy associated with the force, and the force is called conservative: that is, it obeys conservation of energy, a concept at the core of classical mechanics.

If C is a closed curve (a loop), the endpoints are in the same place (either nowhere or any point along the curve, depending on how you think of it), so the line integral of the gradient along the curve is zero. You could say this is why the moon doesn't fall out of the sky or fly away: it is moved in a loop by a conservative force, so it neither gains nor loses energy as it goes around.

If C is a line segment in a 1-dimensional space, the gradient is the same as the first derivative, so the gradient theorem reduces to the fundamental theorem of calculus.

Since a curve's endpoints are its boundary, and the gradient of a function is also its exterior derivative, the gradient theorem is itself a special case of Stokes' Theorem.

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