We have successfully reduced one side of Stokes' theorem to a 2-dimensional formula; we now turn to the other side.
We can now recognize the difference of partials as a (scalar) triple product:
On the other hand, the definition of a surface integral also includes a triple product—the very same one!
We can substitute the conclusion of STEP2 into the left-hand side of Green's theorem above, and substitute the conclusion of STEP3 into the right-hand side. Q.E.D.
In this section, we will discuss the irrotational field (lamellar vector field) based on Stokes' theorem.
In this section, we will introduce a theorem that is derived from Stokes' theorem and characterizes vortex-free vector fields. In fluid dynamics it is called Helmholtz's theorems.
Above Helmholtz's theorem gives an explanation as to why the work done by a conservative force in changing an object's position is path independent. First, we introduce the Lemma 2-2, which is a corollary of and a special case of Helmholtz's theorem.
The claim that "for a conservative force, the work done in changing an object's position is path independent" might seem to follow immediately if the M is simply connected. However, recall that simple-connection only guarantees the existence of a continuous homotopy satisfying [SC1-3]; we seek a piecewise smooth homotopy satisfying those conditions instead.