Nothing in this chapter is new. Every idea here — vectors and their two products, partial derivatives and the gradient, multiple integrals and the Jacobian, line integrals and the conservative-field tests, Green's and Stokes' and the Divergence Theorem, parametric surfaces, and Taylor series — was already taught, carefully, across Calculus I, II, and III. This course's entire premise is that those tools deserve a second look: not "what do I compute," which you already know, but "why does this work, in general, and how far does it actually reach." You cannot ask that question about a tool you have to stop and recall how to use.
The one exception is section 07, which teaches something genuinely new in this course: a matrix as a linear map, not just a grid of numbers you row-reduce. That single reframing — a matrix does something to space, rather than merely storing coefficients — is the piece of linear algebra every chapter from 02 onward leans on, and it is worth ten minutes now rather than a scramble later.
Two threads matter more than the rest for what's ahead. Green's, Stokes', and the Divergence Theorem, reviewed here as three separate statements that happen to look alike, get proven to be one statement in chapter 07 — the biggest payoff this course has to offer. And the Jacobian, reviewed here as "the correction factor a change of variables needs," reappears constantly once curvilinear coordinates (chapter 04) and general coordinate transformations (chapter 06's pullbacks) make the correction factor's origin the entire point, not a fact to memorize.
By the end of this chapter, nothing should feel unfamiliar — and one new idea should feel like it was always obviously true.