Multivariable calculus doesn't replace anything you already know — it relocates it. A vector still has a magnitude and a direction; it just gets a third component. A curve is still traced by a parameter t; it just gets a third coordinate function. Polar coordinates still trade (x,y) for (r,\theta); a later chapter adds a z and calls the result cylindrical. Every one of these ideas survives the jump into space completely intact — the only thing that changes is how many numbers it takes to write down.
This chapter is a fast, deliberate pass over the pieces of Calculus I and II this course leans on hardest, in the exact form they'll reappear: vectors as arrows with components, curves as functions of a parameter, polar coordinates as an alternative to (x,y), arc length as a limit of polygonal approximations, and the two integration techniques — u-substitution and integration by parts — that every double and triple integral ahead will eventually need. Nothing here is new material; every practice question is pulled straight from the course that first taught it.
Two threads get particular attention because the rest of this course is built directly on top of them. The definite integral as an accumulation — a running total that grows and shrinks as its upper bound moves — is the exact picture double and triple integrals generalize to a region instead of an interval. And the chain rule, reviewed here in its ordinary single-variable form, is about to be rebuilt from scratch with partial derivatives standing in for ordinary ones — the single most load-bearing idea in the entire second half of this course.
By the end of this chapter, nothing should feel unfamiliar. What changes, starting next chapter, is the number of coordinates everything gets written with.