Study guide: you need to know everything.
However, here are the highlights (topics not to be missed):

Chapter 14:
You need to be able to
calculate partial derivatives.You need to be able to use the chain rule.

Write the linear approximation of a function at a given point.

The gradient vector and its interpretation (direction of the fastest increase).

Local extrema, the second derivative test. Absolute extrema.

Lagrange multipliers.

Chapter 15: Calculation of double and triple integrals. Application to calculating areas, and volumes.

You need to know the formulas for changing coordinates to cylindrical or spherical and use them for integration.

Chapter 16:
Line integrals: calculation, interpretation as mass, or as work.

The Fundamental Theorem of Calculus for Line Integrals (statement and use in problems).

Be able to recognize when a vector field F is conservative and find its potential f.

Note: from 16.3 only Theorems 2 and 5, the fact that the integral of a conservative field is independent of path, and its integral over a closed curve is zero, calculation of the potential of a conservative field.

Green's theorem: state it and use it.