Study guide: you need to know everything.

However, here are some Highlights:

You need to be able to recognize and plot cylinders, simple quadrics.

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

for integration or substitutions in partial differential equations. You need to know dV in these coordinates.

You need to know the equation of the tangent plane; that grad f points in the direction of the fastest increase,

the formula for a normal vector to a surface given as a graph z=f(x,y) or as an equation F(x,y,z)=0.

Chapter 20: everything!

(Except: the formulas I-IV of 20.3 will be provided if needed; no questions will be asked regarding the gravitational force.)

Chapter 21:

Line integrals:

calculation, interpretation as work or flow, or flux.

Independence on path for F=Li+Mj: the diagram on top of page 762 completed with: implies dM/dx=dL/dy

and we have equivalence if L, M, are continuously differentiable on the domain under discussion.

Gradient field, conservative field, potential.

The Fundamental Theorem of Calculus for Line Integrals (statement with complete conditions, calculation of f, use in problems).

Surface integrals (formula for dA on a sphere or on a cylinder will be provided in the formula sheet).

You need to be able to state Green's, Gauss's and Stokes' Theorems

(formulas, and conditions on the vector field, orientation of paths and normal). You need to be able to use them in problems.

Suggested review problems:

21.5#15, 13 21.4#11 21.3#12, 23, 25, 27 21.2# 16, 17, prove formula (9) which will be provided