Study guide: you need to know
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.
You need to be able to
use the chain rule. Use it in
substitutions in differential
Extrema: local, absolute, with constraints (Lagrange
derivative test is included on the formula sheet.
Use of Taylor polynomials (the formula will be
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.)
calculation, interpretation as work or flow, or
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