Double Integrals:
Understand the definition of a double integral as a volume, and as an
interated integral. Be able to compute double integrals over both
rectangular and general regions. Be able to change the order of
integration in a double integral. Be able to set up and compute
double integrals in polar coordinates in addition to in rectangular
coordinates.
Triple integrals:
Understand the definition of a triple integral, and be able to set up
and compute them. Be able to exchange the order of integration.
Triple integrals in spherical or cylindrical coordinates will not
be explicitly required, but feel free to use them if you know how.
Surface Area:
This is really just the surface integral
,
but you
can treat it separately if it makes you happier.
Vector Fields: Know what a vector field is, and how to tell if
a vector field is conservative or not. If a field is conservative, be
able to find a potential function for it.
Line Integrals: Know what a line integral is, and how to
compute it. If a vector field is conservative, be able to use the
fundamental theorem to compute the line integral in terms of the
potential. Understand the equivalence between independence of path of
the line integral
and the existence of a potential
f such that
.
Green's theorem:
Know the statement of Green's theorem and how to apply it to compute a
line integral.
Surface integrals: Know the definition of a surface integral,
and how to compute them, both for parametric surfaces and surfaces
which are graphs of the form z=f(x,y). Know the relationship
between a surface integral and the flux of a vector field through a
surface.
Curl and divergence: If F is a vector field on
,
know the definition of the curl of F (
)
and
divergence of F (
). Know the interpretation of curl
as a measurement of the rotation of a vector field, and divergence as
measuring its compressability. Know that
,
and that
.
Stokes' Theorem:
Know the statement of Stokes' theorem, and how to apply it.
Gauss' Theorem:
Know the statement of Gauss' theorem, and how to apply it.