Math 311-505 - Final Exam Review
General Information
The Final Exam (Tuesday, December 16, 3:30-5:30 pm) will have 7 to 10
questions, some with multiple parts; it will directly cover material
from the following sections: A3.4-A3.6, A4.1-A4.3, A5.4, A6.1-A6.3,
B3.3, B3.4, B5.5, B6.1-B6.3, B7.1-B7.3. There will be no direct
questions on material prior to section A3.4, although you will need to
know that material well enough to use it to answer questions on any
subsequent material. Problems will be similar to ones done for
homework or as examples in class. You will need to know basic
definitions and theorems well enough to answer such questions; you
will not be asked to state any definitions or theorems. There
will be one question devoted to a proof of a theorem or major
derivation. In addition to things marked in the review sheet or its
links, proofs or derivations assigned for homework are fair game.
You may use calculators to do arithmetic, although you will not need
them. You may not use any calculator that has the capability
of doing either calculus or linear algebra. Please bring two
8½×11 bluebooks.
Topics Covered
Topics covered on Tests II and III
Surface integrals and vector analysis
- Parameterized surfaces (B7.1)
- Given a parameterization for a surface, be able to find these
quantities: standard normal, unit normal, vector area element, area
element.
- Know parameterizations for spheres, cylinders, and planes. For
each of these, know and be able to derive these quantities: standard
normal, unit normal, vector area element, area element.
- Be able to compute both kinds of surface integrals -- flux and
density integrals.
- Stokes's Theorem. (B7.2) Be able to verify Stokes's Theorem or to use it
to compute line or surface integrals.
- Gauss's Theorem. (B7.3) Be able to verify Gauss's Theorem or to
use it to surface or volume integrals. Be able to derive the equation
of continuity for a fluid.
Inner product spaces
- Definition of an inner product (p. B260). Know this.
- Cauchy-Schwarz Inequality (p. B265). Be able to prove this.
- Angle between vectors. Norm of a vector. Orthogonal vectors
(p. B265). Know what these are and be able to find them.