Math 311h - Final Exam Review
General Information
The final Exam (Monday, December 15, 8-10 am) will have 7 to 10
questions, some with multiple parts. The final will directly cover
material from the following sections: 3.6, 3.7, 4.1A&F, 4.4A,
5.2B, 5.3A, 5.4, 6.2, 6.5D, 7.4D, 8.1, 9.1, 9.3, 9.4, 9.5A. There will
be no direct questions on material prior to section 3.6 (eigenvalue
problems), 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 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
Linear Algebra
Eigenvalues and Eigenvectors
- Eigenvalues and eigenvectors. Be able to find the eigenvalues and
eigenvectors for a linear transformation or matrix.
- Diagonalizable. A linear transformation L:V->V is said to be
diagonalizable if and only if there is a basis for V relative to which
the matrix for L is diagonal. Be able to determine whether or not L is diagonalizable. Know that if the eigenvalues of L are distinct, then L is
diagonalizable. The converse is false, however.
- Change of coordinates and similarity transformations. An
n×n matrix A is similar to an n×n matrix B if there is an
invertible matrix S such that B = S-1AS. For an eigenvalue
problem, a matrix A is diagonalizable if and only if it is similar to
a diagonal matrix D. That is, there is an invertible matrix S such
that D = S-1AS. Here, the matrix S has a set of linearly
independent eigenvectors of A for its columns.
- Linear systems and normal modes. Be able to solve simple linear
systems of ODEs. Be able to find the normal modes of a spring system.
Inner products and norms
- Inner product and norm. Be able to define these terms.
- Schwarz's inequality and the triangle inequality. Be able to prove
Schwarz's inequality and be able to state the triangle inequality.
- Angle and length. Be able to find the norm of a vector and to
find the angle between two vectors.
- Orthogonality. Be able to define these terms: orthogonal and
orthonormal sets; orthonormal bases.
- Gram-Schmidt. Be able to use the Gram-Schmidt procedure to find
an orthogonal or orthonormal set of vectors, given a linearly
independent set.
- Least squares. Know the difference between continuous and
discrete least squares problems. Be able to do least-squares problems,
at least to the point of setting them up.
- Rotations and reflections. Know the difference between rotation
matrices and reflection matrices. Given an axis of rotation and an
angle, be able to find a rotation matrix. Also, be able to find the
axis of rotation and angle for a given rotation matrix.
Multivariable calculus
Curves and line integrals
- Parameterization. Smooth curves. Piecewise smooth
curves. (§4.1A, 4.1F, 8.1). Be able to parameterize circles,
lines, and ellipses.
- Line integrals. Be able to compute line integrals using a
parameterization for the curve or, when appropriate, via the
fundamental theorem (§8.1).
Derivatives
- Derivative along a vector and directional derivative. Be able to
find directional derivatives for scalar valued functions. In addition,
be able to find directions of fastest increase and fastest
decrease. (§5.3A)
- Derivative (Jacobian) matrix for a vector valued function and
tangent approximations. Be able to compute the matrix derivative for f
: Rn -> Rm. Be able to find the
tangent approximation for such functions (§5.4).
- Chain rule and inverse function theorem. Know these and be able
to use them to compute tangent approximations. (§6.2)
Change of variables and multiple integrals
- Curvilinear coordinates. See §6.5D.
- Jacobi's Theorem. See §7.4D. Know how to change variables in
multiple integrals.
Surfaces and surface integrals
- Surfaces. Know the parametric form of a surface
(§4.4A). Know how to find the standard normal, unit
normal, vector and scalar surface area elements
(§9.3). Memorize these quantities for a sphere of radius a, a
cylinder of radius a, and a plane.
- Surface integrals. Know the difference between density and flux
integrals. Be able set up and compute surface integrals for oriented
surfaces.
The integral theorems of vector calculus
- Green's Theorem. Green's Theorem has three forms. Know what these
are and be able to verify Green's theorem or to use it to compute line
integrals or integrals or regions in the plane. (§9.1)
- Gauss's Theorem. Know Gauss's Theorem and be able to verify
it. Be able to use it to compute surface integrals or integrals over
certain volumes. You should also know how to derive the equation of
continuity for fluid flow. (§9.4)
- Stokes's Theorem. Know Stokes's Theorem and be able to verify
it. Be able to use it to compute line integrals or certain surface
integrals. (§9.5)