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 -> R^m. 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)