Math 311-505 - Test III Review

General Information

Test I (Monday, November 24) will have 5 to 7 questions, some with multiple parts. It will cover chapter A6 in Leon's Linear Algebra (Part A), and sections B3.3, B3.4, B5.5, and chapter B6 in Colley's Vector Calculus (Part B). Please bring an 8½×11 bluebook. 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.

Topics Covered

Eigenvalue Problems

Definition. Let L:V->V be linear. We say that a scalar λ is an eigenvalue of L if and only if there is a nonzero vector v for which L[v] = λv. The vector v is called an eigenvector of λ.
Matrix formulation. If dim(V) = n and E = {v₁, v₂, ..., vn} is a basis for V, the eigenvalue problem for L shifts to A, the matrix of L relative to E.
- Characteristic polynomial. p_A(λ) = det(A-λI). Know that p_A is a polynomial of degree n and that the eigenvalues of A are the same as the roots of p_A. Know what p_A looks like --
  p_A(λ) = (-1)ⁿ(λⁿ - trace(A)λ + ... (-1)ⁿ det(A))
- Be able to find the eigenvalues and eigenvectors for a matrix. Be able to do simple normal modes problems. (Note: this is not the same as solving the second order system; see your notes for 10/31/03.)
- Be able to handle problems where the eigenvalues are complex numbers.
Bases of eigenvectors.
- Definition. 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.
- Know that eigenvectors corresponding to distinct eigenvalues are linearly independent, and that L is diagonalizable if and only if there is a basis for V comprising eigenvectors of L. 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.
- Be able to diagonalize a matrix A, if that is possible for A. That is, be able to find a matrix X such that XDX^-1 = A, where D is diagonal.
- Be able to compute the matrix exponential e^A and to use it to get solutions to first order linear systems of ODEs.

Multivariable Calculus

Types of derivatives for F:Rⁿ -> R^m
- Derivative (Jacobian) matrix, F'. This is an m×n matrix. It is also denoted by DF.
- Gradient of a scalar-valued function F:Rⁿ -> R. The gradient is a column vector whose components are the same as for the Jacobian derivative, which is a row vector.
  grad F = (F')^T = (DF)^T
- Divergence of a vector field F:Rⁿ -> Rⁿ. The divergence of F is a scalar quantity.
  div F = trace(F').
  For a vector field F = Ai + Bi + Ci on R³, the divergence is given by
- Curl of a vector field F = Ai + Bi + Ci on R³ is given by
  
  The curl of F is also a vector field defined on R³.
- Directional derivative of a scalar-valued function f:Rⁿ -> R along the direction u.
  ∂f/∂u = grad f · u
- Be able to work with the various types of derivatives. In addition, know how to find directions of maximum increase and decrease for a scalar-valued function at a point.
Chain rule. Let g:Rⁿ -> R^p, f:R^p -> R^m, and let h be the composition of f with g. That is, h is f "circle" g or h(x)=f(g(x)). The chain rule rule is given below. Remember, order matters in matrix multiplication! Be able to use the chain rule.
Dh(x)=Df(g(x)) Dg(x) or h'(x)=f'(g(x)) g'(x)
Jacobi's Theorem. Know how to change variables in integrals. Be able to do problems similar to the ones in class and on homework. Know what the Jacobian determinant is (see p. B348).

Line Integrals

Definition. Know the definition of a line integral in Rⁿ. Be able to compute simple line integrals in 2D and 3D.
Green's Theorem. Be able to state use Green's Theorem in the three ways we mentioned in class: the standard form, Stokes's form, and divergence (Gauss's) form. Be able to verify Green's Theorem for specific functions. (We did some of these in class on 11/17/03.)
Path independence.
- Definition. Know the definition of a path independence for a vector field F (Definition 3.1, p. B400) and the equivalent closed path condition (Theorem 3.2, p. B401).
- Scalar potential. A vector field F has path independent integrals where it is defined if and only if a vector field F = grad f, for a scalar valued function f:Rⁿ -> R. (Theorem 3.3, p. B403.) When F = grad f, the vector field F is called conservative. Given a scalar potential, be able to use this theorem to evaluate line integrals.
- Criterion for F to be conservative. The domain on which F is defined must be "nice" (simply connected), and F must satisfy the appropriate condition on its Jacobian matrix DF. The last condition works in 2D and 3D.
  - 2D. F = Mi+Nj. ∂M/∂y = ∂N/∂x
  - 3D. curl F = 0
  - Rⁿ. DF = (DF)^T.