- Exam 2, version B Math 311.501
- Exam 1, version A Math 311.503
- 311 syllabus Sections 501, 503
- Exam 1, version B Math 311.503
- Section 6.2: Systems of Linear ODE's I'm not going to go into as much detail as the book does on this, because
- Section 3.3: Linear Independence We want to throw away "extraneous" vectors from spanning sets, to get
- STABILITY IN A BALL-PARTITION PROBLEM THOMAS I. VOGEL
- Exam 2, version A Math 311.503
- Topics in Applied Mathematics Math 311.501, 311.503
- Section 4.1: Linear transformations You've seen in calculus how important functions are. In linear algebra,
- Chapter 6: Bessel functions (a little bit) Bessel's equation of order 0 is this ODE
- Section 1.3: Matrix arithmetic Start with notation for matrices (note: plural of "matrix" is "matrices").
- Section 5.4: Inner Product Spaces Generalizing dot product.
- PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS
- LOCAL ENERGY MINIMALITY OF CAPILLARY SURFACES
- SUFFICIENT CONDITIONS FOR CAPILLARY SURFACES TO BE ENERGY MINIMA
- On constrained extrema Thomas I. Vogel
- Update: January 25, 2011 Thomas I. Vogel BIOGRAPHICAL SKETCH
- Section 1.1 Much of linear algebra is the study of systems of linear equations. A linear
- Section 1.4: Matrix algebra Rules for matrix arithmetic (Theorem 1.4.1, page 44): various things like
- Section 2.1: Determinants The idea is this: if A is nn, then det (A) will be a number so that det (A) =
- Section 2.2: Properties of determinants First off, computing determinants in practice: Suppose we want the deter-
- Section 3.2: Subspaces Suppose that V is a vector space with operations + and . If S V is also
- Section 3.4 Basis and Dimension Recall that we were looking for minimal spanning sets.
- Section 3.6: Row space, column space Given a matrix A, there are three important subspaces related to it. We've
- Section 4.2: Matrix representations of linear transformations Note: will skip homogeneous coordinates, pitch and yaw.
- Section 6.1: Eigenvalues, eigenvectors Think of an nn matrix A as inducing a linear operation on Rn
- Section 5.1: Orthogonality One thing which you may have noticed that we didn't generalize from Rn
- Chapter 2: Fourier series and applications. Let's look at a slight generalization of a previous example. Suppose that I
- Exam 1, version A Math 311.501
- Exam 2, version A Math 311.501
- Exam 2, version B Math 311.503
- Exam 3, version A Math 311.501
- Exam 3, version B Math 311.501
- 311 course objectives Sections 501, 503
- Exam 1, version A Math 311.502
- Exam 2, version A Math 311.502
- Problems from last semester's 311 final relevant to our third exam. 1. (a) Show that
- CONVEX, ROTATIONALLY SYMMETRIC LIQUID BRIDGES BETWEEN SPHERES
- Exam 1, version A Math 311.502
- Section 3.1: Vector Spaces First, I'll remind you of an elementary idea of a vector space, and then
- Exam 3, version A Math 311.503
- Chapter 1: Boundary value problems We will be looking at partial differential equations (PDE) for the rest of
- Section 1.2: Row echelon form Triangular form is nice, but one can't always get to it, even in square (i.e.,
- Exam 3, version B Math 311.503
- Section 1.5: Elementary matrices Proposition: Suppose that A is an m n matrix, and M is an invertible
- Section 4.3: Similarity Suppose that L : V V is a linear operator on V (recall that all this means
- Exam 2, version A Math 311.502
- Section 3.5 Change of basis The standard basis (if there is one) in a vector space is not always the
- Section 6.3: Diagonalization The general idea is that it's easiest to work with a linear transformation if
- Section 5.5: orthonormal sets Definition: a set {v1, , vn} of vectors in an inner product space is said
- Additional example for chapter 2. Solve the heat equation ut = uxx on [0, L], with initial temperature u (x, 0) =
- Maple in Math 442 We will be using Maple for heavy-duty computations for this class. I'll give you some of the basics in
- 1 Dimensions In general, a dimension is any quantity which can be measured. If we want to
- Latka-Volterra equations and derivative approximation Suppose that we have two species: a prey, and a predator. The simplest model of the populations is the
- The types of population models that we've seen in this class are deterministic, in other words, if you know the population now, you'll know it at any future time. This is an appropriate approximation if we're
- Predator-Prey project due Monday, May 3rd
- Exam 1, version B Math 311.501
- Sufficient Conditions for Multiply Constrained Thomas I. Vogel
- Multivariable optimization We've already seen the following in doing least squares: if we
- Lecture notes for Math 442 T. I. Vogel
- Math 442.500 Midterm exam
- Regression with more general functions Of course, not all phenomena which we want to model can be described by polynomials. For example,
- Ballistics Project Ballistics is the science of projectile motion and impact, phenomena well de-
- Math 442.500 Midterm exam
- A little bit of mechanics One way we get second order ODE's is from Newtonian mechanics. The
- Lecture notes on ODE's Compartmental analysis
- Curve fitting and least squares In modeling, we must often start with observed data, and attempt to describe it with some underlying
- Dimensional analysis problems Math 442.500
- 1. First, we plot both curves on the same set of axes. A little experimentation gives a reasonable range. I'm defining these as functions so that I don't have to type these out again when I want to plug in later.
- Section 2.3: Additional topics (Cramer's rule) Most of this section I'll skip. It gives a formula for A-1
- 309 course objectives Sections 501, 502
- 309 syllabus Sections 501, 502
- Exam 3, version B Math 309.502
- Exam 3, version B Math 309.501
- Exam 1, version A Math 309.502
- Exam 2, version A Math 309.502
- Exam 3, version A Math 309.501
- Exam 2, version A Math 309.501
- Exam 1, version B Math 309.501
- Exam 3, version A Math 309.502
- Exam 1, version A Math 309.501
- Exam 2, version B Math 309.502
- Exam 1, version B Math 309.502
- Exam 2, version B Math 309.501
- Solutions # 3 7.1.1 a) On any interval [a, b], we have |x| max (|a| , |b|), so that x
- + 1) -ln(k2 Example 19. Let f(x) =
- Advanced Calculus II Math 410.500
- Proposition: f : E R is continuous iff for every open set , we have that f-1
- Analytic Functions (7.4) We've seem that if f equals a power series, i.e., f(x) =
- Solutions to homework # 2 6.3.2d. A routine ratio test
- Math 410.500 There are problems on both sides of this sheet!
- Anyway, back to the theorem: (1) Suppose that the collection of open sets is V, A. Take
- Math 251.504 Exam 1, version B
- (x0 -, x0 + ), but of course it converges at every point of (x0 -, x0 + ). It follows that f is differentiable on (x0 -, x0 + ), with
- Math 251.512 Exam 1, version A
- Power Series (7.3) In this section we will look at a special type of series of functions.
- Math 410.500 1. (15 pts.)
- Answers to exam 3, version A 1. If D is the region bounded by x and x2
- Series with Non-negative Terms (6.2) The question which will generally be asked is "Does a series con-
- Answers to exam 2, version A 1. In general, the directional derivative of f at (a, b) in the direction of a unit
- Absolute Convergence (6.3) Definition: A series ak is said to converge absolutely if |ak|
- Math 410.500: answers to exam 3 1. (a) True. (This is Theorem 11.13.)
- Math 410.500 There are problems on both sides of this sheet!
- for all x (a, b). Proof: Since f
- Solutions to assignment # 4 7.3.1b: Writing out the series as 1+x2
- Theorem: Suppose that fn : E R converges uniformly on E to a function f. Suppose that each fn is continuous on E. Then f is
- , it's a little suspicious to just quote the above proof, since we can't really visualize R17
- Engineering Mathematics III Math 251.504
- Algebraic structure of Rn Now we're starting multivariable advanced calculus. We define Rn
- Lecture Notes for Math 410 Thomas I. Vogel
- Solutions to homework #1 6.1.0 a) False. You can use a divergent p series as a counterexample, or
- Proof: Hold x and x0 fixed, and define a function g(z) by g(z) = f(z)-
- (1) Take x E. Suppose BWOC that > 0 so that B (x)E = . Then the complement of B (x) is a closed set (by defi-
- Math 410.500: answers to exam 2 1. (a) U Rn
- Pointwise and Uniform Convergence of Sequences of Functions (7.1)
- Math 410.500: answers to exam 1 1. (a) f is analytic on (a, b) i for each x0 (a, b) there exists a power series
- Math 251.512 Exam 3, version A
- Math 251.504 Exam 1, version A
- Math 251.512 Exam 1, version A
- Alternating Series (6.4) Will not follow book. Instead, I'll give a more elementary proof of
- Interior, closure, and boundary (8.4) More topological definitions: we want to make precise the ideas of