Math 3500/3510: Multivariable Calculus and Linear Algebra

Math 3500/3510 is an integrated introduction to linear algebra and multivariable calculus. It’s designed to be taken as a two semester course which replaces the MATH 2270/MATH 3000 sequence or the MATH 2500/MATH 3300 sequence. This is the fastest-paced and most mathematically rigorous version of this sequence– it’s designed to challenge the best incoming students at UGA.

The course covers the interplay of algebra and analysis which explains the behavior of functions with many inputs and outputs. In the first semester, we will extend many of the ideas from single-variable calculus to this setting: differentiation, critical points, the first and second derivative tests and the solution of max/min problems. In order to do this, we’ll have to reproduce the key insight of single-variable calculus– the derivative describes the behavior of a linear approximation to a function– for linear maps between higher dimensional spaces. This will require us to develop an understanding of linear algebra along the way.

The second semester starts with integration of multivariable functions and then proceeds to study differential forms (a kind of sophisticated generalization of a vector field) and calculus on manifolds. This will take most of the semester, and we will end by pivoting back to linear algebra and studying the singular value decomposition (including the special case of the spectral theorem).

Our text for the course is Shifrin’s Multivariable Mathematics. This is a serious book, and it’s a repository for a lot of important material that you’ll need to do the problems, but we may not cover during lecture. It’s important to have a copy. There’s only one edition, so feel free to get a used copy (if you order early, you can get a paperback copy for about $50 from the Amazon link above; the book is supposed to be in the bookstore, and is on reserve in the Science Library).

Homework will be assigned and collected with Gradescope. You’ll need to log in and make a (free) account. I’ll distribute a course entry code in class. You’ll then scan and upload PDF files of your homework assignments. Exams and quizzes will be taken in class and scanned by the professor and TA. You’ll be able to access and download graded copies of your assignments through Gradescope. We won’t use eLC at all for this course, since Gradescope and this webpage will give us everything we need.

Office hours will be on Wednesday afternoons, 4-6pm in my office (Boyd 405).

Syllabus

Please examine the course syllabus for 3500 and the course syllabus for 3510. If you think you can get by with this copy, save a tree! Don’t print it out. The course syllabus lists the various policies for the course (don’t miss the attendance policy). The course is organized by day. It’s important to read ahead in the book before the corresponding lecture. The format of the class will be mostly lecture, with some in-class activities and problems to check your understanding as we go.

Mathematica.

The course will use Mathematica for in-class demonstrations and (some) homework assignments. It’s a useful language to know, both to check your handwritten math work and to try calculations (especially in linear algebra) which are simply too big to do by hand. You can get a license from EITS. You’ll have to complete this online course: An Elementary Introduction to the Wolfram Language before the start of the second semester. However, it’s wise to finish the online course as soon as you can (preferably during the summer before 3500) so you’ll get the most benefit from this computational tool.

Lecture Notes

Math 3500.

  1. Vectors and Matrices
    1. Introduction to class. Vectors.
    2. The Dot Product. Cauchy-Schwartz and triangle inequalities. Angles. Correlation coefficient.
    3. Subspaces. Linear combinations and Span. Orthogonal complement.
    4. Linear Transformations.
      1. The Identity Matrix and Inverse Matrices.
      2. The Transpose and Orthogonal Matrices.
      3. Example: Visualizing matrix multiplication
    5. Determinant and Cross Product.
  2. Functions, Limits and Continuity
    1. Types of multivariable functions. (No notes.)
    2. Topology in R^n.
    3. Limits and Continuity.
  3. The Derivative
    1. Partial and Directional derivatives.
      1. Example: Partials exist but function not differentiable (Mathematica)
    2. Differentiability and the Jacobian.
    3. Differentiation Rules.
    4. The Gradient
    5. Curves
    6. Higher Partial Derivatives
      1. Example: Harmonic functions (the gravity well) (Mathematica)
      2. The wave equation.
  4. Machine Learning
    1. Introduction to Convolutional Neural Networks, Wu.
    2. Lecture Notes 1.
    3. Lecture Notes 2.
  5. Quadratic Forms and the Second Derivative Test
  6. Lagrange Multipliers Computational example

Math 3510.

  1. 8.2 Differential Forms. (early draft, comments encouraged!)
  2. 8.3 Line Integrals. (early draft, comments encouraged!)
  3. 8.4 Surface Integrals. (early draft, comments encouraged!)
  4. Maxwell’s Equations in Differential Forms.

Supplemental Readings

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