Linear Algebra and Multivariable Calculus (Math 290) — Winter 2019

Welcome to Math 290-2! This is the second in the three-quarter sequence on linear algebra and multivariable calculus for MENU. (My 290-1 page is here.)

Last quarter was all about linear algebra, getting to grips with the tools of vectors and matrices and using them to interpret particular kinds of algebraic problems geometrically. In the first half of this quarter we will continue this journey, and use the tools of linear algebra to parametrise certain kinds of curves and surfaces—this skill will prove to be extremely important in Math 290-3 next quarter!

In the second half of the quarter, we will generalise what you have learnt about differential calculus to functions of more than one variable. You might be wondering, what does this have to do with linear algebra? Here's one example. If $f$ is a nicely behaved function of two variables, then $f(x,y) = 0$ describes a curve in $\mathbb{R}^2$. (For example, if $f(x,y) = x^2-y$, then $f(x,y)=0$ is the parabola $y=x^2$.) At a point $(a,b)$ on the curve, we might ask what is the tangent line to the curve $f(x,y) = 0$ at $(a,b)$. It turns out that this is the subspace of $\mathbb{R}^2$ that is orthogonal to the vector $(f_x(a,b), f_y(a,b))$, where $f_x$ and $f_y$ are the partial derivatives of $f$ with respect to $x$ and $y$, respectively. This example generalises to higher dimensions, and we will make heavy use of the tools of linear algebra in our study of multivariable calculus.

Most administrative aspects of the course will be handled using Canvas.

Time and place. MoWe(Th)Fr 9:00–9:50am in Lunt Hall 103.

Textbooks. We will use two textbooks this quarter:

• For the linear algebra material, we will use the same textbook as in 290-1, Linear Algebra with Applications (5th edition) by Otto Brescher (ISBN 978-0-321-79697-4).
• For the multivariable calulus material, we will use Vector Calculus (4th edition) by Susan Jane Colley (ISBN 978-0-321-78065-2).

Homework. Homework assignments will be due at the beginning of class on Fridays—they can be viewed on Canvas.

Examinations. There will be two midterm exams and a final exam.

• Midterm 1. 6:30–7:30pm on Monday 4th February 2019.
• Midterm 2. 6:30–7:30pm on Monday 4th March 2019.
• Final. 12:00–2:00pm on Monday 18th March 2019.

Handouts. What follows are the handouts from class and solutions to exercises.

1. Orthonormal vectors — handout — solutions
2. Orthogonal projections, lengths and angles — handout — solutions
3. Gram–Schmidt orthonormalisation — handout — solutions
4. Orthogonal transformations — handout — solutions
5. Least squares and data fitting — handout — solutions
6. Orthonormal bases of eigenvectors — handout — solutions
7. Quadratic forms — handout — solutions — supplement
8. Parametric equations — handout — solutions
9. The cross product — handout — solutions
10. Equations of planes, distances — handout — solutions
11. New coordinate systems I — handout — solutions
12. New coordinate systems II — handout — solutions
13. Multivariate functions — guest lecture by job candidate
14. Quadric surfaces — handout — solutions — supplement
15. Limits of multivariate functions — handout — solutions
16. Partial differentiation — handout — solutions
17. Differentiability — handout — solutions
18. Higher-order partial derivatives — handout — solutions
19. The product and chain rules — handout — solutions
20. Directional derivatives I — handout — solutions
21. Directional derivatives II — handout — solutions
22. Taylor polynomials — handout — solutions
23. Local extrema and critical points — handout — solutions
24. Global extrema — handout — solutions
25. Lagrange multipliers I — handout — solutions
26. Lagrange multipliers II — handout — solutions