Rather than have a set rigid schedule, we'll try to be flexible and give each topic the time it deserves. I will always give you a heads up as to what the next few topics will be, but in most cases you can safely assume we’ll be doing every section from each chapter, roughly in the following order, and (very) roughly in/around the indicated weeks.
First half (roughly weeks 1-6)
We will start the semester with Linear algebra, from Strang’s “Introduction to linear algebra”, covering chapters 1-4. You do not need to buy this book, but our topic order will be the same. You can find more detail about sections we'll cover and in depth key ideas here.
Chapter 1 (weeks 1, 2) talks about vectors and matrices, and how they relate to systems of linear equations. Remember, for each of these chapters you can find more detail about sections we'll cover and in depth key ideas here.
Chapter 2 (weeks 2, 3) will teach us how to simplify systems of equations, calculate matrix inverses, and factor matrices into simpler ones. This will give us the techniques to solve systems given by invertible matrices, which always have a unique solution.
Chapter 3 (weeks 4, 5) studies vector spaces, concentrating on those associated with matrices. It will allow us to deal with more complicated systems, recognize when solutions may not exist, and when there might be more than one.
Chapter 4 (weeks 5, 6) works with systems that do not have a solution: we will learn how to find the “best answer”, one that does not fit the equations exactly, but it is as close as possible.
Midterm 1 will cover all of the linear algebra we’ve learned. The expected date of this exam is Tuesday, February 25, in class. Really, we'll take it after we finish chapter 4, which is expected around week 7.
Second half (roughly weeks 8, 11-15)
Using what we have learned about matrices. We will cover chapters 12-15 in McCallum’s “Calculus, Multivariable”. You do not need to buy this book, but our topic order will be the same. A more in depth description of the key ideas in each section will be provided later.
Chapter 12 (weeks 7, 8) introduces functions of multiple variables. We will talk about equations of surfaces, draw cross-sections and topographical maps.
- 12.1 Functions of Two Variables
12.2 Graphs and Surfaces
12.3 Contour Diagrams
12.4 Linear Functions
12.5 Functions of Three Variables
Chapter 13 (week 8) is a review of linear algebra: we will add only the cross product from here, which will allow us to write equations of planes in three dimensions.
Chapter 14 (weeks 11, 12, 13) is about derivatives. We will learn how to calculate x and y derivatives, as well as derivatives in other directions (for example along y = x). The techniques are very similar to those of single variable calculus. Tangent lines to curves will be replaced by tangent planes to surfaces, and we will have three different kinds of second order derivatives instead of one; but everything in this chapter should feel familiar.
- 14.1 The Partial Derivative
14.2 Computing Partial Derivatives Algebraically
14.3 Local Linearity and the Differential
14.4 Gradients and Directional Derivatives in the Plane
14.5 Gradients and Directional Derivatives in Space
14.6 The Chain Rule
14.7 Second-Order Partial Derivatives
Midterm 2: the second exam (not cumulative) will cover the multivariable calculus we’ve learned up through Chapter 14. The expected date of this exam is Tuesday, April 8, in class. Again, we'll really take it after we finish chapter 14, which you should expect around week 13.
Chapter 15 (weeks 14, 15) is about optimization, and finding minima and maxima of multivariable functions. Once again, the material will be quite familiar from single variable calculus, though the second derivative test will be more complicated, as will be the checking of “endpoints” (which will now be entire boundary curves of the region in question). Lastly, we will cover optimization subject to a constraint, using a technique known as Lagrange multipliers.
- 15.1 Critical Points: Local Extreme and Saddle Points
15.3 Constrained Optimization: Lagrange Multipliers
This is where I'll post notes, handouts, quiz solutions, practice exams and more. I'll post an announcement on Canvas whenever this section gets updated.
Economics applications (top)
The originators of this course compiled an expository paper with many applications to economics. I'll take some examples from these to use in class, and assign readings out of it.
Calculus and Linear Algebra Notes - Economics Applications
Useful links/documents (top)
Here are some links and documents that may be usesful, in general, for this course:
- My old lecture notes from a linear algebra course I taught at UConn. A useful source for extra examples/definitions. The topics are in a different order, and some are skipped. In particular, we'll cover nothing that looks like Sections 1.8 or 1.9, Chapters 3 or 5.
- An indepth list of key linear algebra topics and sections we'll cover.
- This website has calculators for many of the matrix operations that we'll be using this semester. It can even show you the steps it uses to reduce a matrix! Try not to rely on it too much, but use it to check your work or if you get stuck.
- The Essence of Linear Algebra series by 3Blue1Brown has excellent visualizations and explanations for some of the concepts we'll be encountering.
Notes and topics from class (top)
The first two documents are a rough outline (unfilled) of what we expect(ed) to cover in class, with definitions and examples. There are two versions. The rest of the bullets include the just the examples that we covered in class, and completed examples (highlighted in red) that we didn't.
The best place to start. Most of the questions on the exam are inspired by the problems below. They're old homework problems (with solutions) from when I taught a similar course. There are some more (untestible) theoretical questions toward the end of each set, and you can ignore anything about transformations. Start with the topic you are least confortable with.
Old exam problems. Look here for exam problems from previous years. I would do these problems after you've exhausted the problems above.
- Here are 25 old exam questions I selected from some exams I wrote for a linear algebra course last year. This is a good place to start.
- Here are more practice problems from 118 last semester. Different material may have been covered. For example, there's nothing on orthogonality here.