Multivariable Calculus

Description of Course Goals and Curriculum

As per the course offerings description:  You will learn about "Vectors in the plane and in space, vector functions and motion, surfaces, coordinate systems, functions of two or three variables and their derivatives, maxima and minima and applications, double and triple integrals, vector fields and Stokes's theorem." This is a pretty accurate summarization of the main topics covered. Essentially you begin to learn mathematics with important applications to optimization, representing complex shapes/patterns mathematically, and solving complicated physics problems. You will have problem sets, take-home quizzes, and exams. The key to exams is to do as many problems as possible. This is because a common struggle (at least that I encountered) was that even when I did know how to solve a problem generally, the course would then throw a difficult integral, limit, derivative, etc. with an ugly/complicated expression that I wouldn't know how to simplify. The purpose of practicing as many old exams as possible is to learn all the possible ways you might get stuck and more importantly how to work through them. Therefore, make sure you have the solutions available and for the exams without posted solutions, take them to office hours. It is these tips, tricks, and shortcuts that you are expected to know (but are impossible to just Google). Math courses here at Princeton require "math intuition."

Learning From Classroom Instruction

MAT 201 is taught in a 3-day lecture/precept style. You will meet M/W/F for 50 mins. The professor mostly lectures posing periodic (non-rhetorical) questions to the class. You are more than welcome to interrupt and ask questions. In fact, this is even encouraged (as this allows a pause for your fellow students to finish writing things down and makes the professor slow down to reexplain something). As is a general theme in the MAT department (and Princeton) at large, you are expected to know the content ahead of time. In this course, you should come to class having read the textbook pages already. More specifically, if you are feeling comfortable with the pace of the course, a simple skim should be more than enough for you to follow along and participate in class. If you struggle to keep up and never feel prepared enough to even ask questions, the night before class you should read through the textbook chapter thoroughly, stopping to do simple example problems (you can refer to the chapter problems, which bonus may also be homework problems) and using the beautiful resource called the internet to look up visuals to any of the concepts (like Quadric surfaces or general multivariable functions). Essentially, class is not the time to learn multivariable calculus. This should be done outside of class. To get the most out of class, you should already be decently comfortable with the concepts. Math tricks are what you should look out for. You will run into many problems where you get stuck because you don't know/remember how to integrate the square root of e to the something. Pay attention to the shortcuts, tricks, and theorems used by the professors to get out those ruts where you would have gotten stuck.

Learning For and From Assignments

This course has weekly p-sets due on Monday generally covering 3 chapter sections (one covered each day you met for class). The smartest/healthiest/least stressful way to do these is to simply complete the section of the homework that you covered in class that night. For instance, if in class on Monday you covered 12.6, that same day after class just do the problems assigned from 12.6. This way you're done by Friday (not Sunday night) and you have ample time to ask for help. This leads me to my next pro-tip: go to McGraw study hall hours or office hours regardless of how you feel about the class. If you have no idea what is going on, go to McGraw/office hours to work through the p-set with other students (you will always find someone else just as confused as you). Likewise, even if you feel pretty confident in the course (or at least about that week's topics) still go to McGraw. During study hall, compare your answers with other students and explain how you solved the problems to someone else. This will help you solidify the concept. Additionally, this year MAT 201 began doing take-home quizzes. Make sure you allow yourself more time those weeks because it's like having two p-sets due. Just don't wait until the last minute or one will suffer.

External Resources

I feel most strongly about this section. While the textbook does a decent job explain the process of problem-solving, I feel it (and the instructors) do not spend enough time explaining things conceptually. MAT 201 gets very abstract (functions with 2, 3, 4 variables or concepts like flux, curl, and divergence). For the latter, prior physics experience could be very useful, if you had a conceptual physics teacher. Otherwise, it can be extremely difficult to grasp mathematical concepts or seem very daunting come finals season. Do a simple Google search for 3D graphing calculators or a program to visualize the oftentimes confusing/vague shapes formed by multivariable functions. It really helps when you can see the quadric surface, vector field, or gradient. I highly recommend the website Better Explained (Vector calculus link here: https://betterexplained.com/articles/category/math/vector-calculus/). It explains in a more conceptual way what it is that you are actually calculating. It was so helpful for me in studying for the final because it helped me realize there were overarching themes in the course. It is so much easier to get a big picture then add in details than it is to memorize details then try to connect them all. While the above resource was great for studying, it would have been even more helpful at the beginning. Khan Academy is another great source to accompany the weekly readings (especially if you prefer watching videos to reading) as you can rewind, slow it down, speed it up, and see a couple of easy examples.

What Students Should Know About This Course For Purposes Of Course Selection

The only reason I imagine you would be taking this course is because it's a prerequisite or engineering requirement, so yeah. However, perhaps in regards to choosing an instructor/time, if you have the flexibility in your schedule try to get a time when you will actually be awake and lucid. I, for one, am not a morning person, so I always go for the 12:30 math classes. Additionally, (not to saying anything against certain professors but this can be a real challenge for some people) if you know you really aren't a "math person" try your hardest to get an instructor who doesn't have too strong an accent. You have the ability to change sections in the first couple of weeks. I started in a section in which I had to focus intensely to follow along (and I already knew the concept being taught), so I switched out (same time) to another instructor. Therefore, if you already know going in that MAT 201 will be a struggle, try to minimize the challenges facing you in understanding the course.