Analysis II, Boston College Spring 2021

Office Hours

My office hours are (tentatively): Tuesdays 6:00p.m-7:00p.m; Thursdays 4:00p.m-6:00p.m. All office hours are held on Zoom for now. Link can be found on Canvas.

Multivariable Calculus, Boston College Autumn 2020

Office Hours

My office hours are: Tuesdays 3:30 p.m.-5:30 p.m.; Thursdays 4:10 p.m. – 5:10 p.m. (after the last recitation); Fridays 1:00 a.m. but by appointment only, i.e. let me know in advanced you would like to meet, I can also do an earlier time on Friday, just ask me. Actually in general if you would like to meet with me outside of these times, let me know, and we can arrange some time that works for both of us.

Course Material

Notes on limits: here.

Midterm 2 practice exam: Link. Solutions: here.

Practice Problems for Midterm 3 (Last updated 11/11 12:35 A.M.): here. Solution.

Notes on Min/Max (Last updated 11/11 12:35 A.M.): here.

Quiz 1

Quiz 2

Quiz 3

Quiz 4

Quiz 5. Solution.

Quiz 6. Solution.

Midterm 3.

Notes from discussion 12/10, final exam review: 12:00 p.m., 2:00 p.m., 3:00 p.m.

Practice Final: here. Solution (last updated: 12/14 5:14 p.m.): here.

What is Multivariable Calculus Good For?

Below are some things to look into to find applications of multivariable calculus based on scientific discipline. I believe that for most of you there is value in looking into them to motivate your studies. Do not worry about not understanding what you see in them, but rather take a laid-back approach. Flip through the books, skim the videos, without the goal of trying to necessarily understand them fully, but to find something interesting, find words, sentences, pictures, diagrams, equations, expositions that peak your interest, or just something that looks cool. Then you can see if there are anything you recognize from multivariable calculus, maybe a derivative, a function, an image, or some description of some ideas that remind if of something you have learned. You can make it a game to find bits and pieces of objects you recognize.

Pure and applied math

The most direct continuation of multivariable calculus is the area called differential geometry. Roughly speaking, you can think of the multivariable calculus that we are doing to be “calculus in \mathbb{R}^n. However, an observation one can make is that properties of functions on \mathbb{R}^n that allow us to do calculus with them, such as differentiability, continuity, etc. are local properties, meaning that they are properties concerning each particular point on the function and some neighborhood around it. Differential geometry takes advantage of this to develop calculus on spaces that “locally look like \mathbb{R}^n. Such a space is called a manifold. Thus some function defined on a manifold would locally look like a function on \mathbb{R}^n, and so all the usual things from multivariable calculus applies.

Another area of math that can also be considered as a continuation of multivariable calculus is the study of differential equations. This is the study of equations involving derivatives of functions, and the goal is to solve theses equations. In this instance this means to find the original functions that give rise to those derivatives. The study of partial differential equations, which study equations involving partial derivatives of multivariable functions, is a very active area of research and one of the biggest area of study of mathematics, both pure and applied. Partial differential equations appears in all kinds of places in applied math and in all the sciences, so a good understanding of them is directly linked to a good understanding of our physical world.

Here are some sources to look into on differential geometry:

  • Calculus on Manifolds by Michael Spivak. Probably the most approachable book on differential geometry suitable for someone who has had a course in multivariable calculus. Note: only the last chapter is really new material compared to multivariable calculus; albeit the treatment of calculus in \mathbb{R}^n in the earlier chapters is much more rigorous compared to a typical undergraduate course.
  • Chapters 9 and 10 of Advanced Calculus by Loomis and Sternberg. Probably my favorite book on calculus. Similar to Spivak’s book, the earlier chapters are devoted to a rigorous treatment of calculus on \mathbb{R}^n. It even goes over some linear algebra and differential equations. You can decide for yourself how much of those earlier chapters you want to read.
  • Introduction to Smooth Manifolds by John M. Lee. This is a real textbook on differential geometry, one of the classics. This requires background in topology and is at the graduate level.

Here are some sources to look into on partial differential equations (PDEs):

  • Introduction to Partial Differential Equations by Peter J. Olver. This is where I personally learned PDEs.


Pretty much all areas of physics that have to do with studying behaviors of the physical world use multivariable functions and their calculus to construct the models we use to describe those behaviors. Some fields that come to mind that use multivariable calculus heavily are mechanics, fluid dynamics, aerodynamics, thermodynamics, electro-magnetism, and most importantly, general relativity, and many others. More specifically, general relativity was built on the foundations of differential geometry mentioned above. In fact, it was the development of differential geometry as a topic in pure mathematics in the late 19th century and early 20th century that created the machinery that allowed Einstein to develop his theory. Not surprisingly, in general relativity the universe is modeled as a manifold.

You can search the internet for fields of physics that interest you and look into the material, more than likely there would be some multivariable functions, their derivatives, and integrals.

Relativity: The Special and General Theory by Albert Einstein. I have been told by someone that this is a non-technical treatment of general relativity by Einstein himself, intended for a general audience.

3Blue1Brown’s video Divergence and curl: The language of Maxwell’s equations, fluid flow, and more


I think the most interesting use of multivariable calculus is in thermodynamics.

Economics and Finance

The Econometrics of Financial Markets by Campbell, Lo, et. al. This one doesn’t really require calculus. This is more background material for economics.

A Primer For The Mathematics Of Financial Engineering by Dan Stefanica.

High Frequency Financial Econometrics by Yacine Ait Sahalia.

Options, Futures, and Other Derivatives by Hull.

Brownian Motion and Stochastic Calculus by Ioannis Karatzas.

Stochastic Volatility Modeling by Lorenzo Bergomi.

Diffusion Processes and their Sample Paths by Kiyosi Itô and Henry P. McKean Jr.

The Misbehavior of Markets: A Fractal View of Financial Turbulence by Richard L. Hudson, Benoit Mandelbrot. This looks like a dope book that’s completely non-technical, definitely check it out; although not technically something directly related to calculus, just something interesting to look into.


Computer Science ?