How can I quickly learn advanced math concepts for my PhD in engineering?

Question

I am a PhD student majoring in Medical Engineering. My interest in Math is high, but my knowledge isn't. I have just a bit of knowledge in Calculus, Linear Algebra and a little Statistics (just can do mean, median and mode). When I entered my PhD, I was really surprised at the complexity of the concepts we were expected to know, like SLAM, Green's Theorem, Cholesky factorization, etc.

How can I catch up? The time constraint is really short (2 months), and I want to really understand all the concepts to the point where I can derive them. However, if I am stuck, I don't want to see the answer since it makes me feel depressed and that makes me realize that I really lack in fundamentals.

I really like Mathematics, but after seeing a lot of Math problems on YouTube where I cannot even get close to the solution, I am questioning whether I am fit for Engineering or not.

Answer 1

As Terrence Tao explains it in his blog, Terrence Tao's Blog:

There are three stages of understanding the mathematical intricacies:

• Pre-rigorous Mathematics (Undergraduate phase), where you understand mathematics by the help of formulae and calculations thus giving you clear insights about what is going on.
• Rigorous Mathematics (Graduate level studies), where you understand proof writing, understanding the formal language of mathematics and
• Post-Rigorous Mathematics, where you've pretty much made peace with the difficulties, bumps, failures and successes and still enjoy problem solving.

Feeling imposter right now is quite normal, since I believe everyone does during several parts of their academic careers. It is based on someone else's understanding of something, based on which we judge our capabilities and want to coop in the same manner and end up rushing and getting badly affected. It is clear that two months are never sufficient to understand something of this magnitude (Would it be called the queen of all sciences, if it was so easy to get the queen?)

I do feel, it is important to prepare a hierarchy of the ideas or concepts that you should be focusing on clearing rather than putting your hands everywhere. Make a clear strategy as to what is necessary at this point, prepare a timeline and don't be harsh on yourself.

Answer 2

The truth is: There is no magic bullet. You didn't learn engineering in two months, and you won't learn math in two months.

If there are specific mathematical concepts you need for your work, you can of course select books from that area. If self-study is not fast enough, you can also consider paying a tutor to walk you through material. But short of that, mathematics is like any other field and language: It requires years or studying and practice to become fluent and productive in.

Answer 3

[As per request by J W, upgraded from comment:]

There is a kind of magic bullet: if you can identify what you need, and a use case, and begin by studying first only what you absolutely need for that, so that you can get it to work - it is sometimes surprisingly easy to understand the surrounding theory when you tried to get something to work and then understand why things need to be the way they are.

It's not always possible, and sometimes you need to do some basic groundwork first, but if you can do it, this "pull" strategy can accelerate understanding and motivation a lot.

Answer 4

Euclid is reported to have said "there is no royal road to mathematics". Live with it.

Becoming proficient in any field, be it math, engineering, swimming, will take a lot of work.

Answer 5

For years now I have struggled with this: having to teach myself complex mathematics years after my CS undergrad, in order to better understand my research topic.

There is no quick solution, but here are some pointers which helped me improve:

1. Figure out what you really need to learn, depending on your field. This comes from looking at many papers on your research topic. For example, I work on applied Natural Language Processing, so the math topics which are really important are mostly multivariate probability, statistics, and Information Theory. Optimization is somewhat important. Calculus is less important. Much less important is proof-writing and learning how to prove convergence.
2. Once you have made a list of topics to learn, to actually learn these topics, find a good textbook/course which teaches them at the right depth for you to understand.
   • For example, a lot of Probability textbooks introduce the topic from the perspective of measure theory. This is great for budding probabalists, but it adds a level of complexity which is totally useless for me. After much searching, I found a great book (Blitzstein & Hwang's Intro to Probability) which was at the perfect depth for me.
   • There's no shame in picking up an undergrad or even high-school textbook or course: even if you have to start from literally 0. Some of them are incredible learning resources. Remember that it's better to end up with a deep understanding, rather than let your pride get in the way of your learning journey. Throughout our careers, everyone ends up learning and re-learning math concepts at increasing levels of depth.
   • Also, refer to multiple textbooks. It's rare for you to ever understand one textbook end-to-end; there's going to be topics where the author's explanation seems too complex or not nuanced enough. In times like this, look at other resources.
3. Dedicate time to complete these topics one-by-one. Each month, try to spend the equivalent of 1-2 days per week to learn a new topic. When doing so, really go deep into it, read from multiple sources, make detailed notes (handwritten on paper/OneNote or typed notes), and solve practice exercises. Reading the material without engaging with it is like learning to be a doctor from reading medical textbooks. The practice of math is the crucial bit.
   • I used to feel like spending time learning these "basic" topics would slow down my research, since I was taking time away from running experiments/reading; the exact opposite is true! By learning the necessary math, I was able to read papers much faster, understand when some work was just a basic application of something well-known and when it was truly surprising, and pre-emptively cull fruitless ideas (which would have otherwise wasted months!). It really opened up a wider spectrum of research for me to understand, and made me a lot more confident engaging with much of the literature.

TL;DR: the math of your research topic formalizes the most crucial ways people are currently thinking about it. It's worthwhile spending a lot of time getting up to speed with it.

Answer 6

You mention having studied single variable calculus and linear algebra. Green's Theorem is a result from multivariable/vector calculus, a course commonly expected to have been taken as part of an undergraduate engineering degree. To get started on vector calculus, you might consider Robert Ghrist's excellent set of video lectures: Calculus BLUE.

The Cholesky factorization is perhaps not emphasized in all introductory linear algebra courses. Have you seen the LU decomposition? If you're familiar with that, it isn't a big step up to Cholesky. See Gilbert Strang's video lectures on MIT OpenCourseware to refresh/learn topics in linear algebra.

Warning: do not assume that just watching videos is enough to understand; it's a starting point, but you'll still need to do exercises and check or get feedback on your answers. For this you'll need a book or other resource with answers and/or a tutor. Note also Michael Field's advice in his preface to Essential Real Analysis, Springer 2017:

On occasions I advise students in my analysis classes not to spend too much time reading mathematics texts. That view is based on my own experience—an effective way to learn mathematics is to do it, play with it but generally avoid spending too much time reading books about it. Reading a mathematics book can give a veneer of superficial understanding that dissolves the moment one tries to use the theory described in the book. An analogy might be learning carpentry, plumbing or a foreign language—knowing the theory is important but not that helpful; knowing how to use the tools is crucial. That takes time, practice and serious effort.

It isn't clear to me how much knowledge of probability and statistics is required for your particular type of engineering, but "mean, median and mode" are indeed unlikely to be enough. At the very least, find out whether your courses need more probability or more statistics and focus on getting up to speed with the basics of the one that's used most or at least first.

This is to get you started and help fill in some gaps or areas where your knowledge might be rusty. It will take a lot of work, but once you have strengthened your foundations, you may find that you are "fit for Engineering" after all. As others correctly say, though, there are no shortcuts*. It will probably be a bumpy and painful road for quite some time to come.

* That said, see Captain Emacs' answer; sometimes a more targeted approach can save you time.