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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.

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    Why two months? you really need to give yourself time. Math is a language. I can't see myself speaking Klingon in two month. May 11, 2021 at 5:48
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    What kind of answer are you expecting? You can catch up by sitting on your butt and studying really hard for as long as you have available - isn’t that the obvious answer? In other words, if there were a magical shortcut for how to study a lot of math effectively in a short amount of time, wouldn’t people have already spread the word about it, and wouldn’t its inventor be a famous billionaire everyone admires, etc?
    – Dan Romik
    May 11, 2021 at 6:03
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    Anyway, good luck with the math studying and the PhD!
    – Dan Romik
    May 11, 2021 at 6:03
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    Many universities will allow grad students to enroll in undergrad math courses (or special "grad" courses that are really just undergrad math). If there are significant gaps in your background, you might want to consider enrolling in some of these courses. You should discuss this with your advisor or department director of graduate studies. May 11, 2021 at 13:53
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    I understand a number of years ago a fellow named Euclid pointed out there's no royal road to geometry. Pretty much states the case for you. Also might want to skip the "wannas" and "dunnos" if you're trying to attract serious answers. May 13, 2021 at 17:24

5 Answers 5

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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.

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    Reference: terrytao.wordpress.com/career-advice/…
    – J W
    May 14, 2021 at 11:36
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    @JW Thanks for pointing out to the reference. Ugggh! How could I possibly forget that it was the mathematical genius Terrence Tao himself. I appreciate it though. Thanks again. May 14, 2021 at 15:59
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    Glad to be able to help. You might want to note regarding "post-rigorous mathematics" that Tao writes: 'The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory.' Is that what you were getting at?
    – J W
    May 16, 2021 at 12:19
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    Completetly loved it! @JW. However, why I rephrased the Post-rigorous definition is to put a perspective in place that even after you've understood all the complexities and subtleties of mathematics, you're still at times stuck at problems that crack you up. So, the only difference that sets you apart in this phase is that you don't lose interest, but rather come back to it, time and again. That's what I meant "you've made peace...". Thanks again! May 16, 2021 at 12:26
  • Thanks for clarifying that.
    – J W
    May 16, 2021 at 12:30
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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.

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[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.

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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.

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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.

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