Crash Course in Math for Grad School

Forum for the GRE subject test in mathematics.
maath
Posts: 5
Joined: Mon Jan 26, 2015 8:45 pm

Crash Course in Math for Grad School

Postby maath » Mon Jan 26, 2015 9:06 pm

Hi I'm an international Engineering graduate interested in interdisciplinary study which involves Applied Math. This year, I will be enrolled in grad school either in my home country (I've already been admitted) or hopefully one of my US applications will work out (no results released yet). In either case, I want to be (at least in part) prepared for graduate study in Applied Math. Kindly help recommend books that will give me just enough knowledge in various fields of Math, so as not to be totally lost in my interdisciplinary study.

My research interests are in the numerical solution of PDEs, preferably meshfree methods. Besides calculus and analysis, what other fields of Math should I be looking at, and what books would you recommend? I need other titles that will not be too vast but can give just enough foundational knowledge to hold a sensible conversation (as if I were, say, a junior Math major) in each of those fields of Math.

For example:
Multivariable calculus, I got Larson's 9th Edition and I am already studying it.

Thanks.

Edit: I took the most recent (October) GRE subject test and it was rather bad at 29%.

unitofobscurity
Posts: 8
Joined: Sun Dec 21, 2014 3:30 am

Re: Crash Course in Math for Grad School

Postby unitofobscurity » Tue Jan 27, 2015 12:07 am

What are you interested in within applied math, specifically? It is a big place :)

Also, I think it's really hard, from the standpoint of a textbook author, to set out to teach you as little as possible about something. In my experience, even applied textbooks give you some extra "big picture" stuff in order to help you understand what's going on in the particular.

For example, if you're doing anything with PDE you should really read Vladimir Arnold's book "Ordinary Differential Equations" — it's very grounded in problem solving and is extremely lucid. It would also be misleading to say that it covers the bare minimum.

maath
Posts: 5
Joined: Mon Jan 26, 2015 8:45 pm

Re: Crash Course in Math for Grad School

Postby maath » Tue Jan 27, 2015 7:25 am

unitofobscurity wrote:What are you interested in within applied math, specifically? It is a big place :)

Also, I think it's really hard, from the standpoint of a textbook author, to set out to teach you as little as possible about something. In my experience, even applied textbooks give you some extra "big picture" stuff in order to help you understand what's going on in the particular.

For example, if you're doing anything with PDE you should really read Vladimir Arnold's book "Ordinary Differential Equations" — it's very grounded in problem solving and is extremely lucid. It would also be misleading to say that it covers the bare minimum.


Thanks! I am interested in numerical solutions of PDE, but the way I see it - it looks like other there are other topics in Math besides calculus and Analysis, that are needed as a foundation for such research. For example, I read an abstract of a paper and it seemed to venture into topology as well. This is what I am hoping to clarify: what other topics I basically need to know besides Calculus and Analysis, and if possible suggestions of books to look out for.

I'm grateful for the tip on Arnold's book. I will seek it out. I was checking to see if he has one on PDE as well but I don't think so. Any recommendations for a well grounded introduction to PDE?

unitofobscurity
Posts: 8
Joined: Sun Dec 21, 2014 3:30 am

Re: Crash Course in Math for Grad School

Postby unitofobscurity » Tue Jan 27, 2015 8:46 am

I am not in PDE, but my impression is that PDE is a bit like number theory, in that almost anything can turn up at the edge of the field. One upon a time I read part of this book by Brezis (http://www.amazon.com/Functional-Analys ... 0387709134) for a functional analysis course, because one of the quickest answers to "why should I care what Hilbert, Banach, Frechet, &c spaces are" is "you can use them in PDE."

I guess if I were going into numerical PDE, I would probably want to know stuff like "what is a measure space," "what is a Banach space," "what is a (Schwartz) distribution," "what does the Fourier transform look like on different spaces," "what is a manifold," and so on in order to be able to talk to more theoretical PDE folks, alongside the more obvious "what are the Euler-Lagrange equations," (did you include calc of variations when you said calculus?), "how do I compute how long this numerical algorithm will take," "what is Stokes' theorem in arbitrary dimension," "how do I quickly find global extrema without the analytic form of the system," et cetera.

Basically, not to condescend, but if you've only done (vector?) calculus and (real and complex?) analysis at this point, all of the other standard upper-division undergraduate math courses contain stuff that will come in handy for some part of PDE. Of course, it's hard to do an entire math degree in your spare time (I should know, I switched from physics close to the end of my undergrad), but even ostensibly useless stuff like Galois theory would come in handy, since it introduces you to modes of argument that are ubiquitous in math after WWII. The Arnold book is definitely where I would start though. "Calculus of variations" by Gelfand and Fomin is also really great, if you have not seen that subject in depth yet.

maath
Posts: 5
Joined: Mon Jan 26, 2015 8:45 pm

Re: Crash Course in Math for Grad School

Postby maath » Tue Jan 27, 2015 8:42 pm

unitofobscurity wrote:I am not in PDE, but my impression is that PDE is a bit like number theory, in that almost anything can turn up at the edge of the field. One upon a time I read part of this book by Brezis (http://www.amazon.com/Functional-Analys ... 0387709134) for a functional analysis course, because one of the quickest answers to "why should I care what Hilbert, Banach, Frechet, &c spaces are" is "you can use them in PDE."

I guess if I were going into numerical PDE, I would probably want to know stuff like "what is a measure space," "what is a Banach space," "what is a (Schwartz) distribution," "what does the Fourier transform look like on different spaces," "what is a manifold," and so on in order to be able to talk to more theoretical PDE folks, alongside the more obvious "what are the Euler-Lagrange equations," (did you include calc of variations when you said calculus?), "how do I compute how long this numerical algorithm will take," "what is Stokes' theorem in arbitrary dimension," "how do I quickly find global extrema without the analytic form of the system," et cetera.

Basically, not to condescend, but if you've only done (vector?) calculus and (real and complex?) analysis at this point, all of the other standard upper-division undergraduate math courses contain stuff that will come in handy for some part of PDE. Of course, it's hard to do an entire math degree in your spare time (I should know, I switched from physics close to the end of my undergrad), but even ostensibly useless stuff like Galois theory would come in handy, since it introduces you to modes of argument that are ubiquitous in math after WWII. The Arnold book is definitely where I would start though. "Calculus of variations" by Gelfand and Fomin is also really great, if you have not seen that subject in depth yet.


This is even more useful than you might probably imagine. Thank you, mate!

P.S. I found these. Can serve as a reference in case anyone in the future is in my shoes. Digging in already.

From http://math.ucr.edu/home/baez/books.html
How to Learn Math

Math is a much more diverse subject than physics, in a way: there are lots of branches you can learn without needing to know other branches first... though you only deeply understand a subject after you see how it relates to all the others!
After basic schooling, the customary track through math starts with a bit of:

Finite mathematics (combinatorics)
Calculus
Multivariable calculus
Linear algebra
Ordinary differential equations
Partial differential equations
Complex analysis
Real analysis
Topology
Set theory and logic
and
Abstract algebra


not necessarily in exactly this order. (For example, you need to know a little set theory and logic to really understand what a proof is.) Then, the study of math branches out into a dizzying variety of more advanced topics! It's hard to get the "big picture" of mathematics until you've gone fairly far into it; indeed, the more I learn, the more I laugh at my previous pathetically naive ideas of what math is "all about". But if you want a glimpse, try these books:
F. William Lawvere and Stephen H. Schanuel, Conceptual Mathematics: a First Introduction to Categories, Cambridge University Press, 1997. (A great place to start.)
Saunders Mac Lane, Mathematics, Form and Function, Springer-Verlag, New York, 1986. (More advanced.)
Jean Dieudonne, A Panorama of Pure Mathematics, as seen by N. Bourbaki, translated by I.G. Macdonald, Academic Press, 1982. (Very advanced - best if you know a lot of math already. Beware: many people disagree with Bourbaki's outlook.)

I haven't decided on my favorite books on all the basic math topics, but here are a few. In this list I'm trying to pick the clearest books I know, not the deepest ones - you'll want to dig deeper later:

Finite mathematics (combinatorics):

Ronald L. Graham, Donald Knuth, and Oren Patshnik, Concrete Mathematics, Addison-Wesley, Reading, Massachusetts, 1994. (Too advanced for a first course in finite mathematics, but this book is fun - quirky, full of jokes, it'll teach you tricks for counting stuff that will blow your friends minds!)
Calculus:

Silvanus P. Thompson, Calculus Made Easy, St. Martin's Press, 1914. Also available free online at http://www.gutenberg.org/ebooks/33283. (Most college calculus texts weigh a ton; this one does not - it just gets to the point. This is how I learned calculus: my uncle gave me a copy.)
Gilbert Strang, Calculus, Wellesley-Cambridge Press, Cambridge, 1991. Also available free online at http://ocw.mit.edu/ans7870/resources/St ... ngtext.htm. (Another classic, with lots of applications to real-world problems.)
Multivariable calculus:

James Nearing, Mathematical Tools for Physics, available at http://www.physics.miami.edu/~nearing/mathmethods/. See especially the sections on multvariable calculus, vector calculus 1, and vector calculus 2. (Very nice explanations!)
George Cain and James Herod, Multivariable Calculus. Available free online at http://www.math.gatech.edu/~cain/notes/calculus.html
Linear algebra:

I don't have any favorite linear algebra books, so I'll just list some free ones:

Keith Matthews, Elementary Linear Algebra, available free online at http://www.numbertheory.org/book/.
Jim Hefferon, Linear Algebra, available free online at http://joshua.smcvt.edu/linalg.html/.
Robert A. Beezer, A First Course in Linear Algebra, available free online at http://linear.ups.edu/.
Ordinary differential equations - some free online books:

Bob Terrell, Notes on Differential Equations, available free online at http://www.math.cornell.edu/~bterrell/dn.pdf. (Does both ordinary and partial differential equations.)
James Nearing, Mathematical Tools for Physics, available at http://www.physics.miami.edu/~nearing/mathmethods/. See especially the sections on ordinary differential equations and Fourier series (which are good for solving such equations).
Partial differential equations - some free online books:

Bob Terrell, Notes on Differential Equations, available free online at http://www.math.cornell.edu/~bterrell/dn.pdf. (Does both ordinary and partial differential equations.)
James Nearing, Mathematical Tools for Physics, available at http://www.physics.miami.edu/~nearing/mathmethods/. See especially the section on partial differential equations.
Complex analysis:

George Cain, Complex Analysis, available free online at http://www.math.gatech.edu/~cain/winter99/complex.html. (How can you not like free online?)
James Ward Brown and Ruel V. Churchill, Complex Variables and Applications, McGraw-Hill, New York, 2003. (A practical introduction to complex analysis.)
Serge Lang, Complex Analysis, Springer, Berlin, 1999. (More advanced.)
Real analysis:

Richard R. Goldberg, Methods of Real Analysis, Wiley, New York, 1976. (A gentle introduction.)
Halsey L. Royden, Real Analysis, Prentice Hall, New York, 1988. (A bit more deep; here you get Lebesgue integration and measure spaces.)
Topology:

James R. Munkres, Topology, James R. Munkres, Prentice Hall, New York, 1999.
Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, Dover, New York, 1995. (It's fun to see how crazy topological spaces can get: also, counterexamples help you understand definitions and theorems. But, don't get fooled into thinking this stuff is the point of topology!)
Set theory and logic:

Herbert B. Enderton, Elements of Set Theory, Academic Press, New York, 1977.
Herbert B. Enderton, A Mathematical Introduction to Logic, Academic Press, New York, 2000.
F. William Lawvere and Robert Rosebrugh, Sets for Mathematics, Cambridge U. Press, Cambridge, 2002. (An unorthodox choice, since this book takes an approach based on category theory instead of the old-fashioned Zermelo-Fraenkel axioms. But this is the wave of the future, so you might as well hop on now!)
Abstract algebra:

I didn't like abstract algebra as an undergrad. Now I love it! Textbooks that seem pleasant now seemed dry as dust back then. So, I'm not confident that I could recommend an all-around textbook on algebra that my earlier self would have enjoyed. But, I would have liked these:

Hermann Weyl, Symmetry, Princeton University Press, Princeton, New Jersey, 1983. (Before diving into group theory, find out why it's fun.)
Ian Stewart, Galois Theory, 3rd edition, Chapman and Hall, New York, 2004. (A fun-filled introduction to a wonderful application of group theory that's often explained very badly.)




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