Exploring Mathematics: A Request for Suggestions

[poobar]

Hey all,
I am finishing the second year of the undergraduate physics program at my school. This involves taking a course which teaches us a lot of upper level (or so I am told) math in the span of one semester at a very fast pace. The class is very difficult, but it has sort of given me a huge appetite for mathematics.

So, I would like to ask for people who read this post to give a list of any and all mathematic subjects which will be good to know, are particularly useful, or even just plain interesting for me to study. It would also be a lot of help if the subjects are listed so that they indicate what must be learned first in order to learn other more complicated subjects. I have already learned Calculus I & II, Multivariable, and Differential Equations. I know Linear Algebra is on the list of things to learn, and I will be exploring that soon.

Thanks!

 

[armolinasf]

I was an econ major but I had to calculus and that's converted me to a physics major and my appetite for math has also increased, so this same question has been on my mind. Interested to hear to what people have to say.

 

[micromass]

linear algebra should be on the top of your list. It's very useful in physics, but I don't consider in "interesting". It's more a handy tool that you should really know.

If you mastered linear algebra, then maybe you can do some functional analysis. This is very useful in quantum mechanics.

Topology and differential geometry can also be worthwile to study if you're going into general relativity.

 

[Robert1986]

Abstract Algebra. With respect to what prerequsites for Abstract Algebra, there really isn't a lot of stuff that you need to know to understand the material, per se. However, this is a class that requires mathematical maturity, which your calculus classes gives you (not only that, examples from calculus are used a lot in Algebra), rather than needing to know stuff about Topic X. Algebra is my favorite. Also, you can't discount good, old fashioned number theory. As Gauss said, mathematics is the queen of the sciences and number theory is the queen of math. There probably isn't a whole lot that you can "use" number theory for, but who cares?Another interesting area is Combinatorics. This encompasses a lot. I have several combo books, but for an introduction, I like this book by a prof from the Georgia Institute of Technology: http://people.math.gatech.edu/~trotter/
the link is on that page.

 

[armolinasf]

can somebody give me explain or give me a link to something that does explain exactly what topology and differential geometry actually are, specifically the latter and how they differ and what prereqs you need to understand it.

 

[micromass]

can somebody give me explain or give me a link to something that does explain exactly what topology and differential geometry actually are, specifically the latter and how they differ and what prereqs you need to understand it.

Differential geometry is the study of geometric objects which are "smooth". For example, a circle and a ball in space are geometric objects and they are very nice/smooth. A donut-shaped form is also an example. An object which is not smooth is a cube: the edges form a rough transisition.
Now, what differential geometry tries to do is to study all of these objects and define things like tangent spaces on the object, tangent vectors, vector fields,...

To study differential geometry, I would guess that just some knowledge of calculus would be enough. A knowledge of analysis would be even better! "A comprehensive introduction to differential geometry" by Spivak is a great book to learn.
Of course, if you want to read more advanced text, then a knowledge of topology is indispensible.

It is hard to say what topology is exactly. In some ways, topology forms the framework for a lot of mathematical subjects. The aim of topology is to generalize how close points are to each other. For example: 1 is closer to 2, then 0 is to 2. This generalization is studied in topology (but it won't be apparent at first that this is what they're trying to do!). Topology defines a lot of things: when spaces are connected, when they are metric spaces,...
To learn topology, a knowledge of metric spaces is necessary to learn topology. In particular, you'll have to be acquanted with open and closed sets, compactness, continuity in metric spaces,... So I guess a basic course in analysis should be a requirement...

 

[Chaostamer]

To learn topology, a knowledge of metric spaces is necessary to learn topology. In particular, you'll have to be acquanted with open and closed sets, compactness, continuity in metric spaces,... So I guess a basic course in analysis should be a requirement...

I don't think that's necessarily true. The topology course at my university starts by introducing metric space theory and generalizes to topological spaces from there. The only strict requirement for a topology course that starts with metric spaces, then, is set theory and a good bit of comfort with mathematical abstraction.

 

[Chaostamer]

As for recommendations, subjects that may be of some use in physics are partial differential equations, linear algebra, numerical analysis, differential geometry, abstract algebra, and complex analysis. I have little experience with physics, but I've been told that those are useful classes.

 

[poobar]

linear algebra should be on the top of your list. It's very useful in physics, but I don't consider in "interesting". It's more a handy tool that you should really know.

could anyone suggest a good linear algebra book or give a link to a good online text? this would be really helpful. thanks!

 

[Whitishcube]

My Elementary Linear Algebra textbook was pretty decent. Linear Algebra by Lay is quite decent for a start. Its pretty easy to understand for a newcomer to linear algebra.