# Importance of these math classes

I'm trying to decide what math classes to take. I have spoken with some of the physics faculty at my school, and each person has given me differing advice (though there is a general consensus regarding some classes). I am a physics and math major, and am trying to decide which math electives best support the physics curriculum and my plans for grad school. So far, I have taken (or will have taken by the end of this semester) Calc 1-3, Differential Equations, Applied Linear Algebra, and Intro to Abstract Math (a required intro proofs course). I would really appreciate it if some of you could rank, from most important to least important, these following classes. (Important here implies usefulness for doing physics).

-Partial Differential Equations
-Numerical Methods
-an upper division Linear Algebra course
-Differential Geometry
-Abstract Algebra 1
-Complex Variables

The general consensus that I mentioned above refers to Partial Differential Equations and Numerical Methods. Every person I have asked so far has recommended both those classes.

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All of these classes are useful for a future physicist. Which classes are the most useful to you depends on what you want to specialize in later. Can you give us that information?

As micromass pointed out, it depends what field of physics you go into!

You'd need to know about group theory if you're planning to do something with quantum gravity, so abstract algebra would be a good thing to pick up.

I would say the best things (no matter what field you go into) are partial differential equations and the upper linear algebra course.

The reason for PDEs is because a lot of equations in physics are themselves PDEs, and if you know what to expect your result to be, you'll make less mistakes.

I may be biased on the Linear algebra, but I believe having too much knowledge of linear algebra can't hurt you as many problems in physics deal with tensors which should be brought up in a upper division linear algebra class.

Good luck.

MarneMath
I tend to agree the PDE and Numerical Analysis are the most important. I can also throw in a more advance understanding of linear algebra would help. Here's the issue though. I can't imagine a math degree (I'm saying this since you say you're also a math major) that doesn't require abstract algebra course, so it seems to me like that should already be something you're going to take. It's such a fundamental part of mathematics.

If you could only pick one I woukd pick Numerical Analysis because it translates well to a lot of different projects and it really open my eyes to flaws in some program and made me more aware. If you already have a good handle on programming, maybe it'll do less for you than it did for me, but even though I didn't particularly enjoy the class, it made me a more aware person.

I would actually say PDEs and then Linear Algebra are the two most important, with numerical methods being a close third. Numerical methods are really important in physics, because many problems are only solvable analytically, but I don't feel you really have to take a whole class on it in the undergrad level.

PDEs and Linear Algebra are useful to a wide range of physics branches.

The rest are useful, and you'll need to know eventually in grad school but are not as important for an undergrad degree. I'd say Complex Variables, then Differential Geometry, then Abstract Algebra, but again, it depends. For example, if you are very interested in General Relativity, as I am, you'll definitely want to take differential geometry!

EDIT: Actually, since I read that you've already taken applied linear algebra, I might suggest just taking the numerical methods class, though to me, upper div LA would be so much more interesting.

All of these classes are useful for a future physicist. Which classes are the most useful to you depends on what you want to specialize in later. Can you give us that information?

At this point in time, I don't think I know enough to be able to make a choice. In a very general sense, I'm leaning towards theory. But whether in condensed matter, particle physics, nuclear, etc., I have no idea.

I tend to agree the PDE and Numerical Analysis are the most important. I can also throw in a more advance understanding of linear algebra would help. Here's the issue though. I can't imagine a math degree (I'm saying this since you say you're also a math major) that doesn't require abstract algebra course, so it seems to me like that should already be something you're going to take. It's such a fundamental part of mathematics.

For whatever reason, the math program at my school does not require abstract algebra. The core classes are Calc 1-3, Diff Eq, App. Linear Algebra, Intro to Abstract Math, Intro Analysis, Applied Combinatorics, Probability and Stats I, Math programming I and II (python and Maple), then we can pick 3 upper division electives. It looks like I will definitely go with PDEs and probably Numerical Methods, as I'm not yet a strong programmer so this will provide some more practice.

Here is a portion of the Linear Algebra syllabus, I don't see any mention of tensors. I appreciate the input, thank you.

The theory of vectors, vector spaces, inner product spaces,
and linear transformations, eigenvalues and canonical forms. Unlike
MATH 2050, this course emphasizes the theoretical aspects of linear
algebra.

I. Vectors and Matrices
1. Introduction to Vectors
2. Lengths and Dot Products
3. Planes
4. Linear Equations by Rows and Columns
II. Systems of Linear Equations
1. Idea of Elimination
2. Elimination Using Matrices
3. Rules for Matrix Operations
4. Inverse Matrices
5. Elimination and Factorization
6. Transposes and Permutations
III. Vectors and Subspaces
1. Spaces of Vectors
2. The Nullspace of A
3. The Rank of A
4. Independence, Basis, and Dimension
5. Dimensions of the Four Subspaces
6. Orthogonality of the Four Subspaces
IV. Determinants
1. Properties of Determinates
2. Cofactors
3. Cramer’s Rule, Inverses, and Volumes
V. Eigenvalues and Eigenvectors
1. Introduction to Eigenvectors
2. Diagonalizing a Matrix
3. Symmetric Matrices and Orthoganal Eigenvectors
4. Positive Definite Matrices
5. Similar Matrices
VI. Linear Transformations
1. Introduction to Linear Transformations
2. Matrix of a Linear Transformations
3. Choice of Basis: Similarity and Diagonalization

If you're interested in theory: I would put added emphasis on Linear Algebra and maybe even Abstract Algebra. Numerical methods are still very important, though I might argue that it's a little less important for theorists? I might be wrong about that, though.

In Linear Algebra, Tensors would be covered by matrices. It's difficult to distinguish matrices, vectors and tensors, they are all very much connected and in some respects, the same idea. A physics math course might spend some time covering tensors specifically, and a few undergrad physics courses such as upper div E/M, Relativity and maybe Classical Mechanics will make use of them.