Master a book on applied mathematical methods used in physics early
Physics classes teach the physics, not the mathematical methods required to solve the problems. Sometimes they will dedicate a small amount of time to mathematical methods, but the limited time available forces them to prioritize the physics.
Physics books have a similar problem: they teach the physics, not the math. These books are rarely self-contained. This is due to the fact that physics more-or-less depends on all applied mathematics and is not really able to restrict itself to a specific domain of mathematics. Instead, any physics course can require arbitrary methods in probability, statistics, linear algebra, infinite series, multi-variable calculus, vector calculus, complex analysis, fourier transforms, integral transformations, differential equations, etc. The consequence of this is that it’s implied that the reader already be acquainted with the mathematical methods, and physics books typically don’t specify the pre-requisites to working through the book.
Physics departments largely leave the responsibilities of teaching math to the math departments, but the math departments are not equipped for this task. Rather, math departments will teach at a level appropriate for the average student taking each course. This is to say if most of the students in a linear algebra course are economics majors, your linear algebra course will be mostly unrigorous for physics students and won’t introduce them to the mathematical methods that they require for their physics courses. Also, math departments tend to lean towards proofs and not applied techniques, so your linear algebra class may be more oriented towards vector spaces and proving properties in linear algebra rather than optimizing for the applied approach that physics students need.
So what are you to do? Unfortunately, you can’t count on getting the right education from the institutions that are supposed to provide it to you. Instead, your best approach is to read a book designed to teach the mathematical methods required for the physical sciences. The first and most famous book with this intent for undergraduates was made by Mary Boas and has the apt name of Mathematical Methods in the Physical Sciences.
Personally I’ve found that the coverage of Mathematical Methods in the Physical Sciences was better preparation for physics than all of the math courses I ever took combined. Boas’ book does not focus on proofs like math courses and is oriented specifically towards students in the physical sciences who need a rigorous foundation in applied mathematics.
It’s not enough to just read this book, you also have to do the practice problems. There is a world of difference between following a derivation explained to you and creating your own. These books should also have a companion book for explaining the answers to the practice questions, not just with correct answers but actually with all the steps. This greatly reduces the feedback loop of not understanding how to use these methods and is arguably where the most important learning occurs.
The earlier you master these methods, the more time you’ll save. Preferably you’d master this within your first year of studying physics, and hopefully even before that.
Master Mathematica
Mathematica is perhaps the best software package for symbolic computation (aka computer algebra). Symbolic computation allows you to use a computer to transform or reduce mathematical expressions. Symbolic expressions are more-or-less the language of physics. Everything is math, but that math is expressed in terms of variables such as x, y, cosine(z), e^{a+b}, etc.
Mathematical expressions in physics can be very large, often requiring more then ten terms a piece, complicated nested functions, lots of subscripts/exponents, and many variables. Reducing these expressions by hand can be very error-prone. Complex expressions are sometimes chosen such that they reduce to a very simple form, at which point you know you’ve finished. Other times, they reduce to some form that cannot be transformed further. If you make one small mistake, you could change a nicely-reducing expression into a much more complex one. Doing this not only makes your answer incorrect, but also acts as a timesink, where you labor down the wrong path working on expressions much more complicated than were actually intended.
Having a computer check your work is invaluable. It ensures correctness, saves time, and makes it easier to understand your mistakes quickly. Short feedback loops allow you to learn faster.
Get at least a minor in mathematics
Physics majors are supposed to be majoring in math, or at least getting a minor in it. This holds regardless of what is explicitly said because there is not enough time for physics courses to teach the mathematical methods required. Most physics programs will have a minimum requirement of math courses for the major, but practically speaking you should go above that requirement. For your benefit, you should take rigorous courses in the following: single-variable calculus, multivariable calculus, linear algebra, complex analysis, differential equations, fourier transforms, and maybe probability. You want these courses to emphasize the applied aspect (using techniques to solve problems), not the pure aspect (using proofs to prove major theorems).
If your math courses are not rigorous, that can be a huge problem. You will save yourself more time by taking a time-intensive, rigorous math course and solving your physics problems faster than taking an easy, unrigorous math course and banging your head against the wall on all your future physics problem sets. In the event your course is not rigorous enough, that’s time that should be spent studying mathematical methods in that field while you still have the time. When your workload becomes harder with your future physics courses, there will probably not be enough time to compensate for any unpreparedness.
Consider typing your homework
Homework is typically done by hand. Because showing your work is required and expressions can be very long, your homework sets will be extremely long. They may be 20 pages, largely comprising long expressions being duplicated with single transformations (to make the work easier to follow). If you type your homework, you can leverage copy-and-paste to save the time it takes to create your assignment and more easily recover from realizing that problem 8c requires a much longer derivation when you’ve only allocated 6 lines. This will save you from possibly developing a repetitive stress injury from constantly writing.
I recommend Mathematica for typing homework. Since it also provides symbolic computation, you can check your work in the same software you use to type your homework. This is a huge advantage over writing on paper where you can’t automatically check your work. Alternatively, you can use LaTeX, the premier typesetting program for physics and mathematics papers.
I’ve used both Mathematica and raw LaTeX to typeset homework, and prefer Mathematica due to the ability to check my work. LaTeX looks more professional but has a high learning curve and is more tedious to use given that you have to constantly compile the program to see the formatting of the document. One typo in LaTeX and things won’t render properly. If you manually trigger compilation, you can cause a repetitive stress injury since you will constantly be pushing the same pattern of keys. With Mathematica it’s what you see is what you get.