I lack talent in pure mathematics. (Although applying mathematical methods has been natural for me.) Im a physics freshman. I can prove and I can self study mathematics given enough time. I guess it's still too early to decide, but, I suppose, I'll be doing theoretical/mathematical physics in grad school because I really like mathematics and physics - and I find experiments to be tedious. I am preoccupied with mathematics. I feel that if I don't learn enough pure math I won't be as good in its applications to physics. The math only courses I'll be taking are Calc I, Calc II, and ODE. The rest are in math methods classes. The curriculum is fixed so I can't do anything about it, besides self studying the gaps. For instance, in Multivariable Calculus Theory and Application by K. Kuttler linear algebra is required for vector calc. But my other book, cookbook calculus, goes up to stoke's theorem without linear algebra. I have two options: 1. Study vector calculus without linear algebra. 2. Study linear algebra, then study vector calculus. The problem is that self-studying is really hard for someone with no formal schooling nor talent for pure mathematics. I also don't have the time. I'm at a loss on which manner should I proceed with my mathematics education. How should I study math? I.e. do I really need to learn the math or just the methods?
Go with: 2. Study linear algebra, then study vector calculus. You need Linear algebra for physics. Partially QM
In my experience, studying pure math doesn't help at all with physics. I've taken a fair amount of pure math courses, even some grad classes, and I still don't see a good connection between the two. A physics course will teach you the math along with the physics anyway. Knowing how to use the math will help immensely more than the proofs of it. Physicists are incredible at approximating and mathematicians are great at generalizing. Choose your poison. It's two different mindsets for each and I haven't seen many make the connection, I know I certainly haven't. On another note, this is my last semester of classes to finish my PhD and I've found that if you thoroughly know Calc 1-3, linear algebra, and differential equations you'll be able to handle anything in your physics courses.
Depends on what kind of physics you are going to do. Studying certain things in pure math gives great insight into physics. I'm more of a mathematician. Knowing some pretty deep bits of math are the key to my understanding of certain physics concepts. This is coming from the perspective of someone who requires a very deep, intuitive understanding of what they study. There are lots of examples. For example, Hamilton's equations in classical mechanics have a beautiful geometrical meaning and are motivated by the analogy between optics and mechanics. As far as I can tell, the best way to get at this meaning is to use a decent amount of pure math stuff. Namely, some differential forms, symplectic geometry, and maybe contact geometry. There is some nice hyperbolic geometry to be found in the space-time of special relativity. Also, in close relation to this, the Mobius transformations of complex analysis show up. General relativity makes heavy use of differential geometry. If you use more sophisticated math, you'll understand it better. In fact, Penrose introduced techniques of differential topology that are now standard in the field because he came from more of a math background. This allowed him to prove a theorem to the effect that black holes always have singularities in them. Gauge theories in particle physics seem to be best understood using the theory of principal bundles and connections. And the list could go on. The way to understand many things in physics on a deep level seems to me to, at least sometimes, involve some pretty deep math. The separation of math and physics seems to me to be pretty detrimental to both fields. Math misses out on a lot of inspiration and physics misses a lot of deep conceptual insights. On the other hand, it is possible for physicists to over-do it with the math. Seems like the physicists are too eager to calculate without wanting to know the deeper meaning and the mathematicians are too steeped in abstraction. That's at least the impression I am under. For a good overview of a lot of the things that I have mentioned, you can take a look at Penrose's book, The Road to Reality.
^ problem is, I find it really hard to study pure mathematics. Sometimes I study at the rate of 1hr a page. It's painful actually. But I really like pure math. It's wonderful. I just find it hard right now, being a complete beginner and all. Are there some fairly easy math books that are rigorous and intuitive at the same time?
Partly, this is due to the silly way that many math books are written. Very inefficient way of learning. One option is to try to come up with parts of the proofs for yourself, rather than reverse-engineer proofs from a book. It's good to have some practice reading formal proofs, though. The best book is Visual Complex Analysis, but it's not that rigorous. But I think that is okay. If you want intuitive and rigorous, your best bet is to try to use more than one book in a lot of cases. But then, sometimes, it can be hard to translate fully from one approach to the other. All of Vladimir Arnold's books are good, but not always easy.
There are indeed. Axler's Linear Algebra done right reads like a novel - its fantastic. Spivak's calculus is also easy to read, and provides some nice examples. Both of these books can easily be studied without any backround, even without previous knowledge of proofs. Vector Calculus by Colley is a good multivariable/vector calc book as well.
If you're concentrating on undergraduate physics only, then it may be so. However, graduate level topics in physics, and even some at the undergraduate level, require knowledge of mathematics beyond these areas as homeomorphic nicely put it. For example, half of the book "Mathematics for Physics" by Stone and Goldbart deals with topics like group theory, topology, differential geometry, lie theory, complex analysis, and some real/functional analysis. For me, knowing the mathematics has often offered deeper insight into the physics. Yes there are quite a few. See this recent thread: https://www.physicsforums.com/showthread.php?t=589045
Like I said earlier, I am finishing up my PhD coursework this semester so clearly I'm talking not about undergrad physics. There are fields within physics that do utilize high level math but it's special cases of the math. Those powerful generalizing theorems in pure math are interesting but physics only uses small chunks of it. After all these years, I believe that if someone wants to learn physics then take every physics class possible then fill in the needed gaps with some math classes. It's completely irrational to think that starting from pure math then applying it to physics is advantageous.
@ homeomorphic You might be one of those rare breeds of mathematicians if you're connecting all those dots within math and physics. That's a very special ability and I haven't seen many do that over the years.
Not true. As I said, that depends on what you want to do. Some people don't really want to be physicists; they want to be mathematicians who work on physics-related problems. And they are not useless people. They have a role to play. Everyone has a role. It's also not just a matter of "wanting to do physics". It's hard to explain what I mean by that, but for example, I had a math prof who said math was always just easier. It seemed like he "wanted" to do physics. But he was better at math, so he did math. In my own case, I was the same way. I wanted to be a physicist. But that didn't really work out that well for me. Neither did math, actually, in some ways, but it seemed more congenial to my way of thinking than the way physics was taught in my undergrad, so I went with math, hoping to eventually get back to physics. It wasn't a question of thinking that that was the best way to do physics. It was more like what I had to do because of my way of doing things. To some extent, it worked. I found a lot of what was missing in my classical mechanics class that was instrumental in turning me away from physics by studying math. It looks like I might not make it in either physics or math because I refused to be a cog in the machine. It had to be my way or the highway. But it's very difficult when you have to swim against the stream. Math and physics are difficult enough even without having to swim against the stream.
I have to agree with lsaldana. I dont see how you cant at least feel crutched without feeling like you know at least some of the topic lsaldana discusses. Group theory is important in solid state and QFT. Functional derivatives are used all the time. Complex analysis is just a given .