Is it good to study math first and then physics?

[joneiljack]

Is it possible to just learn the mathematics first (i.e. multivariable and vector calculus, linear algebra, etc.) and only then start learning physics? (i.e. from a graduate textbook like Classical Mechanics by Golstein)?

Would there be any pitfalls?

I am studying on my spare time as a hobby and it seemed the fastest way to learn the math first. I have a high school physics background and I looked at MIT's OCW syllabus and they have University Physics by Hugh D. Young; Roger A. Freedman as their main book throughout their undergraduate program (more or less). I looked through that book online and it didn't seem like a type of book that would suit my preferences (too many colored pictures, over-explaining, etc.) -- I deeply apologize if this book is held in high regard, but that was just the first impression it gave me and wanted to know if I am mistaken.

I also use "Gerard t'Hooft's Theoretical Physics as a challenge" as a guide. Oddly he doesn't recommend any textbooks of that sort, and he just recommends the math and then straight up Classical Mechanics graduate textbooks. There is actually a link to some textbooks that are similar to Young and Freedman's to which he says "(most of these are rather for amusement than being essential for understanding the World),"

As for mathematics i use "How to Become a Pure Mathematician (or Statistician)" as a guide, and I was wondering:

2. If the ultimate goal is to learn physics, how is it best to approach the learning of mathematics beforehand - learning math in a pure/analytical way, or learning applied math, i.e. here is the formula, plug in, solve this integral, etc.? Or both? If so how much to focus on one or the other? Or does learning mathematics in a pure math way already imply you will know how to do the apply-math part? Or is pure math just a waste of time if the ultimate goal is to study/understand/do physics?

(I am guessing it depends on what type of physics you ultimately want to focus on, but from what I've gathered I understand that the best path is to be a good mathematician in the first place no matter the type of physics.)

And last question. Are Feynman's Lectures on Physics worth getting? I glanced through the books and they don't seem to have exercises. If money is a problem would it be better spent on books that have exercises (like the ones in Gerard t'Hooft's list)?

Thank you in advance and sorry if it's a long read. As many perspectives and opinions I can get would be much appreciated.

 

[R136a1]

Many questions here.

First, I think learning math such as calculus and differential equations before tackling classical mechanics is not a bad idea. But I don't think you should go straight to books such as Goldstein. Instead, take a look at books like Kleppner, Morin or Purcell. These tend to be quite difficult, but amazing from a physics point of view. If you go to grad books immediately, you'll miss out on some things and things might not be well motivated.

There is a huge difference between pure math and the math needed for physics. Nevertheless, I think it's always a good idea to have a good and rigorous understanding of basic calculus and linear algebra. I highly recommend the following books:

Lang's "first course in calculus" https://www.amazon.com/dp/0387962018/?tag=pfamazon01-20
Lang's "introduction to linear algebra" https://www.amazon.com/dp/0387962050/?tag=pfamazon01-20
"Linear algebra done wrong" www.math.brown.edu/~treil/papers/LADW/book.pdf‎[/URL]
Simmons' "Differential equations" [URL]https://www.amazon.com/dp/0070575401/?tag=pfamazon01-20[/URL]
Lang's multivariable calculus [URL]https://www.amazon.com/dp/0387964053/?tag=pfamazon01-20[/URL]

If you want even more math, then Spivak or Apostols calculus are good ideas. But they seem less necessary to me.

Feynman's lectures are awesome. It is something you should absolutely buy. But you shouldn't use it as a textbook. Rather, it is something you should use as a secondary resource. Read a normal textbook and then check Feynman to see what awesome insights he gives.

 

[WannabeNewton]

Is it possible to just learn the mathematics first (i.e. multivariable and vector calculus, linear algebra, etc.) and only then start learning physics? (i.e. from a graduate textbook like Classical Mechanics by Golstein)?

Would there be any pitfalls?

There would be a profuse of pitfalls. This is an absolutely terrible way to learn physics so don't do it. Mathematics is not physics (if anything at the level you're talking about mathematics is much easier than physics); you have to start physics from the ground up. Physics isn't just a body of information, it's a way of approaching and solving problems using a variety of tools that you will only pick up by starting from the basics. So don't skip the foundations.

 

[Hercuflea]

If I were again beginning my studies,
I would follow the advice of Plato and start with mathematics.
- Galileo Galilei

 

[Integral]

I was through Diff Eqs, before I ever set foot in a physics class. I found that being ahead of the game mathematically speaking was very helpful. I could concentrate on the physics and did not have to worry about the math.

 

[ainster31]

I was through Diff Eqs, before I ever set foot in a physics class. I found that being ahead of the game mathematically speaking was very helpful. I could concentrate on the physics and did not have to worry about the math.

I agree. Your physics understanding will be lacking if you don't understand the math behind it. I wish I did a bit more math before learning physics. Without math, the only physics problems you can really do are plug-and-chug problems.

 

[thegreenlaser]

I was through Diff Eqs, before I ever set foot in a physics class. I found that being ahead of the game mathematically speaking was very helpful. I could concentrate on the physics and did not have to worry about the math.

I'll echo this. Physics seems to become much harder when you're trying to learn both the physics and the math at the same time. At the same time, though, there's a balance, because physics can often be a catalyst which helps you learn the math better. For me, in particular, I find that physics gives some context and motivation to subjects that I wouldn't otherwise be hugely interested in. Some people like studying math for math's sake. I enjoy math, even pure math based on axioms and theorems and proofs, but I need to see how it's relevant to physics/engineering or I don't enjoy it nearly as much. So if you're like me, you're not going to want to spend a few years learning ALL the math before you finally start on physics--you'll just get anxious to jump into the physics and you'll probably rush the math and not learn it as well as you should.

Your best bet is probably to jump back and forth between math and physics, with math leading physics slightly. It's actually not as hard as you might think. If you're trying to do physics and you don't have the math skills, it will probably be pretty obvious, and then you just go learn some math. If you're getting bored with math, it may be time to go try to learn some physics just to get some motivation for the math.

You're actually at an advantage in this regard because you're learning in your spare time. When you take a physics class and half way through the semester you realize you don't know enough math, you can't just stop the physics class to take some time to learn the math properly. When you're learning on your own, you have that luxury.

 

[WannabeNewton]

There's a difference between starting from the foundations and starting from a graduate text like Goldstein. It doesn't matter how much math you learned beforehand, this is just a bad idea. OP seriously don't start with a text like Goldstein if your goal is to learn physics. You may or may not scoff at lower-division textbooks now but once you work through them you'll appreciate how important they are. A physics textbook with fancy mathematics isn't necessarily a good one and, like I said, it's the physics at this level that's hard not the math. Also if there's anything the user yuiop has taught me over at the GR subforum it's that mathematics will only take you so far; you have to make sure you have physical intuition and physical concepts firmly grounded in your head. This is what those lower-division texts are for.

 

[thegreenlaser]

There's a difference between starting from the foundations and starting from a graduate text like Goldstein. It doesn't matter how much math you learned beforehand, this is just a bad idea. OP seriously don't start with a text like Goldstein if your goal is to learn physics.

That's also a very good point. It's not just the math that makes Goldstein difficult. There's a lot of physics that he skims over because he's assuming you've learned it before.

I just took a quick look, and he basically spends a single page at the beginning covering Newton's first and second laws at an intermediate level (i.e. Newton's second law as a differential equation, and Newton's first law as a definition for an inertial reference frame). I don't have my undergraduate mechanics book handy, but I'm pretty sure there were entire chapters devoted to each of those concepts, and it definitely took me a few weeks to really understand them, despite the fact that I was already comfortable with the math ahead of time.

Technically he tells you everything you need to know in that one page summary, but if all you've seen is a high school level treatment of Newton's laws, you're going to have a really tough time understanding what he's saying, because he barely gives any explanation/motivation at all. His intention is to quickly review that material, not to teach it to you.

 

[Integral]

Physics is taught in what I call times through. The first time through is critical to the second time through. You get about the same material just at a higher level. The second time through is very hard if you have not done the first time through. Start at the beginning and build your foundations before attempting to build the structure that is Physics.

 

[AlephZero]

I just took a quick look, and he basically spends a single page at the beginning covering Newton's first and second laws at an intermediate level (i.e. Newton's second law as a differential equation, and Newton's first law as a definition for an inertial reference frame). I don't have my undergraduate mechanics book handy, but I'm pretty sure there were entire chapters devoted to each of those concepts, and it definitely took me a few weeks to really understand them, despite the fact that I was already comfortable with the math ahead of time.

Technically he tells you everything you need to know in that one page summary, but if all you've seen is a high school level treatment of Newton's laws, you're going to have a really tough time understanding what he's saying, because he barely gives any explanation/motivation at all. His intention is to quickly review that material, not to teach it to you.

That depends what level of "abstract thinking" (for want of a better description) the reader is at. If you are used to reading advanced math texts with formal definitions and proofs, few examples (you are supposed to be able to invent your own, at that level) and little "motivation", picking up a similar style of physics text is no different.

On the other hand if you gave high school students a book written in that style, they would be unlikely to learn much from it.

 

[Delong]

I don't think math will necessarily make physics problems easier. You just have to know conceptually how force and energy work at some point. But then it's still good to know the math. Try to do both at the same time not one before the other. I don't know that much about pure math but I'm guessing it's not necessary at all.

 

[ZapperZ]

To add, I've written about this earlier:

https://www.physicsforums.com/showthread.php?t=240792#4

Zz.

 

[Intrastellar]

To add, I've written about this earlier:

https://www.physicsforums.com/showthread.php?t=240792#4

Zz.

Your post clarifies a big part of the question, which is that it is definitely highly recommended to learn the maths before one needs it.
The other part of the question asks whether it is a good idea to skip to advanced undergraduate/graduate level classes if one knows the required maths. What do you think of that ?

 

[Vanadium 50]

The other part of the question asks whether it is a good idea to skip to advanced undergraduate/graduate level classes if one knows the required maths. What do you think of that ?

For the reasons expressed by others, this is a terrible idea.

Do you start with Spanish III before Spanish I?

 

[JCA_]

For the reasons expressed by others, this is a terrible idea.

Do you start with Spanish III before Spanish I?

that's not a good example because in case I already know some spanish, I will definately not start with spanish I.
By the other hand I agree that is a good idea to start with basics in case of physics, even if you already know the maths.

 

[ZapperZ]

The other part of the question asks whether it is a good idea to skip to advanced undergraduate/graduate level classes if one knows the required maths. What do you think of that ?

I didn't read that part of the question, and if the OP was really asking this, then that is one of the most ridiculous thing I've ever heard! Just because one knows the mathematics to do QFT doesn't mean that one actually knows how to do QFT! By that logic, once you learn calculus, does that mean you can skip all first-year intro physics?

Zz.

 

[Intrastellar]

This should answer the OP's questions.

Is it possible to just learn the mathematics first (i.e. multivariable and vector calculus, linear algebra, etc.) and only then start learning physics? (i.e. from a graduate textbook like Classical Mechanics by Golstein)?

Would there be any pitfalls?

I didn't read that part of the question, and if the OP was really asking this, then that is one of the most ridiculous thing I've ever heard! Just because one knows the mathematics to do QFT doesn't mean that one actually knows how to do QFT! By that logic, once you learn calculus, does that mean you can skip all first-year intro physics?

Zz.

For the reasons expressed by others, this is a terrible idea.

Do you start with Spanish III before Spanish I?

I am studying on my spare time as a hobby and it seemed the fastest way to learn the math first. I have a high school physics background and I looked at MIT's OCW syllabus and they have University Physics by Hugh D. Young; Roger A. Freedman as their main book throughout their undergraduate program (more or less). I looked through that book online and it didn't seem like a type of book that would suit my preferences (too many colored pictures, over-explaining, etc.) -- I deeply apologize if this book is held in high regard, but that was just the first impression it gave me and wanted to know if I am mistaken.

take a look at books like Kleppner, Morin or Purcell. These tend to be quite difficult, but amazing from a physics point of view.

 

[joneiljack]

I wanted to thank you all for your insights. I understand now that there is physics intuition that needs to be developed which is different from mathematics intuition.

In case anyone is wondering I have decided to get University Physics by Young & Freedman and Kleppner & Purcell. Once I get the first one, I will then alternate between it and the math required to learn the other two books.

Again, thanks a lot for the recommendations and help. Much appreciated! <3

 

[IGU]

I didn't read that part of the question, and if the OP was really asking this, then that is one of the most ridiculous thing I've ever heard! Just because one knows the mathematics to do QFT doesn't mean that one actually knows how to do QFT! By that logic, once you learn calculus, does that mean you can skip all first-year intro physics?

I'm not sure it's that simple. The question ends up being analogous to:
- I know calculus.
- I'm being offered algebra-based introductory physics or calculus-based introductory physics
- Some people tell me I should do the algebra-based one and then the other one after
- The claim is that I'll develop better intuition that way
- But I'm in a hurry and just want to do the calculus-based one

I'm pretty sure in this case everybody here would say to skip the algebra-based physics; it's just there for people who don't have the math to do the real physics intro.

So, analogously, I think the OP is looking for how to go about learning physics if the requisite math is learned first. Is there any dumbed down intro physics that can be skipped, in the same way that algebra-based physics should be skipped if possible?

Personally, I would read Feynman for beauty and intuition regardless. But I don't know the answer to the question as I've reformulated it. What would come next if you don't want repetition? And because it's self-study you can pick and choose just the essential bits.

 

[R136a1]

I'm not sure it's that simple. The question ends up being analogous to:
- I know calculus.
- I'm being offered algebra-based introductory physics or calculus-based introductory physics
- Some people tell me I should do the algebra-based one and then the other one after
- The claim is that I'll develop better intuition that way
- But I'm in a hurry and just want to do the calculus-based one

That's not at all what the question is analogous too. It's actually this
- I know calculus
- I want to learn physics from a grad textbook

 

[joneiljack]

I'm not sure it's that simple. The question ends up being analogous to:
- I know calculus.
- I'm being offered algebra-based introductory physics or calculus-based introductory physics
- Some people tell me I should do the algebra-based one and then the other one after
- The claim is that I'll develop better intuition that way
- But I'm in a hurry and just want to do the calculus-based one

I'm pretty sure in this case everybody here would say to skip the algebra-based physics; it's just there for people who don't have the math to do the real physics intro.

So, analogously, I think the OP is looking for how to go about learning physics if the requisite math is learned first. Is there any dumbed down intro physics that can be skipped, in the same way that algebra-based physics should be skipped if possible?

Personally, I would read Feynman for beauty and intuition regardless. But I don't know the answer to the question as I've reformulated it. What would come next if you don't want repetition? And because it's self-study you can pick and choose just the essential bits.

OMG thank you for translating into better words what I had in mind. The reason I mentioned Golstein (I didn't know about Kleppner) was because it seemed to say the same thing as the algebra based books, i.e. instead of F=ma it was F=m(dv/dt).

So it's exactly like you said. In my mind I just wanted to avoid repetition and save time. However it still seems that Golstein is not what I should go for as it's not a "calculus-based introductory physics". So in that case, I'm sure you can skip the algebra-based text, for something like Kleppner by learning the math needed for it first (calculus, linear algebra, etc.). Either way I still got the algebra-based one as a supplement (for 15$ or so old 11th edition so no loss.)

As for Feynman, I will just get it much later on due to financial issues. :(

 

[CaptainHammer]

I'm currently on the 3rd year of my Physics degree and can I honestly say, I should have taken mathematics first.

It may just be because of teaching method of my university, but the teachers give a lot more importance to the mathematics part than the comprehension of the physical phenomenon.

 

[ZapperZ]

This is a perfect opportunity to highlight what Mary Boas wrote in the Preface to her "Mathematical Methods in the Physical Sciences".

the question of proper mathematical training for students in the physical sciences is of concern to both mathematicians and those who use mathematics in applications. Mathematicians are apt to claim that if students are going to study mathematics at all, they should study it in careful and thorough detail. For the undergraduate physics, chemistry or engineering student, this means either (i) learning more mathematics than a mathematics major or (ii) learning a few areas of mathematics thoroughly and the others only from snatches in science courses. The second alternative is often advocated; let me say why I think it is unsatisfactory. It is certainly true that motivation is increased by the immediate application of a mathematical technique, but ther are a number of disadvantages.

1. The discussion of the mathematics is apt to be sketchy since that is not the primary concern.

2. Students are faced simultaneously with learning a new mathematical method and applying it to an area of science that is also new to them. Frequently the difficulty in comprehending the new scientific area lies more in the distraction caused by poorly understood mathematics than it does in the new scientific ideas.

3. Students meet what is actually the same mathematical principle in two different science courses without recognizing the connection, or even learn apparently contradictory theorems in the two courses! For example, in thermodynamics students learn that the integral of an exact differential around a closed path is always zero. In electricity or hydrodynamics, they run into Int(0 to pi) d(theta), which is certainly the integral of an exact differential around a closed path but is not zero!

Now it would be fine if every science student could take the separate mathematics courses in differential equations (ordinary and partial), advanced calculus, linear algebra, vector and tensor analysis, complex variable, Fourier series, probability, calculus of variation, special functions, and so on. However, most science students have neither the time nor the inclination to study that much mathematics, yet, they are constantly hampered in their science courses for lack of the basic techniques of these subject. It is the intent of this book to give these students enough background in each of the needed areas so that they can cope successfully with the junior, senior, and beginning graduate courses in the physical sciences.

I've stated a similar opinion on this. I truly believe that students shouldn't hear the term "orthornomal" or "Spherical harmonics", etc. for the first time in a physics class. It is a daunting task trying to learn unfamiliar physics while trying to grasp the mathematics at the same time.

Zz.