General Maths & Physics textbooks for the university

[NathanaelNolk]

Hello guys ,

First of all, sorry for my bad english, I'm not a native speaker. :)

I'm a Swiss currently in High School. I first took Philosophy as a specialization when I began my High School, but as I started to think about what kind of study I wanted to do at University, I realized that I was a lot more into physics and maths. I now want to study both and would like to become a (theoretical) physicist.
But, if I want to go to the EPFL (Swiss Federal Institute of Technology), I need some more preparation in maths (Linear Algebra, Complex Numbers, more advanced Calculus with some Differential Equations..) and in physics. (Classical Mechanics, Electricity, Waves..). I've talked with physicists at the university and they told me I needed a good background in mathematics to be able to understand physics at the EPFL.

I'd also like to learn some maths topics that aren't directly related to physics like topology, abstract mathematics or such. I really enjoy both mathematics and physics so I'd like to study on my own these topics, even though I want to graduate in physics.

I've started to learn Classical Mechanics, Calculus and Linear Algebra with the MIT Open CourseWare website. For now, it doesn't seem that hard to me, and the video lectures help a lot. But I'll need some textbooks to go with these lectures to have more in-depth understanding of these subjects. I've made a short list and I wanted to know if it was worth it. By the way, I have a lot of time to study, so that won't be a problem at all.

Here's the list :

Apostol's/Spivak's or Lang's Calculus
Strang's Introduction to Linear Algebra
A textbook about complex numbers/analysis
Bertsekas's Introduction to Probability
Tenenbaum's Ordinary Differential Equations
Rudin's Principles of Mathematical Analysis (Baby Rudin)
Knight's Physics : A Strategic Approach

What do you think about it ? Would you change this list ? Would you add something ?
I would also like to learn about topology, differential geometry and this kind of stuff, do you know if I can do it while studying physics ?

So yeah, that's a lot of questions. I hope you'll have time to give me some tips and recommendations.

 

[micromass]

Here's the list :

Apostol's/Spivak's or Lang's Calculus

If you don't have any experience with calculus, then try Lang. After Lang you can do both volumes of Apostol.

Strang's Introduction to Linear Algebra

I wouldn't recommend this book. What knowledge do you already have of LA? Do you know how to solve systems using matrices? Do you know how to multiply matrices?

I recommend Lang's "Introduction to Linear Algebra" (he also has a "Linear Algebra" which is more advanced).
After Lang, I highly recommend the free linear algebra done wrong: http://www.math.brown.edu/~treil/papers/LADW/LADW.html

A textbook about complex numbers/analysis

I'm a fan of "Introduction of Real Analysis" by Depree and Swartz. Mostly because their theory of integration uses the Henstock integral which is way better than the usual Riemann integral you know from calculus.

Not really good for a first course in analysis, but Carothers is a very very well-written book and is a must-read.

Lang's introduction to analysis is also very solid.

Bertsekas's Introduction to Probability

I recommend Feller's two volume set on probability. It really is one of the best sources of probability out there.

The following website is also superb: http://www.math.uah.edu/stat/index.html

Tenenbaum's Ordinary Differential Equations

I recommend Simmons: https://www.amazon.com/dp/0070575401/?tag=pfamazon01-20

Rudin's Principles of Mathematical Analysis (Baby Rudin)

I highly recommend against this book. The single-variable part is ok (not great), and the multivariable parts are horrible. The measure theory part is disgusting. I don't know how this book could ever have reached such fame.

I would also like to learn about topology, differential geometry and this kind of stuff, do you know if I can do it while studying physics ?

You can certainly do it while studying physics.

For differential geometry, I recommend Pressley https://www.amazon.com/dp/184882890X/?tag=pfamazon01-20 or Milman: https://www.amazon.com/dp/0132641437/?tag=pfamazon01-20 or DoCarmo: https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20 (listed from easy to difficult)

For topology, I really like Lee's book: https://www.amazon.com/dp/1441979395/?tag=pfamazon01-20 It is focused on differential geometry too, which is nice.
You can follow-up with Kelley or Willard, both are excellent (although not easy).

 

[NathanaelNolk]

Thank you for the answer, and yes I do have some basic knowledge about Calculus and LA. I know how to multiply matrices, find the inverse matrix and the determinant, but that's about it. For the Calc, I have done the differentiation part of the OCW course, so I'm pretty much new at integration.

So and updated version of the list would be :

Lang's Calculus (Apostol's after having finished this one)
Lang's Introduction to Linear Algebra
Depree and Swartz's Introduction to Real Analysis
Feller's Probability (2 volumes)
Simmons's Differential Equations

For more advanced study :

Pressley's Elementary Differential Geometry
Lee's Introduction to Topological Manifolds

Do someone know a good textbook about complex variables? Is there some good freshman physics textbook ? I have heard good feedback about Knight's Physics : A strategic approach, but I don't really know if it is useful. Or should I just watch the MIT OCW's video lectures in physics?

 

[micromass]

Thank you for the answer, and yes I do have some basic knowledge about Calculus and LA. I know how to multiply matrices, find the inverse matrix and the determinant, but that's about it. For the Calc, I have done the differentiation part of the OCW course, so I'm pretty much new at integration.

So and updated version of the list would be :

Lang's Calculus (Apostol's after having finished this one)
Lang's Introduction to Linear Algebra
Depree and Swartz's Introduction to Real Analysis
Feller's Probability (2 volumes)
Simmons's Differential Equations

For more advanced study :

Pressley's Elementary Differential Geometry
Lee's Introduction to Topological Manifolds

Do someone know a good textbook about complex variables? Is there some good freshman physics textbook ? I have heard good feedback about Knight's Physics : A strategic approach, but I don't really know if it is useful. Or should I just watch the MIT OCW's video lectures in physics?

My favorite book of complex analysis is Freitag and Busam. But this requires you to know some analysis, so it's not a basic book. Instead, you should look at "visual complex analysis" by Needham. It is a true gem and quite easy to understand.

 

[NathanaelNolk]

Ok thanks!

Lang's Calculus (Apostol's after having finished this one)
Lang's Introduction to Linear Algebra
Depree and Swartz's Introduction to Real Analysis
Needham's Visual Complex Analysis
Feller's Probability (2 volumes)
Simmons's Differential Equations

More advanced :

Pressley's Elementary Differential Geometry
Lee's Introduction to Topological Manifolds

Should I take some physics books too ? Do someone know which one would be good to start with ?