Maths textbooks for self-study

[lizzie96']

I am a 1st-year physics student and over the summer am planning on self-studying some extra maths courses. I was considering geometry, statistics, and algebra as these seem to not require too many prerequisites and are useful in physics. Could anyone recommend some good textbooks/online resources for self study in these areas? I have taken intro courses in algebra and probability this year, and I have very little background in geometry. Thank you for any suggestions.

 

[Greg Bernhardt]

We have some free books listed here
https://www.physicsforums.com/threads/free-math-books.796225/

Read this too
https://www.physicsforums.com/threads/how-to-self-study-mathematics.804404/

 

[lizzie96']

Thank you very much- several of these look really useful. In addition to these pure-maths textbooks, could anyone recommend any books on geometry and algebra written specifically for physics students, i.e. written in a less formal/rigourous style and directed especially towards the areas with physical applications?

 

[micromass]

I was considering geometry, statistics, and algebra

That is extremely vague. Geometry, statistics and algebra are huge fields. If you're somewhat more specific, then I can give recommendations.

 

[lizzie96']

Sorry- I would like to study some 2nd-year courses that I have the prerequisites for but not the time/credits to study next year, so I am looking for material at a fairly introductory level. There are no recommended books and the lecture notes are not made public, but these are some of the main topics included according to the course webpages:

Algebra:
Abstract vector spaces, linear transformations, multilinear algebra of determinants, eigenvectors and eigenvalues, fields, rings and modules, quotients, isomorphism theorems, Sylow theorems, Cayley-Hamilton theorem, inner product spaces, spectral theorem, Jordan normal form, Galois groups.

Geometry:
Curves in Euclidean space, Frenet-Serret frame, curvature and torsion, vector fields, differential forms, Poincare’s lemma, connection forms, structure equations, surfaces, isometries, geodesics on surfaces, integration of forms, Stoke’s theorem, Gauss-Bonnet theorem, Euler characteristic

Statistics:
Random walks, stirling’s approximation, moment generating functions, Fourier transform of probability distribution, central limit theorem, error function, least squares fitting, residuals, error analysis, Kolmogorov-Smirnov test

 

[micromass]

Sorry- I would like to study some 2nd-year courses that I have the prerequisites for but not the time/credits to study next year, so I am looking for material at a fairly introductory level. There are no recommended books and the lecture notes are not made public, but these are some of the main topics included according to the course webpages:

Algebra:
Abstract vector spaces, linear transformations, multilinear algebra of determinants, eigenvectors and eigenvalues, fields, rings and modules, quotients, isomorphism theorems, Sylow theorems, Cayley-Hamilton theorem, inner product spaces, spectral theorem, Jordan normal form, Galois groups.

There are no (beginner) books which cover all of these. You'll have to focus on either linear algebra which covers:
Abstract vector spaces, linear transformations, multilinear algebra of determinants, eigenvectors and eigenvalues, Cayley-Hamilton theorem, inner product spaces, spectral theorem, Jordan normal form

or abstract algebra which covers:
fields, rings and modules, quotients, isomorphism theorems, Sylow theorems, Galois groups.

You can do either of those. For linear algebra, I recommend highly the (free) book by Treil: http://www.math.brown.edu/~treil/papers/LADW/LADW.html
For abstract algebra, I recommend Anderson and Feil: https://www.amazon.com/dp/1584885157/?tag=pfamazon01-20

Geometry:
Curves in Euclidean space, Frenet-Serret frame, curvature and torsion, vector fields, differential forms, Poincare’s lemma, connection forms, structure equations, surfaces, isometries, geodesics on surfaces, integration of forms, Stoke’s theorem, Gauss-Bonnet theorem, Euler characteristic

What you want is (elementary) differential geometry. A good book is: https://www.amazon.com/dp/184882890X/?tag=pfamazon01-20

Statistics:
Random walks, stirling’s approximation, moment generating functions, Fourier transform of probability distribution, central limit theorem, error function, least squares fitting, residuals, error analysis, Kolmogorov-Smirnov test

It's a bit difficult to recommend a book for all of these topics. I think the best probability books require measure theory, which you likely don't know yet. Nevertheless, I think you'll find the classical books by Feller very good: https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20
Another good option would be the following online course: http://www.math.uah.edu/stat/ It is extremely good, but it doesn't quite cover many topics you listed.

 

[lizzie96']

Thank you- that's really helpful!