Does Apostol Calculus Volume 2 cover sufficient multivariate calculus?

[Antineutrino]

Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient multivariable calculus. I looked at the table of contents, and I only see one chapter that looks relevant to multivariable calculus (although I am not 100% sure what to look for). I do like his coverage of linear algebra and ODEs, however I would like to study from Spivak's calculus on manifolds, Munkre's analysis on manifolds, and a PDE book afterwards. Would his multivariate calculus cover what I need for these? My ultimate goal is to be a physicist, and I am most interested in general relativity and particle physics.

 

[pasmith]

Part II (Chapters 8 to 12) covers everything relevant to differential and integral calculus in \mathbb{R}^3 (partial derivatives; div, curl, grad; line integrals, surface integrals, volume integrals; Stokes' theorem and the divergence theorem), which is what an introductory multivariate calculus course would usually cover.

 

[MidgetDwarf]

Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient multivariable calculus. I looked at the table of contents, and I only see one chapter that looks relevant to multivariable calculus (although I am not 100% sure what to look for). I do like his coverage of linear algebra and ODEs, however I would like to study from Spivak's calculus on manifolds, Munkre's analysis on manifolds, and a PDE book afterwards. Would his multivariate calculus cover what I need for these? My ultimate goal is to be a physicist, and I am most interested in general relativity and particle physics.

Both Spivak books are pretty good. Although, I will mention one caveat with Calculus on manifolds. It is a pretty difficult book to work through, since it is a barebones treatment, and proceeds via light speed to the proof of the generalized Stoke's Theorem. Ie., a page can require 3 to 5 pages of notes to justify to oneself why such and such is true.

My course, that I took to graduate (Multivariable Analysis), was based on Spivak's Calculus on Manifolds. I had to ditch it around page 70, and supplemented it with Lang, Munkres, and Bartles books.

I found Munkress covered the same ground as Spivak, but explained more. However, Spivak has the superior exercises.

Apostol's both volumes can be a bit dry, but for self-study, I found the second volume of Apostol better for the multivariable aspect for the layman...

Other books that are more expository than Calculus on Manifolds,

is the books by Shifrin, Hubbard , and for an easier treatment, Marsden .

 

[mathwonk]

In my opinion, Apostol vol2 is ideal preparation for spivak on manifolds. In fact you probably don't need spivak afterwards, but if you do read spivak you will appreciate what he is doing much more than if you just read spivak alone. what you will get from spivak is a unification of the theorems of green, gauss, and stokes from apostol, in terms of differential forms, which apostol omits.

 

[Antineutrino]

In my opinion, Apostol vol2 is ideal preparation for spivak on manifolds. In fact you probably don't need spivak afterwards, but if you do read spivak you will appreciate what he is doing much more than if you just read spivak alone. what you will get from spivak is a unification of the theorems of green, gauss, and stokes from apostol, in terms of differential forms, which apostol omits.

Do you think that Spivak is helpful for learning differential geometry/general relativity, or would it be better to start learning it right after Apostol and skip Spivak?

 

[mathwonk]

I think it would be helpful to learn the language of manifolds and differential forms for differential geometry and relativity. Spivak treats those and thus would be helpful, since apparently apostol does not. I myself read spivak and benefited greatly. Since his treatment is very terse, it might be easier to learn manifolds and forms from one of the other more accessible sources mentioned here, perhaps Munkres, Hubbard, or Shifrin, but I myself am not as familiar with those. People also like books by Lee.

Of course many elementary books on diffrential geometry begin with intro to manifolds and forms. There are also nice books just on differential forms, e.g. one by Henri Cartan, one by Harley Flanders, and one by David Bachman. Since they are kind of unintuitive and abstract as usually presented, it helped me a lot to just see how to calculate with them in a little article iin an AMS monograph by Flanders. I can't find that reference. The point is that just learning how easy it was to add and multiply them was very reassuring considering how abstract their definition was.

You don't really need to worry about all this information at this point however, since it will may well take you quite a long time to master Apostol. I.e. it is more useful to actually read one good book, than to make a long list of books to potentially read. I have not yet read every word of spivak's little book on manifolds, nor much less even volume 1 of his enormous differential geometry book, and I started them in the 1960's.

Still, working through chapters 1-4, especially 4, of spivak's calculus on manifolds, taught me a lot. Unfortunately his notation for the proof of stokes' theorem on about page 103 of calc on manifolds made my eyes glaze over. I never understood how easy it is until I read the little discussion in Lang's Analysis I, on p.442, Stokes theorem for simplices, in the case of a rectangle. Probably you would get this from Apostol, which I had not read then. I.e. stokes theorem is trivial, you just use fubini's theorem to reduce it to a repeated integral in fewer variables, and then it reduces to the one variable fundamental theorem of calculus. But it is hard to understand abstract brief versions of something unless you have first studied the basic classical versions in detail in low dimensions. After that it is nice to see how the details can be made elegant and the concepts unified.

good luck!

 

[Antineutrino]

I think it would be helpful to learn the language of manifolds and differential forms for differential geometry and relativity. Spivak treats those and thus would be helpful, since apparently apostol does not. I myself read spivak and benefited greatly. Since his treatment is very terse, it might be easier to learn manifolds and forms from one of the other more accessible sources mentioned here, perhaps Munkres, Hubbard, or Shifrin, but I myself am not as familiar with those. People also like books by Lee.

Of course many elementary books on diffrential geometry begin with intro to manifolds and forms. There are also nice books just on differential forms, e.g. one by Henri Cartan, one by Harley Flanders, and one by David Bachman. Since they are kind of unintuitive and abstract as usually presented, it helped me a lot to just see how to calculate with them in a little article iin an AMS monograph by Flanders. I can't find that reference. The point is that just learning how easy it was to add and multiply them was very reassuring considering how abstract their definition was.

You don't really need to worry about all this information at this point however, since it will may well take you quite a long time to master Apostol. I.e. it is more useful to actually read one good book, than to make a long list of books to potentially read. I have not yet read every word of spivak's little book on manifolds, nor much less even volume 1 of his enormous differential geometry book, and I started them in the 1960's.

Still, working through chapters 1-4, especially 4, of spivak's calculus on manifolds, taught me a lot. Unfortunately his notation for the proof of stokes' theorem on about page 103 of calc on manifolds made my eyes glaze over. I never understood how easy it is until I read the little discussion in Lang's Analysis I, on p.442, Stokes theorem for simplices, in the case of a rectangle. Probably you would get this from Apostol, which I had not read then. I.e. stokes theorem is trivial, you just use fubini's theorem to reduce it to a repeated integral in fewer variables, and then it reduces to the one variable fundamental theorem of calculus. But it is hard to understand abstract brief versions of something unless you have first studied the basic classical versions in detail in low dimensions. After that it is nice to see how the details can be made elegant and the concepts unified.

good luck!

Thank you for the replies! How long do you think Apostol's two volumes should take to complete? I should be starting them in a few months. I'll know univariate calculus, but not any multivariate calculus or linear algebra.

 

[mathwonk]

actually mastering them could take years, as I think I have already suggested. Don't care about this, just start and hang in there. You can learn a lot in much less time. I once powered through parts of spivak's volumes 1 and 2 of differential geometry in a few days, but mathematics takes time to sink in, and 40 years later I still have not finished them.
It will take a lot longer if you keep asking, instead of just starting reading. As my advisor said when I asked something similar, at some point it is time to "stop dancing around the fire". But I presume you have other concerns at present. take care of those first. best wishes.