Which Books to Read After Spivak's Calculus?

[osnarf]

Sorry if this is in the wrong forum, it wouldn't let me post in the forum i tried first.
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I'm just about to finish up with Spivak's Calculus, and I'm trying to figure out where I should go to next. I've taken Calculus 1-3 and Differential Equations (just an introductory class to ODE's), and would like to repeat this sequence (with more rigor) before I progress any further (I would like to study advanced calc afterwards), which is why i started with Spivak (Seemed to cover Calculus 1-2). I would like to get to differential equations as soon as possible though (unless its not a good idea) because I have some classes coming up in engineering and physics that make use of them. I've pretty much decided on using Morris Tenenbaum and Harry Pollard's Ordinary Differential Equations, but am always open to suggestions.

The likely option for Calculus 3 seemed to be calculus on manifolds at first, but I read more about it and am not so sure anymore. As I understand it, the book is about differential forms, not vector calculus. Is this still multi-variable calculus, just a different way of thinking, or is it something new altogether? More importantly, if it is just a different way of doing multi-variable calc, if I go this route, would I still be well equipped to handle an ODE book and then a PDE book, or should I try to find a good book on vector multi-variable calculus instead?

Also, I was told that in order to study multi-variable calculus from a rigorous standpoint, it is beneficial to study linear algebra first. I've never taken a course on this, so would it be a good idea to go through a linear algebra book before I move to a multi-variable calculus book or is it something the book will develop along the way as needed and i can do linear algebra later?

So, to summarize:
- what are the differences between differential forms and vector calc, and will they both allow me to progress equally in differential equations after?

- is it necessary to study linear algebra first before multi-variable calc or would i be alright to
do linear algebra at a later time?

- what is your suggestion of which books to read and in which order? (In order to get to diff eq as soon as possible, and after multi-variable calc and diff eq be ready to do advanced calc).

Thank you for your help, I appreciate it

 

[Studiot]

My advice

1) Do some linear algebra before anything else.

I recommend Elementary Linear Algebra by Howard Anton and Chris Rorres. There are two versions, plain vanilla and the 'applications version', which includes the former plus more.

2) Vector calculus was developed 200 to 100 years ago for the science and engineering of the time. Using forms is (relatively) modern and includes VC as a special case. You will not get far in modern physics without forms, but they haven't penetrated far into engineering yet. The linear algebra quoted in (1) I would call a prerequisite for either or both.

I like 'Vector Calculus' by Marsden & Tromba they have an introductory section to forms at the end which helps bridge the gap.

A good clear, if old, introductory read is

Div, Grad, Curl and all that by Schey

3) You can never have too many books on Diff Equations. So your suggestions have merit.
Alternatively

An Introduction to Linear Analysis by Kreider, Kuller, Ostberg and Perkins will take you a long way beyond Anton, but it's tough at the beginning as your only linear algebra book. The authors couch offer much applied maths in modern terms, from Fourier series to Laplace transforms and offer a gentle into to PDEs

Hope this goes some way to answering your questions.

 

[bcrowell]

Here's a linear algebra book I like: http://joshua.smcvt.edu/linalg.html/ It also happens to be free.

You will not get far in modern physics without forms [...]

I disagree. I have a PhD in physics, and I never even heard about forms until after I was out of grad school.

 

[Studiot]

I have a PhD in physics, and I never even heard about forms until after I was out of grad school.

Yes but you are out of both grad and postgrad school. I am looking to the future where 'modern' maths will 'intrude' more and more. The idea is to take this stuff in ones stride.

 

[deluks917]

You need some linear algebra to read most good books on calculus in R^n/on manifolds. You don't need teribly much though to reead munkres (anlysis on manifolds) or Spivak. Personally I picked up enough from chapters 1, 3 and 4 of Michael Artin's algebra. I don't really recommend doing that but I'm trying to emphasize how little linear algebra you need. Try Mathwonk's (Roy Smith at UGA) notes, they helped me fill in the holes Artin left for later chapters or omitted.

Also you don't need to know the stuff in calculus on manifolds to read books on DE's. Arnolds book for example does again require some basic linear algebra at times.

 

[osnarf]

Thank you all for your replies.

So would i then be able to do the differential equations book as soon as I finish Spivak, and then go back and do Linear Algebra and then vector calc and forms and then advanced calc; or would I need the linear algebra before I do differential equations?

If it is just bits and peices of linear algebra I can probably pick it up on the way.

Again, i appreciate your help everyone.

 

[atyy]

You'll get to linear algebra in diffeqs straightaway. To solve a first order linear differential equation in N variables, you guess that the solution is an exponential, which makes everything into matrices. Then you rotate the system until the matrix is diagonal - ie. you find eigenvectors and eigenvalues - all that is in linear algebra.

The only thing that isn't covered is the scalar product, which is not essential to the vector spaces. But you already know that from elementary geometry.

 

[jasonRF]

Thank you all for your replies.

So would i then be able to do the differential equations book as soon as I finish Spivak, and then go back and do Linear Algebra and then vector calc and forms and then advanced calc; or would I need the linear algebra before I do differential equations?

If it is just bits and peices of linear algebra I can probably pick it up on the way.

Again, i appreciate your help everyone.

I would do linear algebra before anything else, since you have already had the other subjects at some level and it is truly at the heart of most of differential equations (the linear ones, anyway), and multivariable calculus. Then differential equations, etc. One "rigorous" option is to work through the 2nd edition of Calculus by Apostol. Covers linear algebra, differential equations, and multivariabe/vector calculus at a level similar to spivak, followed by probability theory and some basic numerical analysis. This is a relatively hard route. For the first exposure to linear algebra an easier and perhaps more fruitful approach is Anton's book, as listed by Studiot. I agree with him/her that it is a very good intro book. get a used copy of an old edition for cheap!

good luck,

jason

 

[osnarf]

Okay, I will do linear algebra first, then. I appreciate it.

If I worked through Apostol volume 1 (I have a copy already, I was going to go back and do it at a later time), would Apostol volume 2 be a good option then, since vol 1 includes an intro to linear algebra? Or would you still recommend Anton's book before volume 2?

 

[jasonRF]

Okay, I will do linear algebra first, then. I appreciate it.

If I worked through Apostol volume 1 (I have a copy already, I was going to go back and do it at a later time), would Apostol volume 2 be a good option then, since vol 1 includes an intro to linear algebra? Or would you still recommend Anton's book before volume 2?

Both Volume 1 and Volume 2 have those same chapters on linear algebra, if I recall correctly.

 

[mxbob468]

Okay, I will do linear algebra first, then. I appreciate it.

If I worked through Apostol volume 1 (I have a copy already, I was going to go back and do it at a later time), would Apostol volume 2 be a good option then, since vol 1 includes an intro to linear algebra? Or would you still recommend Anton's book before volume 2?

why would you do apostol volume 1? it's the same thing as spivak's calculus.