Recommended books on Theory of Calculus

[BigFlorida]

Hello all,
I am about to go into vector analysis (next week) and I just wanted to knock out a few books on the subject of integration mainly. I have read through my vector calculus book (P.C. Matthews) and there were a few things that threw me off, but made complete sense when I thought about them.

I.e. Why is it okay to split a double (or triple) integral up into a product?
∫∫∫(xyz)dxdydz = (∫xdx)(∫ydy)(∫zdz)
I never saw this explained in any book explicitly, but some books (as well as professors) do it, and I am just wondering why this, and the opposite of this, is true; I am not comfortable using rules that I do not fully understand, but sometimes I have to do things like this.

I think my main problem is I learned how to do integration one way, and I am stuck in that way, but I see that it is limiting me in my ability to recognize things.

I am very comfortable with integration, and with calculus in general, but I feel as though there are some gaps in my fundamental understanding of integration. I would just like some book/resource recommendations to try to fill these gaps. I have not taken DE yet, so I do not know if that class will have the answers I am looking for.

Anyways, thank you all in advance for your replies.

 

[DEvens]

(snips)
I.e. Why is it okay to split a double (or triple) integral up into a product?
∫∫∫(xyz)dxdydz = (∫xdx)(∫ydy)(∫zdz)

This particular integral can be broken up into a product because the integrand can be split up into a part with only x, a part with only y, and a part with only z. If the integrand could not be so broken up, for example a function $f(x,y,z)$ that did not split up this way, then you can't split it up into a product. At least, not easily.

So, for example, if you were integrating $\int \int \int dx dy dz (x^2 + y^2 + z^2)$ then you can't simply split that up into a product of three integrals.

The reason it works when the integrand is separable is because of the nature of an integral as a limit of sums. The first two are easy to see. Doing the x-integral part gives you an area under a curve, but this area is a function of y and z. But the functional form is just a multiplication. So if you change the value of y, the x-integral is just changed by the corresponding multiplicative factor. So when you do the y-integral, you are summing up areas and getting a volume. And this volume is in turn multiplied by the z value. It gets harder to make a pretty geometric interpretation with the z-integral. But hopefully you get the idea.

As to what books to get: This depends on your budget. At the cheap end there is the Schaums Outline series. These are great if you like worked-examples.

https://www.amazon.com/dp/0071795537/?tag=pfamazon01-20

And search around for other related Schaums books. For $15 it's a good start. Just don't try to let it be your only text because it really isn't appropriate for that. It may help you over the learning curve.

After that, I suggest you check out your school library and see what texts are there. Buying more than the calculus text you need for your classes is probably a waste. Once you get this level of calculus you probably won't be opening these texts every day. You will have a few cheat-sheet type notes for the kind of thing you do every day in your work. And that's about it.

If you have tons of money for books, check out the course catalogue for your class. See if it recommends a text and any alternative texts.

 

[mathwonk]

you might look at the definition of the integral and apply it to this situation.

 

[davidbenari]

there's a course on MIT's OCW that I've always felt intrigued about, maybe you can check it out for yourself... Its called "Calculus with theory", its in the maths department. Also, on threads like these people always recommend books by Spivak on analysis. there's also Rudin's book on analysis which might be of interest.

I'm in the same position as you btw and plan to study calculus more profoundly the next summer.

 

[BigFlorida]

Thank you all for your replies, I shall definitely check out the MIT OCW and the recommended books. @DEvens thank you for that clarification, it makes total sense.

 

[verty]

I like Advanced Calculus by Friedman, it's very readable if you know how to read theorems.

 

[BigFlorida]

@verty Thank you for the recommendation, I shall definitely check it out.