# Any good books in Vector Calculus?

1. Aug 19, 2013

### mathsciguy

I have been restudying vector calculus, especially on topics pertaining to line integrals, surface integrals (and the accompanying vector forms). One problem I have encountered from the book I have been using is that it seems there are some theorems and results that are only restricted to certain coordinate systems. For example, I was studying surface integrals, the derived result for computing surface integrals was developed by using the rectangular coordinate system (though, for some specific problems, you can do an appropriate coordinate transformation). If possible, I want a good textbook that has a more general treatment for calculus of vector fields.

2. Aug 19, 2013

### SteamKing

Staff Emeritus
IDK about certain vector calculus theorems being restricted only to cartesian systems. Most texts I have seen do seem to use cartesian systems for derivation and proofs. Now, some operators like the Laplacian have special forms which depend on the coordinate system being used. It could be that the specific problem in computing surface integrals was much simpler in a cartesian system than another coordinate system, much like spherical coordinates are easier to use with, naturally, spherical problems than cartesian coordinates.

3. Aug 19, 2013

### WannabeNewton

Often, results in vector calculus are proven in specific convenient coordinate systems but written in a way in which you can immediately recast it in coordinate-independent form. The exact same technique carries over to tensor calculus. It isn't any less general. There are certainly coordinate independent ways to prove things; for example, local results can be proven in a coordinate independent fashion using Ricci calculus. But to echo what SteamKing said, most vector calculus texts I know stick to the former.

4. Aug 19, 2013

### mathsciguy

I am saying all those given that my textbook only offers an introduction to calculus of vector fields, so I know asking for a more general approach is a bit unreasonable.

Anyway here's an example of one:

For a function of three variables $G(x,y,z)$, the surface integral of $G$ over the surface $S$, where $S$ is defined by the equation $z = f(x,y)$ can be computed by:
$$\int \!\!\! \int _S G(x,y,z) d\sigma = \int \!\!\! \int_D G(x,y,f(x,y)) \sqrt{f_x^2(x,y)+f_y^2(x,y)+1} dA$$
Where $D$ is the projection of $S$ in the xy-plane, $d\sigma$ is the differential surface area element.

I think this is an example of a result that is quite coordinate specific.

Last edited: Aug 19, 2013
5. Aug 19, 2013

### WannabeNewton

Well sure but the integral itself is coordinate independent; you're just choosing a set of coordinates to evaluate it with so what's the problem there?

Anyways, are you looking for a more theoretical treatment of vector calculus e.g. calculus on submanifolds in $\mathbb{R}^{n}$? Or perhaps something more along the lines of Hubbard: https://www.amazon.com/Vector-Calculus-Linear-Algebra-Differential/dp/0130414085?

Last edited by a moderator: May 6, 2017
6. Aug 19, 2013

### mathsciguy

Yes, exactly. But the problem here is, he didn't actually tell that the surface integral is a double integral (at least from what I currently know) that can be solved by actually integrating over the surface. In the textbook, after constructing something that's starting to look like a Riemann sum using the function values of $G$ at the sub regions $\Delta _i \sigma$ he denoted it by: $\int \!\!\! \int _S G(x,y,z) d\sigma$. Which of course looks like a double integral over the surface S intuitively. But he only denoted the limit of such sum by this symbol, and he didn't actually tell that it can exactly be solved by a straightforward integration using the surface, or is it? Actually I'm not very sure either, but I have seen this in my EM textbook. That confused me for a bit.

Anyway, it's just some trivial rambling. I am indeed trying to look for books with more theoretical treatment of the subject. Thanks.

Last edited: Aug 19, 2013
7. Aug 19, 2013

### WannabeNewton

Last edited by a moderator: May 6, 2017
8. Aug 19, 2013

### yungman

I have at least 5 calculus books for first 3 semesters. They only use rectangular coordinates. For example:
$$\hat x r\sin\theta \cos\phi+ \hat y r\sin\theta\sin\phi +\hat z r\cos \theta$$
is not Spherical Coordinates by any stretch. I even had 3 Vector Calculus book which is more in depth into vector calculus than any of the beginner's text books. They have nothing on the true cylindrical and spherical coordinates.

Look up Curvilinear coordinates, this is where they started. SP and Cy coordinates are part of the orthogonal coordinate system. But you'll get much more than what you bargained for!!!

I went through a big learning curve myself looking for information. You get to learn some in Electromagnetic if you are EE major. My Partial Differential Equation book touch a little into this topics. Here is a book that has a chapter in this, but the rest is EM.https://www.amazon.com/Fundamentals-Applied-Electromagnetics-Textbook-Only/dp/0005368138/ref=sr_1_14?s=books&ie=UTF8&qid=1376929468&sr=1-14&keywords=ulaby. Buy used for \$34. It is one of the best I found, still you need to do some more digging.

Here is a link, but it really does not explain much:http://en.wikipedia.org/wiki/Spherical_coordinate_system. Good luck, I feel for you as I had to really dig around for information. You would think the vector calculus book should cover this, they don't!!!

Last edited: Aug 19, 2013
9. Aug 19, 2013

### SteamKing

Staff Emeritus
Another thing you encounter quite frequently is that the vector calculus often uses different notation for the same problem.

For example, in elasticity, most of the vector calculus appears with a single integral sign, but, depending on the region of interest, the integral sign may imply a single, double, or triple integral. The difference is usually noted by the region of integration: C for a closed curve, A for an area, and V for a volume. This method is used, I suspect, because it simplifies the equation typesetting, and, once you are used to it, it makes reading and understanding a particular equation much easier.

10. Aug 19, 2013

### mathsciguy

Oh this, I did some searching. Apparently, what I said above about straight forward integrating over the surface S isn't exactly correct. You parametrize the surface (to vector valued function), I think, usually using spherical/cylindrical coordinates then transform it to rectangular components. I think though, that this parametric method can almost intuitively be thought as the same as 'integrating over the surface', especially for simple cases in which the surface can be parametrized by spherical or cylindrical coordinate system.

Anyway, can anyone link me to a book that discusses this parametric technique in depth?

11. Aug 19, 2013

### yungman

I don't know exactly what you mean. But it is not straight forward to translate between rect and SP or Cy. What you see in those calculus books are just simple rectangle coordinates, nothing more. The only thing they do is using $(r,\theta,\phi)$ representation of the scalar components
$$\vec F=\hat x F_x+\hat y F_y+\hat z F_z\;\hbox { where }\; F_x=r\cos\phi\sin\theta,\;F_y=r\sin\phi\sin\theta,\;F_z=r\cos\phi\theta$$
This is rectangular coordinates, not at all spherical coordinates.

Spherical coordinates uses $(\hat R,\hat \theta,\hat \phi)$, not $(x,y,z)$ at all. Those calculus book talk NOTHING about spherical coordinates. For simple position vector( coming out from the origin to a point P(x,y,z)), you will find only the $\hat R$ component remain after transforming to spherical coordinates. It is only vector fields that all the other components appear.

I am not math expert, I know how to deal with it. It is very tricky to translate between coordinates, not as simple as using the formulas. Two different points in space have their own spherical coordinates. For example two points $P(x_p,y_p,z_p) \; and\;Q(x_q,y_q,z_q)$ has different $(\hat R,\hat \theta,\hat\phi)$. It is not like x,y and y that has absolute position no matter what. You really need to read one of the chapter in the book I posted to get to know it, not from the calculus book.

You really need to study the spherical coordinates.

If you are interested in vector calculus, enroll in an Electromagnetic class. It is 70% vector calculus. You'll see all these coordinates in action.

Last edited: Aug 19, 2013
12. Aug 19, 2013

### WannabeNewton

An intermediate electromagnetism class (e.g. a Griffiths based class) will not deal with vector calculus in a pure math setting. OP, it is unclear to me what you mean by "more theoretical" treatment, if not vector calculus in a pure math setting which is naturally construed as calculus on submanifolds of $\mathbb{R}^{n}$ for which books like Munkres and Spivak are quite good.

Honestly there is very little to vector calculus as seen in say an electromagnetism class. It's basically just a bunch of simple tricks and techniques, mainly with regards to integration over 2-manifolds embedded in $\mathbb{R}^{3}$.

13. Aug 19, 2013

### yungman

That's the reason I did not mention Griffiths, Griffiths has nothing. But the few Engineering EM books has detail explanation of CY and SPH coordinates. Both https://www.amazon.com/Fundamentals-Applied-Electromagnetics-Textbook-Only/dp/0005368138/ref=sr_1_14?s=books&ie=UTF8&qid=1376942855&sr=1-14&keywords=ulaby and https://www.amazon.com/Field-Wave-Electromagnetics-David-Cheng/dp/0201128195/ref=sr_1_1?s=books&ie=UTF8&qid=1376942898&sr=1-1&keywords=david+k+cheng have a big section on CY and Spherical coordinates.

I bag the difference that EM class has not much to do with vector calculus. Of cause it's not going to teach you vector calculus, you need to have basic vector calculus to enroll in that. I spent 5 years studying EM and it's all vector calculus. Yes, its is not a complete vector calculus, but it sure use Green's Stoke's Divergence theorem in all their glory. all the EM fields are vector fields. I have to keep referring back to vector calculus time after time in EM studies.

I for one on here asking about vector calculus for my EM study at this very moment.

14. Aug 19, 2013

### WannabeNewton

I'm not saying that intermediate EM classes don't use vector calculus; I'm saying that they don't use it in a theoretical setting. When I say "honestly there is very little to vector calculus as seen in say an electromagnetism class" what I mean is that vector calculus as used in intermediate EM classes is not at all deep in a theoretical sense; it's just used as a computational tool and in that sense there is very little to it.

15. Aug 19, 2013

### yungman

I am an EE, I am not really qualify to give advice in this section. I just ran across this thread and I just want to share my experience. I gone through this very thing and I did struggle through it.

Most of the articles all get way deep into the subject, like the curvilinear coordinates has so much more than just the spherical and cylindrical coordinates. It is very hard to go through all the advanced math just to get to this. Only place I found the simplest representation is in the few Engineering EM book I posted. It's just a watered down, simplified way to explain the two coordinates without getting into the deep stuff. It is just my experience that I found they serve me well in understanding spherical and cy coordinates.

Yes, EM books does not give the theoretical end of it, I should say it's the continuation of the vector calculus and really twist things around and work you over. Actually when I worked through chapter 10 and 11 of Griffiths, it is getting quite deep into application of vector calculus. It is a lot more involve than the first 9 chapters.

I should quit talking here!!!

16. Aug 19, 2013

### lurflurf

I cannot tell exactly what you want, but I share your frustration with vector calculus books. Calculus books do not clearly indicate which topics are actually difficult. Vector calculus can bring up a number of questions that require algebraic topology and existence uniqueness of partial differential equations to settle. There is the issue of how coordinates should be used. The best vector calculus books like Philips are out of print. One book that is in print, but very expensive is General Vector and Dyadic Analysis by Chen-To Tai. Maybe your library has it. The book is solid, but not comprehensive. I feel it goes a bit too critical of the common notation and methods, but it is a reaction to common errors. Some of the authors related essays are here
http://deepblue.lib.umich.edu/browse?value=Tai,+C.+T.&type=author
you can also search http://deepblue.lib.umich.edu/ for
(title:vector) AND (author:tai)

Spencer and Moon have written several related books. They have many strange things like a silly diatribe about Laplacians. As is often the case there do not seem to exist comparable books so one might as well flip through them.

Any vector calculus book should talk about curvilinear coordinates with examples like prolate spheroidal coordinates. They tend not to give much pratice with them so you can do that on your own.

Last edited: Aug 19, 2013
17. Aug 19, 2013

### SteamKing

Staff Emeritus
18. Aug 20, 2013

### yungman

Gauss Law here is a simplified way that assume the charge is at the origin. The electric field at any point distance r from the origin is:
$$\vec E=\hat r \frac{q}{4\pi \epsilon r^2}$$
Which is in spherical coordinates. But since the charge is assumed to be at the origin, the $\vec E$ is like a position vector that only contain $\hat r$. There is not $\hat\phi \;and \;\hat\theta$.
Then Gauss assume a spherical surface. the outward normal of the surface is therefore also $\hat r$ and it is not position dependent ( not depending on $\phi \; \hbox { and } \; \theta)$. That's the reason the equation become very simple. This make it almost like what you see in the regular vector calculus books.

$$d\vec s=\hat r (r\sin\theta d\theta)(rd\phi)\;\hbox { where } \;(r\sin\theta d\theta)\;\hbox { and }\; (rd\phi)\;\hbox { and the two sides of the differential area.}$$

If you are physics major, it's really worth your while to get the Engineering Electromagnetics by Ulaby. Engineering electromagnetics is quite different than physics major with books like Griffiths. Griffiths spend less time on wave and no transmission lines at all. I am from the EE side, I did studied Griffiths. YOu really need both.

Last edited: Aug 20, 2013
19. Aug 20, 2013

### mathsciguy

^ I can understand these, at least mechanically, but well, I guess, what I am desperately trying to find is a general method that applies to all cases; something in which the usual basic methods is actually a special case of it. What you outlined above is intuitive, but it doesn't seem to fit any of the basic method schemes that I have seen. What I find are usually coordinate specific methods. This is just me trying to be really nitpick-y though. Anyway, I'll check out the books, thanks, those were helpful.

20. Aug 20, 2013

### yungman

I don't think you should try to find a "general" one shoe fit all formula. I look at every problem as individual. This particular example assumes the $\vec E$ is omni directional, in real life, there are very very few cases that field is like this. Also it is a very simplistic view to assume a point charge where everything is at the origin.

For example, I am studying antenna theory, unless the dipole antenna is impractically short, we have to assume that the dipole is along say z axis and current distribution is not constant along the wire. Now you have to calculate off the origin and non constant current.

And in loop antenna, the loop only center at the origin, but the current element is off origin and going in circle with non constant current!!!

So!!! Those are just two cases that the field is directional, off origin and non constant intensity!!! That's what I was referring about the EM really work you over on the vector calculus. When I studied cpt 10 and 11 of Griffiths, I was just laughing!!! Did I just studying a continuation of vector calculus? They even introduce delay field where they consider that it takes time for a field at the source to reach a certain point.

If you are a physics major, you really need to get an Engineering Electromagnetic text book on top of the one used in your major ( a lot of people use Griffiths). This, to me, is just as important for any EE student interested in EM, should study the Griffiths on top of whatever book they use. EM is a very difficult subject, you really have to study multiple times to get the feel of it. And when you think you understand, you'll be surprised the next time you read it. I studied Ulaby cover to cover first, then David K Cheng cover to cover. Then I study Griffith. I am still studying and every time I study, I learn more and I start moving to more advance topics.

Back to your OP about spherical co., I don't know enough math to comment on the books other people suggested. I can only tell you that I got the info I needed between Ulaby and Cheng's books. Then you have to work through the problems to learn about the tricks of spherical and cylindrical coordinates, that you have to be careful in the reference point of those two coordinates that you never have to for rectangular coordinates.....That you don't just blindly add or do anything until you are clear where the reference points are.

Here is the book by David K Chenghttps://www.amazon.com/Field-Wave-Electromagnetics-David-Cheng/dp/0201128195/ref=sr_1_1?s=books&ie=UTF8&qid=1377016357&sr=1-1&keywords=david+k+cheng. I use this and Ulaby. I have many EM books, other than Griffiths, none can compare to these two. Ulaby is very simple, good for reading but very limited. Cheng is very strong in Field and Wave. I have all the exact edition of the books.

Last edited: Aug 20, 2013