Question

I'd like someone to check me on this since the textbook answer appears quite different.

Answer 1

\[
\iint_{S}d\sigma \,x^2z=\pi
\]
Where:
\[
S:x^2+y^2=1,\; 0\le z\le 1
\]

Answer 2

@Kainui

Answer 3

So you're integrating over the entire surface?

Answer 4

Well, over one of the orientations. Note that it's not a closed surface.

Answer 5

Wait, I'm not sure I fully understand what context this is in. Is this a surface integral over a cylinder? In what way is this not closed?

Answer 6

You are correct, but note that we have the strict equality \(x^2+y^2=1\).

Answer 7

Sure, so this represents just a circle of radius 1, correct?

Answer 8

Yessir.

Answer 9

Essentially, I just changed coordinates to polar, and evaluated the integral.

Answer 10

I'm sorry, I don't understand what you're saying. Why are you saying this isn't closed?

Answer 11

Wait I have confused myself haha. Sorry I realize now that it's just the curved part and not the top and bottom. Alright one second let me rewind.

Answer 12

The surface isn't actually closed, it's essentially just the label (it is missing the top and bottom ends).

My approach was fairly simple, I allowed:
\[
\langle \cos(u),\sin(u), v\rangle=\langle x, y, z\rangle
\]
And integrated from there.

Answer 13

Ahh, gotcha. And it's quite all right, thanks.

Answer 14

That looks fairly solid, what happened afterwards?

Answer 15

I'm missing some factor somewhere, and landing with \(I=\pi\) instead of \(\frac{\pi}{2}\), for some reason or another.

Answer 16

I've just solved it and got pi. Do you want me to show you what I did or do you want to walk me through what you did and I'll point it out to you?

Answer 17

No, I'm good, I'm getting \(\pi\), as well. But (for some reason or another), it's supposed to be \(\frac{\pi}{2}\). I feel I'm dropping something somewhere that's absolutely obvious, but I can't quite see where.

Answer 18

Wait, I thought you said it was supposed to be pi earlier...? I feel like my reading skills are terrible today haha.

Answer 19

Jaja, I meant to say I got \(\pi\), even though it's supposed to be \(\pi/2\).

Probably my writing skills...

Answer 20

Oh I see what it is haha. The integral of z is not z.

Answer 21

Ok I'll show you, since there are two places where 1/2 crops up and that's just where my mistake was.

Answer 22

\[\Large \int\limits_0^1\int\limits_0^{2\pi}\cos^2\theta zd \theta dz=\int\limits_0^1zdz *\int\limits_0^{2\pi}\cos^2\theta d \theta\] Use the trig identity\[\Large \cos^2\theta = \frac{1}{2}(1+\cos(2\theta))\]

Answer 23

OH. Jeez. That was absolutely stupid. Thank you!

Answer 24

All good, any time! I don't see as many good calculus questions like this as I'd like!

Answer 25

Wow, totally missed the fact that I had that:
\[
\int_\gamma dz\, z
\]There....

Answer 26

This is sad, this is the review part from my complex analysis text... and I'm missing this stuff... jeez.

Answer 27

Thanks again, mate.

Answer 28

Haha no problem, I just got out of vector calculus last semester and I've just started to realize I had been mistaking all of calculus as being in the equations, but really we have all these different ways of representing the same thing and that the mathematical objects are actually kind of separate in a way. It makes it easier to do stuff in my opinion this way at least. I don't know if that helps you in any way or not haha.

Answer 29

Oh, yeah, definitely. Have you taken PDEs, yet? Or Fourier Analysis? That stuff will blow your mind. Also, it's going to happen that the better you get at math, the more crappy you become at the really basic computations (case 1 found above).

Answer 30

Also, of course, Tensors and Geometric Algebra. Stuff is fantastic, you'll enjoy it, if you like what you describe.

Answer 31

Like I used to think, "OK this is y=mx+b, what does that look like?" to thinking "I have this line that looks like this, where do I want to put it on the graph?"

Yeah, I took PDE last semester and I'm considering taking either fourier analysis or complex analysis next semester. Geometric Algebra is like the coolest thing in the whole world.

Answer 32

Seriously, I wish they just taught Geometric Algebra, it's basically just everything I ever learned, but it makes sense and it's super easy. Seriously, calculating determinants is just simply distributing a polynomial for crying out loud. The scalar and vector (bivector) product make so much sense after I started reading about it on the internet. I've not formally been taught it so I have a weird knowledge about it.

Answer 33

GA is possibly one of the prettiest subjects out there. The problem with teaching it early is that, while there are really fantastic explanations out there that make it fairly simple to understand, it's hard to observe and truly apply the power of it without a lot of work on vector analysis and differential forms, but I wish they did a little earlier, too--at least introduce it some time before hand.

I'd recommend going with Complex before Fourier since a lot of the deeper points in FA are kind of hard to get without complex analysis. But, you'll be fine either way you choose.

Answer 34

The problem is I'm a chemistry major and next semester will be my last. So I have to choose one or the other and that's it. I am trying to weigh which one will be easier to teach myself.

Answer 35

Oh, no, then you should definitely be fine either way. You can teach yourself both quite easily, though I must say that Fourier Analysis is the most clearly applicable of the two (though having Complex under your belt makes it a lot easier to understand some stuff). I personally did Fourier then Complex (working on it, now), which I believe turned out okay, but I think would have been better the other way around.

Doesn't really matter, though: if you've got the motivation, both should come fairly easily to you; plus there's plenty of great texts out there for self-study.

Answer 36

Do you know of any good resources for either on the web? I might have to just take both at the same time haha.

Answer 37

Jaja. Well, I'm working off of an older text from '56 (Danese's "Advanced Calculus"), but the Dover series for Fourier was very, very good, and it's fairly cheap; maybe about $4 when I got it?

If you're a Chem major, you might enjoy Boas's Mathematical Methods in the Physical Sciences, which introduces you to essentially all of the math you'll need to know for pretty much every science. It's a fantastic book for application, as I went through it once and still have it around, but it's very much application-focused, so the math is well-presented, but you have to keep in mind it's not a rigorous mathematics text.

Answer 38

Let me find you the other book, it's in my shelf.

Answer 39

It is Fourier Series by Tolstov

Answer 40

I think I've seen that and read some things out of Mathematical Methods actually. I recall learning some stuff about complex phase from there.

Answer 41

Great, great book. I'm an EE/CM Physics and definitely still go back to it every so often for techniques, I'd very much recommend getting it. There's also a similar type of book out there for free, I'd need to find it for you, though, but it's fairly nice, too.

Answer 42

I think I found one of them http://people.stfx.ca/x2011/x2011bhf/Mathematical%20Methods%20in%20the%20Physical%20Sciences-Third%20Edition.pdf

Answer 43

Aha. Found it, here:

http://www.physics.miami.edu/~nearing/mathmethods/

Answer 44

Thanks! This is great stuff, perfect!

I'm fairly comfortable with Fourier Series, the idea actually makes sense to me. I am not so sure about Fourier Transforms, but I am under the impression that it's essentially an infinite dot product.

I think the concept of convolution and dirac delta functions sort of mystify my a little bit still.

I'd really like to understand these ideas intuitively, so hopefully this helps.

Answer 45

Kind of, it's more like orthonormal sets of functions. But yes, go through some of these (which are more intuitive than rigorous) and you should definitely see the beauty in them.
Fantastic tools.

And, glad to help, man!

Answer 46

Yeah, this will be great. I think the main thing stopping me from understanding quantum mechanics is fourier analysis and the main thing stopping me from relativity is a reliable source of information and tensors.

Alright, I guess I'll go back to studying haha. Thanks again, you're saving my life over here.

Answer 47

Jaja, sure thing.

Answer 48

forget to integrate z's more often, ok? lol =P

Answer 49

Lol, I will try my best ! :)

Perhaps I'll put up some more interesting questions once I'm done with the review.