Vector Calc Question

[Sage]

I don’t understand, when doing vector analysis, when you can invoke symmetry. The only pattern I’ve noticed is that it seems to have to do with odd functions, but I’m not positive.

Example:

In my book, there’s an example verification of Green’s Theorem: P(x, y) = x, Q(x, y) = xy, D is the unit disk x^2 + y^2 ≤ 1. They evaluate it directly and get zero, then evaluate it using Green’s theorem, like so:

∫∫_D (∂Q/∂x – ∂P/∂y)dxdy = ∫∫_D ydxdy

which is zero by symmetry.

So, is it zero because y is an odd function? And, if so, does that only apply because the unit disk is a symmetrical region, or would that apply for any region D? And does this apply for vector integrals in general, or only when you’re using certain theorems, or…?

 

[Danoff]

Yea, Vector Cal was the most advanced math course I took - and I don't remember it well enough to answer your questions. I think you're better off with a TA to be honest. I'm assuming you're in this course right now.

Now if you had a question about Matrices I might be able to help. That one I actually use.

I can have my wife look at it later. She did better in Vector Cal than I did and really gets into this stuff.

Edit: If it's any help, I can tell you that the equation looks right.
Edit#2: What' the underscore represent in front of the D? Is that supposed to be a gradient?

 

[Sage]

Ah, that’s okay, I’ve figured it out – it is dependent on the region being integrated over.

Mrs. Danoff still does mathy stuff? Wow.

 

[Danoff]

Ah, that’s okay, I’ve figured it out – it is dependent on the region being integrated over.

Mrs. Danoff still does mathy stuff? Wow.

Not still, but she used to - and when she did, she was way better at it than I ever was. She gets excited when she does get to do math because it doesn't happen often.

 

[Sage]

That’s very interesting… I would’ve bet money that vector calc is the bread and butter of what you do.

Edit#2: What' the underscore represent in front of the D? Is that supposed to be a gradient?

That’s to represent a subscripted character (so, in this case, it’s to show that the function is being integrated over the region D). BTW, on that note, I hate vector calc notation so much – took me forever to figure out that ∂D means the boundary of D. And that the difference between ds and dS is a dotted unit vector. The math itself isn’t hugely challenging – the hard part is trying to figure out what the hell they want out of you.

 

[Danoff]

That’s to represent a subscripted character (so, in this case, it’s to show that the function is being integrated over the region D).

Oh I see. Yea that's starting to come back to me. But I've longsince forgotten what it means to integrate over a region. I should get a refresher or something. Green's theorem definitely rang a bell.

That’s very interesting – I would’ve bet money that vector calc is the bread and butter of what you do.

Not really. Matrix math is bread and butter. I do a lot of least-squares processing of massive and usually borderline singular matrices.

BTW - since I see that you're a math wiz, if you can figure out how to scale portions of a matrix without breaking correlations let me know. I basically tried scaling eigenvectors to no avail (still not totally sure why that didn't work).

Also I use a lot of optimization theory. Hamiltonian, Jacobian, Lagrangian,... other things that end in "ian".

 

[Sage]

No no no, “math wiz” I am most definitely not. (Actually, that should be a sad emoticon – I just took my final for this class, and I really don’t think I passed. Ugh.) I only sounded like I knew what I was talking about because I was studying for it.

 

[Brett]

If this had been a couple years ago when I was in calculus, then I may have been able to help you, as Green's theorem rings a bell, but I have not used any of it since finishing my calculus classes. Good luck! 👍