Question

How to attack it, please, help

Answer 1

@oldrin.bataku

Answer 2

I am supposed to use integral by parts here.

Answer 3

@dan815

Answer 4

@imqwerty

Answer 5

@sleepyjess

Answer 6

@♪Chibiterasu

Answer 7

hi loser

you're not having much luck getting this done Loser so i'll throw in my 2 cents, though it may not be helpful. first of all look at this equation taken straight from wiki:

\[\Large \iiint_V \left[\mathbf{G}\cdot\left(\nabla\times\mathbf{F}\right) - \mathbf{F}\cdot \left( \nabla\times\mathbf{G}\right)\right]\, dV = \Large \iint_S(\mathbf F\times\mathbf{G}) \cdot d\mathbf{S}\]

https://en.wikipedia.org/wiki/Divergence_theorem

this is written in language i understand better (eg dV is a plain old volume element) and you should recognise the first two terms as your first two terms, though written the other way around

if i re-write it, you actually have this:

\[\Large \iiint_V \left[ \mathbf{F}\cdot \left( \nabla\times\mathbf{G}\right)\right] - \mathbf{G}\cdot\left(\nabla\times\mathbf{F}\right) \, dV = -\Large \iint_S(\mathbf F\times\mathbf{G}) \cdot d\mathbf{S} \\ = \Large \iint_S(\mathbf G\times\mathbf{F}) \cdot d\mathbf{S}\]

so we are using the divergence theorem so we need to establish that \(\nabla.(\vec F \times \vec G) = \mathbf{G}\cdot\left(\nabla\times\mathbf{F}\right) - \mathbf{F}\cdot \left( \nabla\times\mathbf{G}\right)\).

Well that's an identity. you'll find it here somewhere:
https://en.wikipedia.org/wiki/Vector_calculus_identities

the last bit then is reconciling this with your nomenclature

in Wiki, the \(\mathbf {dS}\) means \(\hat {dS}\), it is the area element wrapped in a unit normal vector, very handy thing to do. and so the wiki thing can be written as

\(\Large \iint_S(\mathbf G\times\mathbf{F}) \cdot \hat n \; dS \) where \(\hat n \) is the outward normal vector to the surface S, dS is just a scalar area element, ..... and \(\hat n\) called v in your terminology

you can then re-arrange that scalar triple so that is is \((\vec F \times \hat n). \vec G\) and you get what they want

in the languange of my dodgy applied maths though so maybe not that helpful....:-(

[forgive any typos]

Answer 8

Thanks @IrishBoy123 Actually, I have it "almost done", just stuck at integral of the boundary. Let me post my work.

Answer 9

For the first part, the left hand side:
\[\int_{Omega}<\vec F(x), curl \vec G(x)>dx\]

Answer 10

this is the second part

Answer 11

I got it right already for part a)
Now I try to attack part b by applying part a) . Any idea would be appreciated. :)

Answer 12

medal for bravery loser!

and i still don't know what any of \(C^1, C^2, \bar{\Omega}\) actually means.....!