Thread: Vector Calculus: Divergence theorem and Stokes' theorem

Hi =)
I've been working on some exercises, but unfortunately we are not given the answers, so have no idea if I am going wrong or not. I attemted part a) which reads:
Consider the vector field
V= 3x^2y i - 2xy^2 j - xyz k

Determine the divergence of this vector field and evaluate the integral of this quantity over the interior of a cube of side a in z> or equal to 0, whose base has the vertices (0,0,0), (a,0,0), (0,a,0), (a,a,0).
Evaluate the flux of V over the surface fo the cube and thereby verify the divergence theorem.

So for this I got both sides to be 1/4 a^5

Its the next part that has got me completely confused! (its long, here goes..! if anyone can teach me how to use the integral symbols and stuff, i will edit this post to make more sense!)

[B]b) Given V = (3x-2y) i + x^2z j + (1-2z) k
Find del.V and del x V and evaluate:[B]
I found them to be (3x-2y) i + x^2y j + (1-2z)k
and -x^2 i + (2xz-2) k respectively.

[B]i) double integral of V.dS
over the circular region in the xy-plane bounded by x^2 + y^2 = a^2. Regard the area element as positive in the positive z direction.[B]

What is dS supposed to be?!

ii) Evaluate
the double integral of (del x V) . dS
over the same region.

iii) Evaluate
the closed loop integral of V.dr
clockwise looking in the positive z direction around the circle x^2 + y^2 = a^2 in the xy-plane.

iv) Evaluate
the triple integral of (del x V)dV
over the volume of the hemisphere bounded by the spherical surface x^2 + y^2 + z^2= a^2 for z>0 and the xy-plane.

Which two answers provide a verification of Stokes' Theorem?

Thank you SO much

Can no one help me?

first you can quote other's post, so that you can see the source of the LaTex content. It's only a simple tag "math" warrping the LaTex source code.



and dS is the volume element vector.
is the surface integral with V.n as the integrand, where n is the outer pointing unit normal vector of the surface.

And I think if your results (surface integral and volume integral) agree, they are supposed to be both correct.