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^2yi- 2xy^2j- xyzk

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 ofVover the surface fo the cube and thereby verify the divergence theorem.

So for this I got both sides to be1/4 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) GivenV= (3x-2y)i+ x^2zj+ (1-2z)k

Find del.V and del x V and evaluate:[B]

I found them to be (3x-2y)i+ x^2yj+ (1-2z)k

and -x^2i+ (2xz-2)krespectively.

[B]i) double integral ofV.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 xV) . dS

over the same region.

iii) Evaluate

the closed loop integral ofV.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 xV)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