Vector Calculus: Divergence theorem and Stokes' theorem
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?!
the double integral of (del x V) . dS
over the same region.
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.
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