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Question 4 (a) Use the divergence theorem to evaluate SS F.ndS S where F = (2x + 1y)i+(2+y)j+3zk and S is the part of the cylinder x²+y² = 4 between the surfaces z=0 and Z = 5. (b) The parametric equation of a surface S is r =ucosvi+usinvj+vk (0 susl; OSVSN / 2) Sketch the surface and use Stoke's theorem to evaluate the integral $ F. dr C where C is the border of S and F(r) = zi+xj+yk Question 5 (a) (i) A vector field B is solenoidal. Define a vector field A(r) by the integral 1 where i is a scalar parameter. Show that B - V ^ A and hence that A a vector potential that describes B. (Note: This is a general method for finding vector potentials of a field that has been shown to be solenoidal - it is an alternative to the differential equation approach) (ii) Use (i) to calculate a vector potential for the irrotational field - (b) Use the divergence theorem to show that JS (ax2 + + CZ - 4.t S where S is the closed surface formed by the unit sphere x2 + y2 + z2 - 1.

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