(a) Use the divergence theorem to evaluate
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
(0 susl; OSVSN / 2)
Sketch the surface and use Stoke's theorem to evaluate
$ F. dr
where C is the border of S and
F(r) = zi+xj+yk
(i) A vector field B is solenoidal. Define a vector field A(r) by
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
where S is the closed surface formed by the
unit sphere x2 + y2 + z2 - 1.
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