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# Multiple Integrals, Vector Fields, Hemispheres and Divergence Theorem

B10. (a) State the Divergence Theorem, being careful to explain any notation you use and any conditions that must apply.
The vector field B is given by
B = Rcos &#952;(cos &#952;R - sin &#952; ^&#952; )
in spherical polar coordinates (R; &#952;; &#966;). This field exists in a region which includes the hemisphere x2 + y2 + z2  a2; z  0. By direct evaluation of the two surface integrals,
Find the
ux
R R
B  ^n dS of B out of the (closed) surface of the hemisphere.
[Hint: this surface consists of two parts, a hemispherical cap and a at base, which are respectively R = const and &#952; = const.]
Now use the given formula (3) for the divergence to show that div B = 1, and confirm
ux by an application of the Divergence Theorem. [8 marks]
(b) The vector field F is given by
F = (x2y2 + 2xe&#56256;&#56320;y)i + (2x3y &#56256;&#56320; x2e&#56256;&#56320;y + 1)j
where  is a constant. For a certain value of , which you should find, the line integral
S(r) =
Z r
0
F  dr
is independent of the path from 0 to r. Find S(r) for this particular value of .
[4 marks]

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Multiple Integrals, Vector Fields, Hemispheres and Divergence Theorem are investigated.

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