3. (a) Sketch and fully describe the surface Sph given, in terms of Cartesian coordinates

(x, y, z), by

Sph = {(x, y, z) : (x − xo)² + (y − yo)² + (z − zo)² = a²},

where a, xo, yo and zo are constants, and show that it can be parameterized by

r(0,0) = (xo + a sin cos , yo + a sin sind, zo + a cos 0), giving the ranges of

the parameters and p.

(b) For the surface Sph, defined in part (a), show that a vector surface element is given

by ds = a sin 0 [r - (To, yo, 2o)] dedo, and justify the sign convention associated

with this normal vector. Hence evaluate the flux

ƒ= $₁₂

Sph

G.dS

of the vector field G = (1,0, z²) out of the surface.

(c) Using your answer to part (b), evaluate L = lima-03f/(4πa³).

(d) State the (coordinate-invariant) integral form of the definition of divergence. By

referring to your answer to part (c), demonstrate that it is consistent with the

Cartesian definition of divergence in this case.

(e) By applying the Divergence Theorem to the vector field u = ow, where is an

arbitrary differentiable scalar field and w is a constant vector, derive the identity

JI

føds,

where the closed surface S is the boundary of the volume V.

VodV=