1.   (a) Let r = xi + yj + zk and E = r/|r|p , where p is positive. Find ∇.E. For what value of p is E divergence free?

(b) Let G = −yi + xj + zk, and S be the part of the sphere x2  + y2  + z2  = 25 below the plane z = 4. Let S have an the outward-pointing unit normal n e.g. at (5, 0, 0), n = i. Use Stokes’Theorem to calculate the lux integral

\\S curlG.n dS.

2.   (a) Find the Fourier series of 1 + sin2 x on the interval −T ≤ x ≤ T .

(b) Consider the odd function F(x) on [T,T] which is equal to x(T − x) on 0 ≤ x ≤ T . Show it has Fourier series

F(x) =   之   .

n  odd

Use Parseval’s theorem to show that

 +  +  + ··· =  .

3.   (a) Let L  =  L(x,y/ ) where y/   =  dy/dx be a Lagrangian satisfying the Euler- Lagrange equation which is independent of y . Show that

∂L

= constant.

(b) Let L =  ^1 + y/2/x be a Lagrangian.   Show that the extremal curves are circles with centre on the y-axis.

(c) Let L*  =^1 + y/2/y be a Lagrangian. Use Beltrami’s identity and the result in (b) above, or otherwise, to show that the extremal curves for this functional are circles with centre on the x-axis.

4. Use the method of characteristics to solve

(y + )     + y      = x − y,

in the region y > 0 subject to  = 1 + x on y = 1. Express your solution explicitly in terms of x and y .

5.   (a) Show that Euler-Lagrange equation for the functional L =  x(2)+y(2)  is equivalent to solving Laplace’s equation.  If (x,y) represents temperature, give a brief physical interpretation of this equivalence.

(b) Solve the PDE

?2           ?2              ?2

?x2        ?x?y       ?y2

subject to (x,0) = y (x, 0) = x.