2.5a) On R1 let f(x) be continuous function which may change sign and such that
. Define fk(x) by compression as fk(x) = kf(kx) and set
. Show that rk(x) teands pointwise to the Heaviside function H(x) and that
, where c is a constant independent of k and x. It therefore follows the Lebesque Dominated Theorem that
or, equivalently, rk(x) -> H(x) distributionally.
let:
Therefore, if y > 0 then rk = 1
, if y < 0 then rk = 0
2.14) Coordinate Transformations. Show that if u1,u2, u3are the spherical coordinates in R3 usually denoted by r, q, f, respectively, then J = r2sinq and
Apply....(2.24)
(b) For the spherical coordinates of part (a), show that a point source at the orgin can be represented as:
by considering the limit as r` -> 0 of a uniform simple layer of unit strength on the sphere r = r`
(c) For cylindrical coordinates r, q, z, show that:
Apply....(2.24)
Show that a uniform ring of sources of unit strength on r = r', z = 0 has representation:
and that for unit source at the origin
Integrate around ring
unit source at origin...
apply a limit to previous to bring the ring down from a radius of r' to zero
5.7) Consider one-dimensionl unsteady diffusion in an absorbing medium. The causal fundamental solution E with pole at x = 0, t=0 satisfies:
Reduce the problem to ordinary diffusion by the transformation E = e-q^2*t * F
