In n + 1 space dimensions we have Cartesian coordinates a = (1, n+1) and spherical...

 

Transcribed Image Text:

In n + 1 space dimensions we have Cartesian coordinates a = (1, n+1) and spherical polar coordinates (r. 2), where :-) (...) are n angles. They are related by ra]. The integral over all space can be written in either set of coordinates as 1x=1 " r" dr. Here S" is the n-sphere || = 1, and the angles 2 serve as coordinates on S". Let A = - d"n (2) be the area of S. [In this problem we do not need explicit expressions for d" or An, but you know that A = 2 and A = 4m, and d' = d0 and = sin 0 de de. ] a) For n 1, show that the n+1-dimensional 6-function is given in polar coordinates by 8(x):= 8(x)...(n+1) = lim (3) [Hint: Show that 8(a)o(a)d"+= 6(0) for any test function . Integrate first with e > 0, then take the limit e 0.] b) For n 21, the Laplace operator in n +1 space dimensions is 8 n 1 A = + + dr. r dr 8(re) Ar For n 22, show that it is given by (derivatives in f). [You do not need the complete expression, but you know it for n = 1 and n = 2.] The Green's function for the Poisson equation Au(x) = f(x) in n + 1 dimensions is defined by A,G(x, y) = 8(x - y). H(r-e) 40 C-1 (1) G(x, y)=G(z)=G(r) = -lim (5) where z:- z-y and r:= z), and find the value of the constant C Hint: Express (5) in spherical polar coordinates. Multiply by a spherically symmetric test function (r) and integrate over all space, use integration by parts in r (several times), then take the limit 0 of both sides at the end to show that (6) holds.] ->> In n + 1 space dimensions we have Cartesian coordinates a = (1, n+1) and spherical polar coordinates (r. 2), where :-) (...) are n angles. They are related by ra]. The integral over all space can be written in either set of coordinates as 1x=1 " r" dr. Here S" is the n-sphere || = 1, and the angles 2 serve as coordinates on S". Let A = - d"n (2) be the area of S. [In this problem we do not need explicit expressions for d" or An, but you know that A = 2 and A = 4m, and d' = d0 and = sin 0 de de. ] a) For n 1, show that the n+1-dimensional 6-function is given in polar coordinates by 8(x):= 8(x)...(n+1) = lim (3) [Hint: Show that 8(a)o(a)d"+= 6(0) for any test function . Integrate first with e > 0, then take the limit e 0.] b) For n 21, the Laplace operator in n +1 space dimensions is 8 n 1 A = + + dr. r dr 8(re) Ar For n 22, show that it is given by (derivatives in f). [You do not need the complete expression, but you know it for n = 1 and n = 2.] The Green's function for the Poisson equation Au(x) = f(x) in n + 1 dimensions is defined by A,G(x, y) = 8(x - y). H(r-e) 40 C-1 (1) G(x, y)=G(z)=G(r) = -lim (5) where z:- z-y and r:= z), and find the value of the constant C Hint: Express (5) in spherical polar coordinates. Multiply by a spherically symmetric test function (r) and integrate over all space, use integration by parts in r (several times), then take the limit 0 of both sides at the end to show that (6) holds.] ->>