Math 509: Advanced Analysis

Homework 10

Lawrence Tyler Rush

<me@tylerlogic.com>

April 14, 2015

http://coursework.tylerlogic.com/courses/upenn/math509/homework10

According to the notes, an infinitesimal displacement has

dxi+ dyj+ dzk = dℓ = drˆr+ rdθˆθ+ rsin θdφˆφ

(3.1)

which gives us a relationship between cartesian and spherical coordinates. Now we also know that

x = rsin θcosφ y = rsin θsin φ z = rcosθ

Differentiating each of these equations yields

dx	= sinθ cosφdr + r cosθ cosφdθ - r sinθ sinφdφ
dy	= sinθ sinφdr + r cosθ sinφdθ + r sinθ cosφdφ
dz	= cosθdr - r sinθdθ

We can then plug these values into the left-hand side of equation 3.1 to obtain

dr + rdθ+ r sinθdφ	= (sinθ cosφdr + r cosθ cosφdθ - r sinθ sinφdφ)i+
	(sinθ sinφdr + r cosθ sinφdθ + r sinθ cosφdφ)j+
	(cosθdr - r sinθdθ)k

Grouping the right-hand side by dr, rdθ, and r sinθdφ we then obtain

dr + rdθ+ r sinθdφ	= (sinθ cosφi + sinθ sinφj + cosθk)dr
	+ (cosθ cosφi + cosθ sinφj - sinθk)rdθ
	+ (cosφj - sinφi)r sinθdφ

implying that