math guide - 34.49

z

z

( x, y, z) ↔ ( r, θ , z)

z

z

y

r x

x	y			θ
x	= r cos θ	r	=	x2 + y2
y	= r sin θ	θ	=	y
				atan -
				x

34.4.3Spherical Coordinates

•This system replaces the cartesian box with a sphere,

math guide - 34.50

z		z
	( x, y, z) ↔ ( r, θ ,φ )	φ

z

r

y

x

x

y

θ

x

= r sin θ cosφ

r =

x2 + y2 + z2

y

= r sin θ sinφ

θ

y

= atan -

x

z

= r cos θ

φ

z

= acos -

r

34.4.4 Practice Problems

1. Simplify the following expressions.

a) 16

---------------------

( 4j + 4) 2

b)3j + 5

---------------------

( 4j + 3) 2

c) ( 3 + 5j) 4j

where, j = –1