MTH623 Modules 22-45

Transformation equations between spherical and rectangular coordinates

The transformation equations expressing the

rectangular coordinates in terms of

spherical polar coordinates are

sin cos

sin sin

cos

squaringandaddingabove equations

sin cos sin sin cos

sin (cos sin cos

sin cos (sin cos x r

y r

z r

x y z r r r

r r

r r r r

r x y z sin cos sin sin cos ˆ sin cos sin sin cos

ˆ sin cos sin sin cos

cos cos cos sin sin ˆ cos cos cos sin sin

ˆ cos cos cos sin r

r r

i j k r e r

r

e i j k

r

r i r j r k e r r r r

e i j sin

sin sin sin cos ˆ sin sin sin cos

ˆ sin cos

In matrix notation

ˆ sin cos sin sin cos

ˆ cos cos cos sin sin

ˆ sin cos r k

r

r i r j e r r r

e i j

i e

e j

e k

Scale factors:

The scales factors for the spherical coordinate system are given by.

sin cos sin sin cos

sin (cos sin cos

sin cos

cos cos cos sin sin

cos (cos sin sin

cos sin

sin sin s r

r r h h r

h h

h h r r r

r r

r r r

h h r

h h r r in cos

sin (sin cos

sin sin

sin r

r r

h h r