handout of vectors

MAS2104/MAS3104 Handout 5

Grad, Div and Curl in Cylindrical and Spherical Coordinates

In applications, we often use coordinates other than Cartesian coordinates. It is important to remember that expressionsfor the differential operators of vector calculus are different in different coordinates. Here we give explicit formulaefor cylindrical and spherical coordinates.

Cylindrical Polar Coordinates (, , z)

x = cos , y = sin , z = z , r = (x, y, z) = (, 0, z) ,

f = f

+ 1

f

+ z

f

z,

u = 1

(u)

+

1

u

+uzz

,

u = 1

z

zu u uz

=

(1

uz

uz

)+

(uz

uz

)+ z

1

(u)

u

.NB! The cylindrical radius =

x2 + y2 is often denoted s or r; in the latter case, it can be confused with the length

of the position vector, r = |r| = x2 + y2 + z2. Therefore, the notation r for the cylindrical radius, if used, is alwaysexplained explicitly.

MAS2104/MAS3104 Handout 5 2

Spherical Polar Coordinates (r, , )

x = r sin cos, y = r sin sin, z = r cos , r = (x, y, z) = (r, 0, 0) ,

f = r fr

+ 1

r

f

+

1

r sin

f

,

u = 1r2

(r2ur)

r+

1

r sin

(u sin )

+

1

r sin

u

,

u = 1r2 sin

r r r sin

r

ur ru ru sin

= r

1

r sin

[

(u sin ) u

]+

1

r

[1

sin

ur

r

(ru)

]+

1

r

[

r(ru) ur

].

http://www.mas.ncl.ac.uk/nas13/mas2104/