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 expressions for the differential operators of vector calculus are differentin different coordinates. Here we give expli cit formul ae for cylindrical and spherical coordinates. Cylindrical Polar Coordinates (ρ , φ ,z ) x= ρ cos φ , y= ρ sin φ , z= z , r= (x,y,z ) = (ρ, 0, z) , ∇f= ρ ∂ f∂ρ + φ 1 ρ ∂f∂φ + z ∂ f∂z, ∇ · u = 1 ρ ∂ (ρu ρ ) ∂ρ + 1 ρ ∂u φ ∂φ + ∂ u z ∂z, ∇ × u = 1 ρ ρ ρ φ z ∂ ∂ρ ∂ ∂φ ∂ ∂zu ρ ρu φ u z = ρ 1 ρ ∂u z ∂φ − ∂u φ ∂z+ φ ∂u ρ ∂z− ∂u z ∂ρ + z 1 ρ ∂ (ρu φ ) ∂ρ − ∂u ρ ∂φ . NB! The cylindrical radius ρ= √ x 2 +y 2 is often denoted s or r; in the latter case, it can be confused with the length of the position vector, r =|r| = √ x 2 +y 2 +z2 . Ther efor e, the notationr for the cylindrical radius, if used, is always explained explicitly.
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) ,
