Appendix B
Vector calculus in orthogonal curvilinear
coordinates
Derivation of generalized relations
Letx 1 ,x 2 ,x 3 be a right-handed set of orthogonal coordinates and letsbe the distance
along a curve in three dimensional space. Lethibe the scale factor relating an increment
in theithcoordinate to an increment in length in that direction so
dli=hidxi; (B.1)
an example is theφdirection of cylindrical coordinates wheredlφ=rdφ.Since the co-
ordinates are assumed to be orthogonal, the increment in distance along a curve in three
dimensional space is
(ds)^2 =h^21 (dx 1 )^2 +h^22 (dx 2 )^2 +h^23 (dx 3 )^2. (B.2)
The{x 1 ,x 2 ,x 3 }coordinates and corresponding{h 1 ,h 2 ,h 3 }scale factors for Cartesian,
cylindrical, and spherical coordinate systems are listed in Table B.1 below.
coordinate system distance along a curve {x 1 ,x 2 ,x 3 } {h 1 ,h 2 ,h 3 }
Cartesian (dx)^2 +(dy)^2 +(dz)^2 {x,y,z} { 1 , 1 , 1 }
cylindrical (dr)^2 +(rdφ)^2 +(dz)^2 {r,φ,z} { 1 ,r, 1 }
spherical (dr)^2 +(rdθ)^2 +(rsinθdφ)^2 {r,θ,φ} { 1 ,r,rsinθ}
Table B.1: Coordinate systems and their scale factors
The differential of the scalarψfor a displacement bydl 1 of the coordinatex 1 is
dψ=dl 1 ˆx 1 ·∇ψ (B.3)
so the component of the gradient operator in the direction ofˆx 1 is
xˆ 1 ·∇ψ=
dψ
dl 1
=
1
h 1
∂ψ
∂x 1
. (B.4)
The partial derivative notation is invoked in the right-most expression because the displace-
ment is just in the direction ofx 1.
Generalized gradient operator
Because the coordinates are independent, a displacement in the direction of one coor-
dinate does not affect the dependence on the other coordinates and so the gradient operator
is just the sum of its components in the three orthogonal directions, i.e.,
∇=
ˆx 1
h 1
∂
∂x 1
+
xˆ 2
h 2
∂
∂x 2
+
xˆ 3
h 3
∂
∂x 3