Fundamentals of Plasma Physics

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

. (B.5)

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