Orthogonal Curvilinear Coordinates - www.personal.psu.edu

Orthogonal Curvilinear Coordinates

Often it is convenient to use coordinate systems
other than rectangular ones. You are familiar, for
example, with polar coordinates (r, θ) in the plane.
One can transform from polar to regular (Cartesian)
coordinates by way of the transformation equations

x = r cos θ, y = r sin θ

One can summarize these as p = r cos θi + r sin θj,
where p is the position vector of the point P = (x, y).

Notice that the r coordinate curves and the θ
coordinate curves are orthogonal (at right angles)
to one another. This convenient property may be
expressed by saying that the vector partial derivatives
∂p/∂r and ∂p/∂θ are orthogonal, i.e., have zero dot
product.

The unit vectors er and eθ in the directions of
∂p/∂r and ∂p/∂θ form the moving frame associated to
the polar coordinate system. Vector-valued functions
may be expressed in terms of this moving frame.

– Typeset by FoilTEX – 1

By analogy, we define an orthogonal curvilinear
coordinate system in three dimensions to be a ‘non-
degenerate’ system of transformation equations

x = x(u, v, w), y = y(u, v, w), z = z(u, v, w)

or simply p = p(u, v, w), where the partial derivatives
∂p/∂u, ∂p/∂v, and ∂p/∂w should be mutually
orthogonal at each point.

Example 1 Cylindrical Coordinates (r, θ, z) (book
page 842)

x = r cos θ, y = r sin θ, z = z.

Example 2 Spherical Coordinates (ρ, φ, θ) (book
page 844)

x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.

– Typeset by FoilTEX – 2

Moving frame and scale factors

We put

∂p = hueu, ∂p = hvev, ∂p = hwew
∂u ∂v ∂w

where the e’s are (orthogonal) unit vectors — the
moving frame associated to the coordinate system —
and the h’s are scale factors given by

hu = ∂p , hv = ∂p , hw = ∂p .
∂u ∂v ∂w

For example consider cylindrical coordinates (r, θ, z).
From the transformation equations,

er = cos θi + sin θj, hr = 1

eθ = − sin θi + cos θj, hθ = r

ez = k, hz = 1. 3

– Typeset by FoilTEX –

Calculations in spherical coordinates

For spherical coordinates
x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.
Calculation gives
eρ = sin φ cos θi + sin φ sin θj + cos φk, hρ = 1,
eφ = cos φ cos θi + cos φ sin θj − sin φk, hφ = ρ,

eθ = − sin θi + cos θj, hθ = ρ sin φ.

– Typeset by FoilTEX – 4

The gradient in curvilinear coordinates

Theorem Let f be a function. In orthogonal
curvilinear coordinates (u, v, w) the gradient of f can
be expressed by

∇f = 1 ∂f eu + 1 ∂f ev + 1 ∂f ew.
hu ∂u hv ∂v hw ∂w

Proof Look at the chain rule to see

∂f = ∇f · ∂p = hu∇f · eu
∂u ∂u

with similar calculations for v and w.
See Exercises 11,12 on book p.999

– Typeset by FoilTEX – 5

The Laplacian in curvilinear coordinates

The Laplacian of f is the expression

∇2f = ∂2f + ∂2f + ∂2f
∂x2 ∂y2 ∂z2

that appears in Laplaces equation. In curvilinear
coordinates this becomes

∇2f = 1 ∂ hvhw ∂f
huhvhw ∂u hu ∂u

+ ∂ hwhu ∂f
∂v hv ∂v

+ ∂ huhv ∂f
∂w hw ∂w

The proof will be easier once we have studied the
Divergence Theorem.

– Typeset by FoilTEX – 6