Unit 4: Orthogonal Curvilinear Coordinates

Definitions and Properties
Coordinate Transformations
Vector Elements (Displacement, Arc, Area, Volume)
General Expressions for Grad, Div, Curl, and Laplacian
Cylindrical Coordinates (ρ, φ, z)
Spherical Coordinates (r, θ, φ)

Cartesian coordinates (x, y, z) are simple but not always convenient. For problems with cylindrical or spherical symmetry, using coordinates that match the symmetry (like cylindrical or spherical coordinates) simplifies the math significantly. These are examples of orthogonal curvilinear coordinates.

Definitions and Properties

Curvilinear Coordinates (u₁, u₂, u₃)

A point P in space can be defined by the intersection of three surfaces:

u₁(x,y,z) = c₁, u₂(x,y,z) = c₂, u₃(x,y,z) = c₃

The values (u₁, u₂, u₃) are the curvilinear coordinates of P.

Orthogonal System

The system is orthogonal if these three coordinate surfaces are mutually perpendicular at every point of intersection. This means the tangent vectors to the coordinate lines are orthogonal.

Scale Factors (h₁, h₂, h₃)

The coordinates u₁, u₂, u₃ may not have units of length (e.g., angles). The scale factors (or "metric coefficients") `h₁, h₂, h₃` convert an infinitesimal change in a coordinate `duᵢ` into an actual physical displacement `dsᵢ`.

Let r = x(u₁,u₂,u₃)î + y(u₁,u₂,u₃)ĵ + z(u₁,u₂,u₃)k̂ be the position vector.

The scale factors are defined as the magnitudes of the tangent vectors:

h₁ = |∂r/∂u₁|, h₂ = |∂r/∂u₂|, h₃ = |∂r/∂u₃|

An infinitesimal displacement `dsᵢ` along the uᵢ-axis is `dsᵢ = hᵢ duᵢ`.

The total displacement `dr` is `dr = h₁du₁ ê₁ + h₂du₂ ê₂ + h₃du₃ ê₃`, where `ê₁, ê₂, ê₃` are the orthogonal unit vectors in the direction of increasing u₁, u₂, u₃.

The square of the arc length `ds` is:

ds² = |dr|² = (h₁du₁)² + (h₂du₂)² + (h₃du₃)²

Coordinate Transformations

This involves the set of equations that relate (x,y,z) to (u₁,u₂,u₃) and vice versa.

Curvilinear to Cartesian (Vice Versa)

We need the functions `x(u₁,u₂,u₃)`, `y(u₁,u₂,u₃)`, `z(u₁,u₂,u₃)` and their inverses `u₁(x,y,z)`, `u₂(x,y,z)`, `u₃(x,y,z)`.

Example: Cylindrical Coordinates (ρ, φ, z)

Cylindrical to Cartesian:
- x = ρ cos(φ)
- y = ρ sin(φ)
- z = z
Cartesian to Cylindrical:
- ρ = (x² + y²)^1/2
- φ = arctan(y/x)
- z = z

Example: Spherical Coordinates (r, θ, φ)

Spherical to Cartesian:
- x = r sin(θ) cos(φ)
- y = r sin(θ) sin(φ)
- z = r cos(θ)
Cartesian to Spherical:
- r = (x² + y² + z²)^1/2
- θ = arccos(z / (x² + y² + z²)^1/2)
- φ = arctan(y/x)

Vector Elements (Displacement, Arc, Area, Volume)

Once we have the scale factors `h₁, h₂, h₃`, we can define all the differential elements needed for integration.

Displacement Vector (dr)

An infinitesimal displacement vector `dr` is given by:

dr = h₁du₁ ê₁ + h₂du₂ ê₂ + h₃du₃ ê₃

Arc Length (ds)

The magnitude of the displacement is the arc length `ds`. From the Pythagorean theorem in 3D:

ds² = |dr|² = (h₁du₁)² + (h₂du₂)² + (h₃du₃)²

Area Elements (dA)

An area element `dA` is a small patch on a coordinate surface. It's a vector `dA = n̂ dA`, where `n̂` is the normal unit vector.

Area element on the u₁=const surface (normal is ê₁):
dA₁ = (ds₂) (ds₃) = (h₂du₂) (h₃du₃) = h₂h₃ du₂du₃
Area element on the u₂=const surface (normal is ê₂):
dA₂ = (ds₁) (ds₃) = (h₁du₁) (h₃du₃) = h₁h₃ du₁du₃
Area element on the u₃=const surface (normal is ê₃):
dA₃ = (ds₁) (ds₂) = (h₁du₁) (h₂du₂) = h₁h₂ du₁du₂

Volume Element (dV)

The infinitesimal volume element `dV` is a small box formed by the three displacements.

dV = (ds₁) (ds₂) (ds₃) = (h₁du₁) (h₂du₂) (h₃du₃) = h₁h₂h₃ du₁du₂du₃

The term J = h₁h₂h₃ is called the Jacobian of the transformation.

General Expressions for Grad, Div, Curl, and Laplacian

Using the scale factors, we can write general expressions for the vector operators. Let `f` be a scalar field and `V = V₁ê₁ + V₂ê₂ + V₃ê₃` be a vector field.

Gradient (∇f)

∇f = (1/h₁)(∂f/∂u₁)ê₁ + (1/h₂)(∂f/∂u₂)ê₂ + (1/h₃)(∂f/∂u₃)ê₃

Divergence (∇ · V)

∇ · V = (1 / h₁h₂h₃) [ ∂/∂u₁(h₂h₃V₁) + ∂/∂u₂(h₁h₃V₂) + ∂/∂u₃(h₁h₂V₃) ]

Curl (∇ × V)

This is best remembered as a determinant:

        |  h₁ê₁    h₂ê₂    h₃ê₃   |
∇ × V = (1 / h₁h₂h₃) * | ∂/∂u₁   ∂/∂u₂   ∂/∂u₃  |
        |  h₁V₁    h₂V₂    h₃V₃   |

Laplacian (∇²f)

The Laplacian is the divergence of the gradient (∇²f = ∇ · (∇f)).

∇²f = (1 / h₁h₂h₃) [ ∂/∂u₁( (h₂h₃/h₁)(∂f/∂u₁) ) + ∂/∂u₂( (h₁h₃/h₂)(∂f/∂u₂) ) + ∂/∂u₃( (h₁h₂/h₃)(∂f/∂u₃) ) ]

These formulas are the "master templates". You just need to plug in the scale factors for any specific coordinate system.

Cylindrical Coordinates (ρ, φ, z)

The coordinates are `(u₁, u₂, u₃) = (ρ, φ, z)`.

`ρ`: radial distance from the z-axis (0 to ∞)
`φ`: azimuthal angle from the x-axis (0 to 2π)
`z`: height along the z-axis (-∞ to ∞)

Transformations: `x = ρ cos(φ)`, `y = ρ sin(φ)`, `z = z`

Scale Factors for Cylindrical

From `r = ρcos(φ)î + ρsin(φ)ĵ + zk̂`:

∂r/∂ρ = cos(φ)î + sin(φ)ĵ → h_ρ = |∂r/∂ρ| = 1
∂r/∂φ = -ρsin(φ)î + ρcos(φ)ĵ → h_φ = |∂r/∂φ| = ρ
∂r/∂z = k̂ → h_z = |∂r/∂z| = 1

So, (h₁, h₂, h₃) = (1, ρ, 1).

Jacobian: J = h₁h₂h₃ = (1)(ρ)(1) = ρ

Vector Elements

Arc Length: ds² = (1dρ)² + (ρdφ)² + (1dz)² = dρ² + ρ²dφ² + dz²
Volume Element: dV = (1)(ρ)(1) dρ dφ dz = ρ dρ dφ dz

Vector Operators

Plug (h₁, h₂, h₃) = (1, ρ, 1) into the general formulas:

Gradient: ∇f = (∂f/∂ρ)ê_ρ + (1/ρ)(∂f/∂φ)ê_φ + (∂f/∂z)ê_z
Divergence: ∇ · V = (1/ρ) [ ∂/∂ρ(ρV_ρ) + ∂/∂φ(V_φ) + ∂/∂z(ρV_z) ]
= (1/ρ)(∂(ρV_ρ)/∂ρ) + (1/ρ)(∂V_φ/∂φ) + (∂V_z/∂z)
Laplacian: ∇²f = (1/ρ) [ ∂/∂ρ( ρ(∂f/∂ρ) ) + ∂/∂φ( (1/ρ)(∂f/∂φ) ) + ∂/∂z( ρ(∂f/∂z) ) ]
= (1/ρ)∂/∂ρ(ρ ∂f/∂ρ) + (1/ρ²) (∂²f/∂φ²) + (∂²f/∂z²)

Spherical Coordinates (r, θ, φ)

The coordinates are `(u₁, u₂, u₃) = (r, θ, φ)`.

`r`: radial distance from the origin (0 to ∞)
`θ`: polar angle from the z-axis (0 to π)
`φ`: azimuthal angle from the x-axis (0 to 2π)

Transformations: `x = r sin(θ) cos(φ)`, `y = r sin(θ) sin(φ)`, `z = r cos(θ)`

Scale Factors for Spherical

From `r = rsinθcosφ î + rsinθsinφ ĵ + rcosθ k̂`:

∂r/∂r = sinθcosφ î + sinθsinφ ĵ + cosθ k̂ → h_r = |∂r/∂r| = 1
∂r/∂θ = rcosθcosφ î + rcosθsinφ ĵ - rsinθ k̂ → h_θ = |∂r/∂θ| = r
∂r/∂φ = -rsinθsinφ î + rsinθcosφ ĵ → h_φ = |∂r/∂φ| = r sin(θ)

So, (h₁, h₂, h₃) = (1, r, r sin(θ)).

Jacobian: J = h₁h₂h₃ = (1)(r)(r sin(θ)) = r² sin(θ)

Vector Elements

Arc Length: ds² = (1dr)² + (rdθ)² + (r sinθ dφ)² = dr² + r²dθ² + r²sin²θ dφ²
Volume Element: dV = (1)(r)(r sinθ) dr dθ dφ = r² sin(θ) dr dθ dφ
Area Element (on surface of sphere, r=const): dA = (ds_θ)(ds_φ) = (rdθ)(r sinθ dφ) = r² sin(θ) dθ dφ

Vector Operators (Laplacian is most important)

Plug (h₁, h₂, h₃) = (1, r, r sin(θ)) into the general formulas:

Gradient: ∇f = (∂f/∂r)ê_r + (1/r)(∂f/∂θ)ê_θ + (1/(r sinθ))(∂f/∂φ)ê_φ
Divergence: ∇ · V = (1/r²sinθ) [ ∂/∂r(r²sinθ V_r) + ∂/∂θ(r sinθ V_θ) + ∂/∂φ(r V_φ) ]
= (1/r²)∂/∂r(r²V_r) + (1/(r sinθ))∂/∂θ(sinθ V_θ) + (1/(r sinθ))∂V_φ/∂φ

Laplacian in Spherical Coordinates (HIGHLY important for exams):
This operator is the key to solving the Schrödinger equation for the Hydrogen atom.
Using the general formula `∇²f = ∇ · (∇f)`:

∇²f = (1/r²)∂/∂r(r² ∂f/∂r) + (1/(r²sinθ))∂/∂θ(sinθ ∂f/∂θ) + (1/(r²sin²θ))(∂²f/∂φ²)

You should be able to derive this from the general formula or, in many cases, memorize it directly.

Unit 4: Orthogonal Curvilinear Coordinates

Table of Contents

Definitions and Properties

Curvilinear Coordinates (u₁, u₂, u₃)

Orthogonal System

Scale Factors (h₁, h₂, h₃)

Coordinate Transformations

Curvilinear to Cartesian (Vice Versa)

Vector Elements (Displacement, Arc, Area, Volume)

Displacement Vector (dr)

Arc Length (ds)

Area Elements (dA)

Volume Element (dV)

General Expressions for Grad, Div, Curl, and Laplacian

Gradient (∇f)

Divergence (∇ · V)

Curl (∇ × V)

Laplacian (∇²f)

Cylindrical Coordinates (ρ, φ, z)

Scale Factors for Cylindrical

Vector Elements

Vector Operators

Spherical Coordinates (r, θ, φ)

Scale Factors for Spherical

Vector Elements

Vector Operators (Laplacian is most important)