Unit 4: Orthogonal Curvilinear Coordinates
Table of Contents
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 |