Unit 4: Orthogonal Curvilinear Coordinates

1. Definition and Examples

Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These systems are orthogonal if the coordinate surfaces intersect at right angles at every point.

Common examples used in physics include:

• Cylindrical Coordinates: Useful for problems with axial symmetry (e.g., flow in a pipe).
• Spherical Coordinates: Essential for problems with central symmetry (e.g., gravitational fields, hydrogen atom).

2. Coordinate Transformations

Transformation involves converting coordinates from a Curvilinear system (u1, u2, u3) to the standard Cartesian system (x, y, z) and vice versa.

x = x(u1, u2, u3), y = y(u1, u2, u3), z = z(u1, u2, u3)

The scale factors (h1, h2, h3) are critical for these transformations as they relate the differential change in a coordinate to a physical length.

3. Line, Surface, and Volume Elements

In an orthogonal curvilinear system, infinitesimal elements are defined using scale factors h1, h2, and h3:

• Line Element (dl): dl² = h1²du1² + h2²du2² + h3²du3²
• Surface Elements (dS): dS1 = h2*h3 du2*du3, dS2 = h1*h3 du1*du3, etc.
• Volume Element (dV): dV = h1*h2*h3 du1*du2*du3

4. Vector Operators (Grad, Div, Curl, Laplacian)

The general expressions for vector operators in orthogonal curvilinear coordinates depend on the scale factors:

Gradient (∇f)

∇f = (1/h1)(df/du1) e1 + (1/h2)(df/du2) e2 + (1/h3)(df/du3) e3

Divergence (∇·A)

∇·A = [1/(h1h2h3)] * [ d/du1(A1h2h3) + d/du2(A2h1h3) + d/du3(A3h1h2) ]

5. Cylindrical Coordinate System (ρ, φ, z)

Used for systems with a central axis.

• Transformations: x = ρ cos φ, y = ρ sin φ, z = z
• Scale Factors: hρ = 1, hφ = ρ, hz = 1
• Volume Element: dV = ρ dρ dφ dz

6. Spherical Coordinate System (r, θ, φ)

Used for systems with a central point.

• Transformations: x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ
• Scale Factors: hr = 1, hθ = r, hφ = r sin θ
• Volume Element: dV = r² sin θ dr dθ dφ