In advanced electromagnetism and fluid dynamics, standard Cartesian coordinates often complicate problems with spherical or cylindrical symmetry. To solve these efficiently, we use Orthogonal Curvilinear Coordinates. This guide derives the generalized expressions for Gradient, Divergence, and Curl using scale factors \( h_1, h_2, h_3 \).
Figure 1: An elementary volume in curvilinear coordinates defined by \( u, v, w \) with edges \( h_1 du, h_2 dv, h_3 dw \).
Generalized Vector Operators: Gradient, Divergence, and Curl
1. Scale Factors and Coordinates
Let the coordinates be \( (u, v, w) \). The position vector is \( \mathbf{r}(u, v, w) \). Since coordinate lines may be curved, a small change \( du \) does not necessarily correspond to a length \( du \). We introduce Scale Factors (Metric Coefficients) \( h_i \) to convert coordinate changes to physical arc lengths.
The squared arc length \( ds^2 \) is given by:
\[ ds^2 = (h_1 du)^2 + (h_2 dv)^2 + (h_3 dw)^2 \]
The elementary volume is:
\[ dV = (h_1 du)(h_2 dv)(h_3 dw) = h_1 h_2 h_3 \, du \, dv \, dw \]
2. The Gradient (\( \nabla \psi \))
The gradient relates the change in a scalar field \( \psi \) to displacement. The total differential is:
\[ d\psi = \nabla \psi \cdot d\mathbf{r} \]
In curvilinear coordinates, the displacement vector is \( d\mathbf{r} = h_1 du \hat{e}_1 + h_2 dv \hat{e}_2 + h_3 dw \hat{e}_3 \).
Writing \( \nabla \psi \) as components \( (\nabla \psi)_1 \hat{e}_1 + (\nabla \psi)_2 \hat{e}_2 + (\nabla \psi)_3 \hat{e}_3 \), the dot product becomes:
\[ d\psi = (\nabla \psi)_1 h_1 du + (\nabla \psi)_2 h_2 dv + (\nabla \psi)_3 h_3 dw \]
Comparing this with the chain rule expansion \( d\psi = \frac{\partial \psi}{\partial u} du + \frac{\partial \psi}{\partial v} dv + \frac{\partial \psi}{\partial w} dw \), we equate coefficients to find the components:
3. The Divergence (\( \nabla \cdot \mathbf{A} \))
Divergence is the net flux per unit volume. Consider a small volume element with faces perpendicular to the coordinate axes.
Let vector \( \mathbf{A} = A_1 \hat{e}_1 + A_2 \hat{e}_2 + A_3 \hat{e}_3 \).
Derivation:
Consider flux through the faces perpendicular to the \( u \)-axis.
Input Area (at \( u \)): \( dA_u = (h_2 dv)(h_3 dw) \)
Flux In: \( [A_1 h_2 h_3]_{u} \, dv \, dw \)
Flux Out (at \( u+du \)): \( [A_1 h_2 h_3]_{u+du} \, dv \, dw \approx \left( A_1 h_2 h_3 + \frac{\partial}{\partial u}(A_1 h_2 h_3)du \right) dv \, dw \)
Net Flux in \( u \)-direction: \[ d\Phi_u = \frac{\partial}{\partial u}(A_1 h_2 h_3) du \, dv \, dw \]
Adding contributions from \( v \) and \( w \) directions and dividing by the volume \( dV = h_1 h_2 h_3 du dv dw \):
4. The Curl (\( \nabla \times \mathbf{A} \))
Curl is derived from the circulation \( \oint \mathbf{A} \cdot d\mathbf{l} \) per unit area. Consider a small rectangular path in the \( uv \)-plane.
Figure 2: Calculation of circulation around a loop with sides \( h_1 du \) and \( h_2 dv \).
The circulation around the boundary of area \( dA_{uv} = h_1 h_2 du dv \) gives the component of Curl along the normal (direction \( \hat{e}_3 \)).
Summing \( \mathbf{A} \cdot d\mathbf{l} \) along the four sides:
\[ \Gamma_{uv} \approx A_1 h_1 du + A_2 h_2 dv \bigg|_{u+du} - A_1 h_1 du \bigg|_{v+dv} - A_2 h_2 dv \]
Using Taylor expansions, this simplifies to:
\[ \left( \frac{\partial (A_2 h_2)}{\partial u} - \frac{\partial (A_1 h_1)}{\partial v} \right) du \, dv \]
Dividing by area \( h_1 h_2 du dv \), we get the \( \hat{e}_3 \) component. The full result is best expressed as a determinant:
5. Summary Table: Common Systems
Here is how the scale factors translate for common coordinate systems:
| System | Coords (\(u,v,w\)) | \(h_1\) | \(h_2\) | \(h_3\) |
|---|---|---|---|---|
| Cartesian | \(x, y, z\) | 1 | 1 | 1 |
| Cylindrical | \(\rho, \phi, z\) | 1 | \(\rho\) | 1 |
| Spherical | \(r, \theta, \phi\) | 1 | \(r\) | \(r \sin\theta\) |