Covariant Expressions for Vector and Tensor Operators
The metric tensor for orthogonal coordinate systems reads
where h1,h2 and h3 are scale factors or Lamé coefficients. Depending on the specific coordinate system, they take different forms (see Table-1 below).
Table-1 Scale factors and derivatives in various coordinate systems.
| scale factors | Cartesian | Cylindrical | Spherical |
|---|---|---|---|
| h1 | 1 | 1 | 1 |
| h2 | 1 | r | r |
| h3=h31· h32 | 1 | 1 | r·sinθ |
| h31 | 1 | 1 | r |
| h32 | 1 | 1 | sinθ |
| ∂h2/∂x1 | 0 | 1 | 1 |
| ∂h31/∂x1 | 0 | 0 | 1 |
| ∂h32/∂x2 | 0 | 0 | cosθ |
∂