Orthogonal Coordinates - Differential Operators in Three Dimensions

Differential Operators in Three Dimensions

Since these operations are common in application, all vector components in this section are presented with respect to the normalized basis.

Operator Expression
Gradient of a scalar field \nabla \phi = \frac{\hat{ \mathbf e}_1}{h_1} \frac{\partial \phi}{\partial q^1} + \frac{\hat{ \mathbf e}_2}{h_2} \frac{\partial \phi}{\partial q^2} + \frac{\hat{ \mathbf e}_3}{h_3} \frac{\partial \phi}{\partial q^3}
Divergence of a vector field \nabla \cdot \mathbf F = \frac{1}{h_1 h_2 h_3} \left[ \frac{\partial}{\partial q^1} \left( F_1 h_2 h_3 \right) + \frac{\partial}{\partial q^2} \left( F_2 h_3 h_1 \right) + \frac{\partial}{\partial q^3} \left( F_3 h_1 h_2 \right) \right]
Curl of a vector field \begin{align} \nabla \times \mathbf F & = \frac{\hat{ \mathbf e}_1}{h_2 h_3} \left[ \frac{\partial}{\partial q^2} \left( h_3 F_3 \right) - \frac{\partial}{\partial q^3} \left( h_2 F_2 \right) \right] + \frac{\hat{ \mathbf e}_2}{h_3 h_1} \left[ \frac{\partial}{\partial q^3} \left( h_1 F_1 \right) - \frac{\partial}{\partial q^1} \left( h_3 F_3 \right) \right] \\ & + \frac{\hat{ \mathbf e}_3}{h_1 h_2} \left[ \frac{\partial}{\partial q^1} \left( h_2 F_2 \right) - \frac{\partial}{\partial q^2} \left( h_1 F_1 \right) \right] =\frac{1}{h_1 h_2 h_3} \begin{vmatrix} h_1\hat{\mathbf{e}}_1 & h_2\hat{\mathbf{e}}_2 & h_3\hat{\mathbf{e}}_3 \\ \dfrac{\partial}{\partial q^1} & \dfrac{\partial}{\partial q^2} & \dfrac{\partial}{\partial q^3} \\ h_1 F_1 & h_2 F_2 & h_3 F_3 \end{vmatrix} \end{align}
Laplacian of a scalar field \nabla^2 \phi = \frac{1}{h_1 h_2 h_3} \left[ \frac{\partial}{\partial q^1} \left( \frac{h_2 h_3}{h_1} \frac{\partial \phi}{\partial q^1} \right) + \frac{\partial}{\partial q^2} \left( \frac{h_3 h_1}{h_2} \frac{\partial \phi}{\partial q^2} \right) + \frac{\partial}{\partial q^3} \left( \frac{h_1 h_2}{h_3} \frac{\partial \phi}{\partial q^3} \right) \right]

Read more about this topic:  Orthogonal Coordinates

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