The divergence of a vector \( \mathbf{B} = i B^\lambda + j B^\vartheta + k B^r \) is \[ \nabla \cdot \mathbf{B} = \frac{1}{\cos \vartheta} \left[ \frac{1}{r} \frac{\partial B^\lambda}{\partial \lambda} + \frac{1}{r} \frac{\partial}{\partial \vartheta} (B^\vartheta \cos \vartheta) + \frac{\cos \vartheta}{r^2} \frac{\partial}{\partial r} (r^2 B^r) \right] \] The vector gradient of a scalar \( \phi \) is \[ \nabla \phi = \frac{1}{r \cos \vartheta} i \frac{\partial \phi}{\partial \lambda} + \frac{1}{r} j \frac{\partial \phi}{\partial \vartheta} + k \frac{\partial \phi}{\partial r} \] The Laplacian of a scalar \( \phi \) is \[ \nabla^2 \phi \equiv \nabla \cdot \nabla \phi = \frac{1}{r^2 \cos \vartheta} \left[ \frac{\partial}{\partial \lambda} \left( \frac{1}{\cos \vartheta} \frac{\partial \phi}{\partial \lambda} \right) + \frac{\partial}{\partial \vartheta} \left( \cos \vartheta \frac{\partial \phi}{\partial \vartheta} \right) + \cos \vartheta \frac{\partial}{\partial r} \left( r^2 \frac{\partial \phi}{\partial r} \right) \right] \]
The curl of a vector is \[ \text{curl} \, \mathbf{B} = \nabla \times \mathbf{B} = \frac{1}{r^2 \cos \vartheta} \begin{vmatrix} i r \cos \vartheta & j r & k \\ \frac{\partial}{\partial \lambda} & \frac{\partial}{\partial \vartheta} & \frac{\partial}{\partial r} \\ B^\lambda r \cos \vartheta & B^\vartheta r & B^r \end{vmatrix} \] The vector Laplacian \( \nabla^2 \mathbf{B} \) (can be used for calculating viscous terms in the momentum equation) is \[ \nabla^2 \mathbf{B} = \nabla (\nabla \cdot \mathbf{B}) - \nabla \times (\nabla \times \mathbf{B}) \] Only in Cartesian coordinates does this take the simple form \[ \nabla^2 \mathbf{B} = \frac{\partial^2 \mathbf{B}}{\partial x^2} + \frac{\partial^2 \mathbf{B}}{\partial y^2} + \frac{\partial^2 \mathbf{B}}{\partial z^2} \]
1 Vallis, G.K. (2010) Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, Cambridge.