Azimuthal derivative of a cylindrical vector Fourier mode (${\partial }_{\varphi }\mathbf{v}=im\mathbf{v}+\hat{\mathbf{z}}\times \mathbf{v}$)

If the cylindrical components of a vector field have dependence $e^{im\varphi }$, differentiation of the rotating basis adds the displayed cross product term. Scalar-component differentiation alone misses this contribution. For $\mathbf{B}=(J/2)s\hat{\mathbf{\varphi }}$, one also has $(\mathbf{v}\cdot \nabla )\mathbf{B}=(J/2)\hat{\mathbf{z}}\times \mathbf{v}$. These identities are useful in cylindrical magnetohydrodynamic waves.