Expressing this result in different coordinates

Question

Is there a neat way to express this in Cylindrical and Spherical coordinate systems?


$(\vec{A}\boldsymbol{\cdot }\nabla)\vec{B}$

Reference: this occurs quite frequently in Electrodynamics books including Griffiths. Everywhere it is evaluated in Cartesian - Coordinate form.

Answer 1

In arbitrary coordinates the inner product is $$\mathrm u\boldsymbol\cdot\mathrm v=\langle \mathrm{u},\mathrm v\rangle=u^iv_i$$ And the nabla operator is $$\nabla=\sum_{i}\frac{1}{\sqrt{g_{ii}}}\mathrm{e}_i\partial_{i}$$ Combine these expressions to yield what you want in any coordinate system you choose.