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Product Rules
We digress briefly to discuss product rules for vector derivatives, which are discussed in §1.2.6 of Griffiths, and summarized on the inside front cover.
All types of derivatives have product rules, all of which take the form
- The derivative of a product is the derivative of the first quantity times the second plus the first quantity times the derivative of the second.
For more complicated functions, the only trick is figuring out which derivative to take, and what multiplication to use! Here are the product rules for the various incarnations of the del operator: \begin{eqnarray*} \grad(fg) &=& (\grad f) \, g + f \, (\grad g) \\ \grad\cdot(f\GG) &=& (\grad f) \cdot \GG + f \, (\grad\cdot\GG)\\ \grad\times(f\GG) &=& (\grad f) \times \GG + f \, (\grad\times\GG) \end{eqnarray*} Care must be taken with the order of the factors in the last of these rules, since the cross product is not commutative.
How do you prove these rules? The simplest way is to work out the components of both sides in rectangular coordinates, using the ordinary product rule for partial derivatives.