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Section A.2 Product Rules

All types of derivatives have product rules. Many of these 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 example, the familiar product rule for functions of one variable is

\begin{equation*} \frac{d}{dx}(fg)=\frac{df}{dx}g+f\frac{dg}{dx} . \end{equation*}

For products involving at most one vector field, the only trick is figuring out which derivative to take, and what multiplication to use! Remember that you can only take the divergence and curl of a vector field. Here are the simple product rules for the various incarnations of the del operator when at most one vector field is involved:

\begin{align*} \grad(fg) \amp= (\grad f) \, g + f \, (\grad g) ,\\ \grad\cdot(f\GG) \amp= (\grad f) \cdot \GG + f \, (\grad\cdot\GG) ,\\ \grad\times(f\GG) \amp= (\grad f) \times \GG + f \, (\grad\times\GG) . \end{align*}

Care must be taken with the order of the factors in the last of these rules, since the cross product is not commutative.

There is a messier, less intuitive set of product rules for products of vector fields. In these formulas, the definition of the gradient of a vector field is the gradient of each of the (rectangular) components.

\begin{align*} \grad(\GG\cdot\HHv) \amp= \GG\times(\grad \times\HHv) + \HHv\times (\grad \times\GG) +(\GG\cdot\grad)\HHv +(\HHv\cdot\grad)\GG ,\\ \grad\cdot(\GG\times\HHv) \amp= \HHv \cdot (\grad\times\GG) - \GG \cdot (\grad\times\HHv) ,\\ \grad\times(\GG\times\HHv) \amp= (\HHv\cdot\grad)\GG-(\GG\cdot\grad)\HHv +\GG(\grad\cdot\HHv)-\HHv(\grad\cdot\GG) . \end{align*}

Again care must be taken with the order of the factors in these rules.

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.