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17.2 The Product Rule and the Divergence

We now address the question: how can we apply the product rule to evaluate such things?

The ∇ ⟶ or "del" operator and the dot and cross product are all linear, and each partial derivative obeys the product rule.

Our first question is: what is ∇ ⟶ · ( f * v ⟶ ) ?

Applying the product rule and linearity we get

∇ ⟶ · ( f * v ⟶ ) = ∇ ⟶ ( f ) · v ⟶ + f * ( ∇ ⟶ · v ⟶ )

And how is this useful?

With it, if the function whose divergence you seek can be written as some function multiplied by a vector whose divergence you know or can compute easily, finding the divergence reduces to finding the gradient of that function, using your information and taking a dot product.

Exercise 17.1 What is the divergence of the vector field ( x , y , z ) ? Of ( − y , x , 0 ) ?

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