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
Applying the product rule and linearity we get
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)?