
The second important combination of partial derivatives of a vector field $\stackrel{\u27f6}{v}$ is its curl.
This is the cross product of the differential operator $\stackrel{\u27f6}{\nabla}$ with the vector $\stackrel{\u27f6}{v}$
Though again we will defer full explanation of the meaning of this entity we can observe some important properties which give it great importance in themselves.
Since we can create a vector from a scalar by taking its gradient, we can ask:
What happens if we now take the curl of the resulting gradient? That is, what is $\stackrel{\u27f6}{\text{curl}}\left(\stackrel{\u27f6}{\text{grad}}f\right)$ ? We can also ask. What is $\text{div}\left(\stackrel{\u27f6}{\text{curl}}\stackrel{\u27f6}{v}\right)$ ? And what is $\text{div}\left(\stackrel{\u27f6}{\text{grad}}f\right)$ ?
The first two of these questions have wonderfully simple answers which are extremely important. The third has an important answer though one whose meaning will not yet be apparent.
Exercises:
9.3 Evaluate $\text{div}\left(\stackrel{\u27f6}{\text{curl}}\stackrel{\u27f6}{v}\right)$ and $\stackrel{\u27f6}{\text{curl}}\left(\stackrel{\u27f6}{\text{grad}}f\right)$ in general; these answers in themselves make curl and div important operators.
9.4 Write down the operator $\text{div}\stackrel{\u27f6}{\text{grad}}$ in terms of partial derivatives. (It is often called "the Laplacian". )
