]> 9.2 The Curl

9.2 The Curl

The second important combination of partial derivatives of a vector field v is its curl.

This is the cross product of the differential operator with the vector v

curl v = × v = ( v z y v y z ) i ^ + ( v x z v z x ) j ^ + ( v y x v x y ) k ^

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 curl ( grad f ) ? We can also ask. What is div ( curl v ) ? And what is div ( grad f ) ?

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 div ( curl v ) and curl ( grad f ) in general; these answers in themselves make curl and div important operators.

9.4 Write down the operator div grad in terms of partial derivatives. (It is often called "the Laplacian". )