]> 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". )