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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". )

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