9.2 The Curl

]> 9.2 The Curl

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9.2 The Curl

The second important combination of partial derivatives of a vector field $\overset{⟶}{v}$ is its curl.

This is the cross product of the differential operator $\overset{⟶}{\nabla}$ with the vector $\overset{⟶}{v}$

\overset{⟶}{curl} \overset{⟶}{v} = \overset{⟶}{\nabla} \times \overset{⟶}{v} = (\frac{\partial v_{z}}{\partial y} - \frac{\partial v_{y}}{\partial z}) \hat{i} + (\frac{\partial v_{x}}{\partial z} - \frac{\partial v_{z}}{\partial x}) \hat{j} + (\frac{\partial v_{y}}{\partial x} - \frac{\partial v_{x}}{\partial y}) \hat{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 $\overset{⟶}{curl} (\overset{⟶}{grad} f)$ ? We can also ask. What is $div (\overset{⟶}{curl} \overset{⟶}{v})$ ? And what is $div (\overset{⟶}{grad} f)$ ?

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 $div (\overset{⟶}{curl} \overset{⟶}{v})$ and $\overset{⟶}{curl} (\overset{⟶}{grad} f)$ in general; these answers in themselves make curl and div important operators.

9.4 Write down the operator $div \overset{⟶}{grad}$ in terms of partial derivatives. (It is often called "the Laplacian". )