Question

Find \(\operatorname{curl}(\operatorname{curl} \mathbf{F})=\nabla \times(\nabla \times \mathbf{F})\). \(\mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k}\)

Short answer

The short answer will be the result from Step 3 - the simplified vector from the second curl operation.

Step-by-step solution

Compute the First Curl

First, calculate the curl of the vector \(\mathbf{F}\). The curl of a vector field \(\mathbf{F}(x, y, z) = P(x, y, z) \mathbf{i}+ Q(x, y, z) \mathbf{j}+ R(x, y, z) \mathbf{k} \) is defined as:\[ \nabla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\mathbf{i} - \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\mathbf{k} \] Substituting \(\mathbf{F} = x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k} \) into the above equation will yield the first part of the solution.

Compute the Second Curl

After calculating the first curl, the next step is to calculate the second curl on your result from Step 1. For this step, use the same formula for the curl as you did in Step 1 and apply the formula again with the output from the first step being your new \(\mathbf{F}\) vector

Simplify the Result

The last step is to simplify the result. The output from the second curl operation in step 2 may be complicated. Try to simplify it by combining like terms and reducing the component functions to their simplest form