Question

Find \(\operatorname{div}(\operatorname{curl} \mathbf{F})=\nabla \cdot(\nabla \times \mathbf{F})\) $$ \mathbf{F}(x, y, z)=x^{2} z \mathbf{i}-2 x z \mathbf{j}+y z \mathbf{k} $$

Short answer

After performing the above steps, your final answer to the problem will be the function for the divergence of the curl of the given vector field.

Step-by-step solution

Find the Curl of the vector field

The curl of a vector field is given by \(\nabla \times \mathbf{F}=(\frac{\partial}{\partial y} F_z - \frac{\partial}{\partial z} F_y) \mathbf{i} - (\frac{\partial}{\partial x} F_z - \frac{\partial}{\partial z} F_x) \mathbf{j} + (\frac{\partial}{\partial x} F_y - \frac{\partial}{\partial y} F_x) \mathbf{k}\). When you substitute the given vector field into the curl formula, you will find the curl of your vector field.

Find the Divergence of the Curl

Next, to find the divergence of the curl, use the formula \(\nabla \cdot(\nabla \times \mathbf{F}) = \frac{\partial}{\partial x}(\nabla \times \mathbf{F})_x + \frac{\partial}{\partial y}(\nabla \times \mathbf{F})_y + \frac{\partial}{\partial z}(\nabla \times \mathbf{F})_z\). Substitute the curl, calculated in previous step, into the divergence formula.

Perform the Computations

Now, perform the calculations for partial derivatives to find the divergence of the curl