Examples

1. Given \( \vec{F} = xy^2\hat{i} + xy \hat{j} + xy \hat{k} \), find \( \mathrm{curl} (\vec{F}) \).
2. Given \( \vec{E} = e^{xy}\hat{i} + \sin (xy)\hat{j} + \cos (yz^2)\hat{k} \), find \( \mathrm{curl} (\vec{E}) \).
3. Consider an arbitrary vector field \( \vec{V}(x,y,z) = V_x(x,y,z) \hat{i} + V_y(x,y,z) \hat{j} + V_z(x,y,z) \hat{k} \), show that \( \vec{\nabla} \circ (\vec{\nabla} \times \vec{V}) = 0 \).
4. Find the time derivative of the magnetic field at a location where the induced electric field is given by

   \( \vec{E}(x,y,z) = E_0\big[\big(\frac{z}{z_0}\big)^2 \hat{i} + \big(\frac{x}{x_0}\big)^2 \hat{j} + \big( \frac{y}{y_0} \big)^2 \hat{k} \big] \)

5. Show that any conservative vector field is irrotational, i.e. it has zero curl.