Vector Calculus

The curl of $\vec{F}$ is given by: $\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 4y & x & 2z \end{vmatrix}$ Components: $\hat{i}: \frac{\partial(2z)}{\partial y} - \frac{\partial(x)}{\partial z} = 0 - 0 = 0$ $\hat{j}: -\left(\frac{\partial(2z)}{\partial x} - \frac{\partial(4y)}{\partial z}\right) = 0 - 0 = 0$ $\hat{k}: \frac{\partial(x)}{\partial x} - \frac{\partial(4y)}{\partial y} = 1 - 4 = -3$ $\nabla \times \vec{F} = -3\hat{k}$ Magnitude: $|\nabla \times \vec{F}| = |-3| = 3.0$