Vector Calculus

The divergence of the curl of a vector field is

Given a vector $\vec{u} = \frac{1}{3}(-y^3 \hat{i} + x^3 \hat{j} + z^3 \hat{k})$ and $\hat{n}$ as the unit normal vector to the surface of the hemisphere $(x^2 + y^2 + z^2 = 1; z \ge 0)$, the value of integral $\iint_S (...

The divergence of the vector -yi+x j is

The surface integral $\iint_S \mathbf{F} \cdot \mathbf{n}\,dS$ over the surface $S$ of the sphere $x^2 + y^2 + z^2 = 9$, where $\mathbf{F}=(x+y)\mathbf{i}+(x+z)\mathbf{j}+(y+z)\mathbf{k}$ and $\mathbf{n}$ is the unit out...

For the vector $\vec{V} = 2yz \hat{i} + 3xz \hat{j} + 4xy \hat{k}$, the value of $\vec{V} \cdot (\nabla \times \vec{V})$ is _________