Example 9.8.1

Apply the Divergence theorem to the vector field $\mathbf{F}\=x \mathbf{i}\+y \mathbf{j}\+z \mathbf{k}$ and the surface $S$,  the unit sphere centered at the origin.

Example 9.8.2

Apply the Divergence theorem to the vector field $\mathbf{F}\=x \mathbf{i}\+y \mathbf{j}\+z \mathbf{k}$ and $R$,  the surface and interior of a cylinder of height 1, with central axis along the $z$-axis, and base in the plane $z\=0$.

Example 9.8.3

Apply the Divergence theorem to the vector field $\mathbf{F}\=x \mathbf{i}\+y \mathbf{j}\+z \mathbf{k}$ and $R$, the rectangular box defined by $x\in \left[1\,2\right]\,y\in \left[3\,4\right]\,z\in \left[5\,6\right]$.

Example 9.8.4

Apply the Divergence theorem to the vector field $\mathbf{F}\=x y^3\mathbf{i}\+y z \mathbf{j}\+x^{2}z \mathbf{k}$ and $R$, the region bounded above by the upper hemisphere of the unit sphere centered at the origin, and below by the plane $z\=0$.

Example 9.8.5

Apply the Divergence theorem to the vector field $\mathbf{F}\=x y^3\mathbf{i}\+y z \mathbf{j}\+x^{2}z \mathbf{k}$ and $R$, the region bounded by two spheres centered at the origin, one with radius 1, the other with radius $1\/2$.

Example 9.8.6

Apply the Divergence theorem to the vector field $\mathbf{F}\=x y^3\mathbf{i}\+y z \mathbf{j}\+x^{2}z \mathbf{k}$ and $R$, the region bounded by the paraboloid $z\=x^2\+y^2$ and the plane $z\=1$.

Example 9.8.7

Apply the Divergence theorem to the vector field $\mathbf{F}\=x y^3\mathbf{i}\+y z \mathbf{j}\+x^{2}z \mathbf{k}$ and $R$, the region bounded by the paraboloid $z\=x^2\+y^2$ and the paraboloid $z\=2-x^2-y^2$.

Example 9.8.8

Apply the Divergence theorem to the vector field $\mathbf{F}\=x y^3\mathbf{i}\+y z \mathbf{j}\+x^{2}z \mathbf{k}$ and $R$, the region bounded by the elliptic paraboloid $z\=5-3 x^2-2 y^2$ and the plane $z\=0$.

Example 9.8.9

Apply the Divergence theorem to the vector field $\mathbf{F}\=x y^3\mathbf{i}\+y z \mathbf{j}\+x^{2}z \mathbf{k}$ and $R$, the region bounded below by the elliptic paraboloid $z\= x^2\+4 y^2$ and above by the plane $y\+z\=3$.

Example 9.8.10

Apply the Divergence theorem to the vector field $\mathbf{F}\=x \mathbf{i}\+y \mathbf{j}\+z \mathbf{k}$ and $R$, the first-octant region bounded by the coordinate planes and the plane $x\+y\+z\=1$.