Use Stokes' Theorem to evaluate int int_S CURL f * dS. F(x,y,z)=x^2y^3zi+sin(xyz)j+xyzk, S is the part of the cone y^2=x^2+z^2 that lies between the planes y = 0 and y = 2, oriented in the direction of the positive y-axis.

Use the Divergence Theorem to calculate the surface integral ${\displaystyle \int {\int }_{S}F·dS}$, that is, calculate the flux of F across S.
${\displaystyle F(x,y,z)=(x^3+y^3)i+(y^3+z^3)j+(z^3+x^3)k}$, S is the sphere with center the origin and radius 2.

Use Green's Theorem to find ${\displaystyle {\int }_C\overrightarrow{F}\cdot d\overrightarrow{r}}$ where ${\displaystyle \overrightarrow{F}=⟨y^3,-x^3⟩}$ and C is the circle ${\displaystyle x^2+y^2=3}$.

As Mark Ellon is getting ready for a meeting, the room goes dark. He fishes for a green shirt in his drawer. He wears only blue, green and white colors. His drawers have identical shirts in these colors: 28 blue, 25 green, and 13 white shirts. How many shirts did he throw on the floor to be a cent percent sure that he has a taken out a green shirt?

Let ${\displaystyle F=[x^2,0,z^2]}$, and S the surface of the box ${\displaystyle \left|x\right|\le 1,\left|y\right|\le 3,0\le z\le 2}$.
Evaluate the surface integral ${\displaystyle \int {\int }_{S}F\cdot ndA}$ by the divergence theorem.

Let F = curl(A). Which of the following are related by Stokes' Theorem? (a) The circulation of A and flux of F (b) The circulation of F and flux of A

Use Stokes' theorem to evaluate the line integral ${\displaystyle {\oint }_{C}F\cdot dr}$ where A = -yi + xj and C is the boundary of the ellipse ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,z=0}$.