Apply Green’s theorem to find the outward flux for the field F(x,y)=tan^(−1)(y/x)i+ln(x^2+y^2)j

Use the Divergence Theorem to calculate the surface integral F · dS, that is, calculate the flux of F across S.
 $F(x,y,z)=x^3+cosyi+y^{3}sinxzj+z^3+2ek$
S is the surface of the solid bounded by the cylinder $y^2+z^2=4$ and the planes $x=0$ and $x=4$.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The work required to move an object around a closed curve C in the presence of a vector force field is the circulation of the force field on the curve.
b. If a vector field has zero divergence throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is zero.
c. If the two-dimensional curl of a vector field is positive throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is positive (assuming counterclockwise orientation).

Use Green’s Theorem to evaluate around the boundary curve C of the region R, where R is the triangle formed by the point (0, 0), (1, 1) and (1, 3).
Find the work done by the force field F(x,y)=4yi+2xj in moving a particle along a circle ${\displaystyle x^2+y^2=1}$ from(0,1)to(1,0).

Randomly drawing three green skittles in a row from a bag that contains eight green skittles out of 25 skittles total.

Use the Divergence Theorem to calculate the surface integral ${\displaystyle \int {\int }_{S}F·dS}$, that is, calculate the flux of F across S.
${\displaystyle F(x,y,z)=(\cos \left(z\right)+x{y}^2)i+x{e}^{-z}j+(\sin \left(y\right)+x^{2}z)k}$
S is the surface of the solid bounded by the paraboloid ${\displaystyle z=x^2+y^2}$ and the plane z = 9.

Let ${\displaystyle F(x,y)=⟨x{y}^2+8x,x^{2}y-8y⟩}$. Compute the flux ${\displaystyle \oint F\cdot nds}$ of F across a simple closed curve that is the boundary of the half-disk given by ${\displaystyle x^2+y^2\le 7,y\ge 0}$ using the vector form Greens

A camera shop stocks eight different types of batteries, one of which is type A7b.Assume there are at least 30 batteries.
of each type.a. How many ways can a total inventory of 30 batteries be distributed among the eight different types? b.
How many way can a total inventory of 30 batteries be distributed among the eight different types of the inventory must
include at least four A76 batteries?c. How many ways can a total inventory of 30 batteries be distributed among the eight
different types of the inventory includes at most three A7b batteries