Use the Divergence Theorem to calculate the surface

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).

A particle moves along line segments from the origin to the points(3,0,0),(3,3,1),(0,3,1), and back to the origin under the influence of the force field ${\displaystyle F(x,y,z)=z^{2}i+3xyj+4y^{2}k}$.
Use Stokes' Theorem to find the work done.

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 ${\displaystyle x^2+y^2=3}$.

Evaluate the line integral by the two following methods. y) dx + (x+y)dy C os counerclockwise around the circle with center the origin and radius 3(a) directly (b) using Green's Theorem.

Use Greens

How do Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem relate to the Fundamental Theorem of Calculus for ordi-nary single integral?

Use Green's Theorem to evaluate ${\displaystyle {\int }_C(e^x+y^2)dx+(e^y+x^2)dy}$ where C is the boundary of the region(traversed counterclockwise) in the first quadrant bounded by ${\displaystyle y=x^2\text{and}y=4}$.