Master Green's Theorem Problems with Plainmath

Divergence Theorem for more general regions Use the Divergence Theorem to compute the net outward flux of the following vector fields across the boundary of the given regions D.
${\displaystyle F=⟨z-x,x-y,2y-z⟩}$, D is the region between the spheres of radius 2 and 4 centered at the origin.

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

Suppose that the plane region D, its boundary curve C, and the functions P and Q satisfy the hypothesis of Greens

use Green’s Theorem to find the counterclockwise circulation and outward flux for the field F and the curve C. ${\displaystyle F=(y^2-x^2)i+(x^2+y^2)j}$
C: The triangle bounded by y = 0, x = 3, and y = x

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 ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,z=0}$.

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 Theorems to evaluate ${\displaystyle {\oint }_{C}xydx+x^2{y}^{3}dy}$, where C is the triangle with vertices(0,0),(1,0)and (1,2).