Become a Multivariable Calculus Pro!

a) Find the functions

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

What tis the complete domain D and range R of the following multivariable functions: ${\displaystyle w(x,y)=\sqrt{y-4x^2}}$

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).

Relative extrema of multivariables:
${\displaystyle f(x,y)=\frac{xy}{7}}$
find critical points and relative extrema, given an open region.

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).

What tis the complete domain D and range R of the following multivariable functions: ${\displaystyle w(x,y)=\frac{1}{x(y-1)}}$