09 Green’s Theorem and Planar Vector Calculus Online Quiz Questions

For a counterclockwise-oriented boundary $C$ and $\mathbf F=\langle P,Q\rangle$, which expression correctly represents the outward flux form of Green’s theorem?

When applying Green’s theorem to a region with holes, how should the boundary components be oriented?

Let $\mathbf F(x,y)=\langle 2x,2y\rangle$, with potential $\phi(x,y)=x^2+y^2$. What is $\displaystyle\int_C\mathbf F\cdot d\mathbf r$ along any path from $(1,0)$ to $(0,2)$?

Select all statements that correctly describe Green’s theorem in circulation form for $\mathbf F=\langle P,Q\rangle$.

Select all statements that are valid consequences of the relationship between curl, conservative fields, and path independence.

For the planar vector field $\mathbf F(x,y)=\langle 3x,-2y\rangle$, what is the constant value of $\operatorname{div}\mathbf F$?

What is the standard term for the scalar quantity $\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}$, which measures local rotational tendency in a planar vector field?

Explain why the hypothesis that a domain be simply connected is needed when using zero curl to conclude that a planar vector field is conservative. Include the consequences for line integrals and a counterexample involving a domain with a hole.

For $\mathbf F(x,y)=\langle x^2,xy\rangle$, what is $\operatorname{div}\mathbf F$ at the point $(1,2)$?

For a positively oriented simple closed curve enclosing a planar region, in which direction is the outer boundary normally traversed?

For a counterclockwise-oriented boundary $C$ and $\mathbf F=\langle P,Q\rangle$, which line integral represents the outward flux?

True or false: Reversing the orientation of a boundary changes the sign of both its circulation integral and its oriented outward flux integral.

Let $\mathbf F(x,y)=\langle x^2,3xy\rangle$. What is $\operatorname{curl}\mathbf F$ at the point $(1,2)$?

In Green’s theorem for a region with holes, how should each inner boundary component be oriented?

Let $C$ be the counterclockwise unit circle and let $\mathbf F(x,y)=\langle -y,x\rangle$. What is $\oint_C \mathbf F\cdot d\mathbf r$?

What term describes a vector field whose line integral is independent of the path and depends only on the starting and ending points?

True or false: If $C$ is a positively oriented simple closed curve and $\oint_C (x\,dy-y\,dx)=14$, then the area enclosed by $C$ is $7$.

True or false: Green’s theorem cannot be applied directly to the unit disk bounded by the unit circle for the vector field $\mathbf F(x,y)=\left\langle -\frac{y}{x^2+y^2},\frac{x}{x^2+y^2}\right\rangle$.