5. Vector Calculus

Green’s Theorem — Quiz

Test your understanding of green’s theorem with 5 practice questions.

Practice Questions

Question 1

Which of the following is the correct statement of Green’s Theorem for a vector field $\mathbf{F} = \langle P(x,y), Q(x,y) \rangle$ on a region $D$ with boundary $C$?

Question 2

If $\mathbf{F}(x,y) = \langle y^2, x^2 \rangle$, what is the integrand of the double integral in Green’s Theorem?

Question 3

Which of the following conditions must hold for Green’s Theorem to be applicable?

Question 4

Evaluate the line integral $\oint_C (y \ dx + x \ dy)$ where $C$ is the circle $x^2 + y^2 = 1$ oriented counterclockwise using Green’s Theorem.

Question 5

Suppose $\mathbf{F}(x,y) = \langle -y, x \rangle$. What is the curl of $\mathbf{F}$, i.e., $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$?