5. Vector Calculus

Stokes’ Theorem — Quiz

Test your understanding of stokes’ theorem with 5 practice questions.

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Practice Questions

Question 1

Which of the following best describes the fundamental idea behind Stokes’ Theorem?

Question 2

Which of the following is a necessary condition for applying Stokes’ Theorem to a vector field $\mathbf{F}$?

Question 3

If $\mathbf{F}(x,y,z) = (y, -x, z)$ and the curve $C$ is the unit circle in the $xy$-plane, oriented counterclockwise, what is the value of the line integral $\oint_C \mathbf{F} \cdot d\mathbf{r}$?

Question 4

Given the vector field $\mathbf{F}(x,y,z) = (x^2, y^2, z^2)$ and the boundary curve $C$ is a square in the $xy$-plane with vertices at $(0,0,0)$, $(1,0,0)$, $(1,1,0)$, and $(0,1,0)$, what is the curl of $\mathbf{F}$?

Question 5

For the vector field $\mathbf{F}(x,y,z) = (-y, x, 0)$, if the curve $C$ is the boundary of the surface defined by $z = 1 - x^2 - y^2$, what is the value of the surface integral of the curl of $\mathbf{F}$ over that surface?