13. Stokes’ Theorem

Applications — Quiz

Test your understanding of applications with 5 practice questions.

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Practice Questions

Question 1

What is the main relationship described by Stokes' theorem?

Question 2

If a surface is oriented upward, how should its boundary be traversed to match the positive orientation?

Question 3

If $\nabla \times \mathbf{F} = \mathbf{0}$ on a surface $S$, what is $\oint_C \mathbf{F}\cdot d\mathbf{r}$ for the boundary curve $C$ of $S$?

Question 4

What is the boundary of the upper hemisphere $x^2+y^2+z^2=1$, $z\ge 0$?

Question 5

If two surfaces $S_1$ and $S_2$ have the same boundary curve $C$ and are oriented consistently, what does Stokes' theorem imply about $\iint_{S_1} (\nabla\times\mathbf{F})\cdot\mathbf{n}\,dS$ and $\iint_{S_2} (\nabla\times\mathbf{F})\cdot\mathbf{n}\,dS$?