Application of Stoke's Theorem

Question

$\newcommand{\unit}[1]{\hat{\mathbf{#1}}}$ Verify the truth of Stokes' theorem in the case when $\mathbf{v} = y\unit{i} + 2x\unit{j} + z\unit{k}$, if the path $C$ is a circle $x^2 + y^2 = 1$ in the $xy$ plane and the surface is the plane area bounded by $C$.

I am having trouble determine what the surface is. Is it cylinder cut by the plane formed by $\mathbf{v}$?

S L · Answer 1 · 2013-09-09T22:24:24.093

3

Anything that is protruding out of closed surface. Stokes theorem guarantees that flux through each surface will be constant depending only on the path of the closed curve. enter image description here
For your particular case, unit disc, i.e. $x^2+y^2 \le 1 , z=0$, would be the best choice because it's easiest.

edited Sep 09 '13 at 22:24

answered Sep 09 '13 at 22:16

S L

11,731

score 2 · Accepted Answer · answered Sep 09 '13 at 22:14

2

No. The surface in this case is the unit disk in the $xy$-plane (i.e. the part of the plane $z=0$ that satisfies $x^2+y^2\leq 1$).

answered Sep 09 '13 at 22:14

Nick Peterson

32,430

Yep! That's correct. – Nick Peterson Sep 09 '13 at 22:15
I don't know. It explicitly says "the surface is the plane area bounded by $C$". However, his (potential) misreading of the problem doesn't make his answer wrong -- Stokes' Theorem guarantees us that the integral will be the same. – Nick Peterson Sep 09 '13 at 22:21

Application of Stoke's Theorem

2 Answers2