Verifying Stoke's Theorem for a Triangular Region

[yosimba2000]

Homework Statement

The goal is to verify Stoke's Theorem. I've uploaded the image showing the problem and diagram. I'd like to get a double check on my work as I work on part b.

Homework Equations

Curl in cartesian coords and vector E.
Integral of E dot dl = Integral of (Curl of E) dot dS

The Attempt at a Solution

Performing the line integral first. I've uploaded my paper work because I don't know how to write it all in here.

To clarify some points, the bottom of the triangle is at y=0, the right side is y = -x+2, and the left side is y = x.

The arrows you see next to the equations represent which side of the triangle it references.

Anyway, my solution comes out to -3, but I can't seem to get a matching solution when doing the curl. I'll upload the curl work when I finish it soon.

*edit* So I've done the curl and it comes out to 1.5

What am I doing wrong? I've uploaded to Imgur because you can zoom in easier.

Problem: https://imgur.com/a/GCPYH
Line integral: https://imgur.com/a/9StaQ
Curl part 1: https://imgur.com/a/CmWlO
Curl part 2: https://imgur.com/a/ujK6F[/B]

 

[jtbell]

I don't know how to write it all in here.

https://www.physicsforums.com/help/latexhelp/

 

[yosimba2000]

https://www.physicsforums.com/help/latexhelp/

It would have been really arduous to write it out, I think. I made my written work as neat as possible, so I'd appreciate it if you could check that out.

 

[jtbell]

The problem with putting a lot of equations into a single image is that it's difficult to refer to a specific part of one equation in order to point out an error. With the LaTeX code, we can simply "quote" the equation in question.

In the integral of the curl, you split the surface integral into two triangular pieces. I suggest you double-check the y-limits on the second piece. Third line from the bottom of "Curl part 1".

Note that you can check this integral by doing the x and y integrations in the opposite order, that is, first x then y. In this case you can do the integral in one piece! (I did it both ways, myself.)

I haven't tried the line integral yet, but I think if you fix the surface integral, you'll find that they now agree.

 

[yosimba2000]

Sorry, I'm not seeing what's wrong with the limits of integration. I assume I could also put the upper bound as y = -x+2 and the lower bound as 0, wihch would give me just for that integral, 2.

I'm not seeing why the limits aren't equivalent, though.

 

[jtbell]

I assume I could also put the upper bound as y = -x+2 and the lower bound as 0

That's not what I see:

(Now do you see why we dislike embedding equations in images? )

 

[yosimba2000]

That's not what I see:
View attachment 212338
(Now do you see why we dislike embedding equations in images? )

Oh right, thanks! I messed up thinking about the height of the triangle.