# Verifying Stoke's Theorem for a Triangular Region

• yosimba2000
Thanks for the help!In summary, the conversation was about verifying Stoke's Theorem. The individual had uploaded an image of the problem and their work for a double check. They were struggling to get a matching solution for the curl and asked for help in identifying their mistake. The summary also includes a reminder to use LaTeX code instead of embedding equations in images for easier reference and accuracy. The final solution was found to be incorrect due to an error in the limits of integration.
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]

#### Attachments

• em.png
3 KB · Views: 472
• line integral.jpg
37 KB · Views: 484
Last edited:
jtbell said:
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.

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.

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.

yosimba2000 said:
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? )

jtbell said:
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.

## 1. What is Stokes' Theorem?

Stokes' Theorem is a fundamental theorem in vector calculus that relates the surface integral of a vector field over a surface to the line integral of the same vector field along the boundary of the surface.

## 2. How is Stokes' Theorem used in science?

Stokes' Theorem is used in a variety of scientific fields, such as physics, engineering, and fluid dynamics, to calculate the flux of a vector field through a surface and to solve problems involving circulation and vorticity.

## 3. What is the mathematical formula for Stokes' Theorem?

The mathematical formula for Stokes' Theorem is ∫S(∇ x F) · dS = ∫CF · dr, where S is a surface bounded by a closed curve C, F is a continuous vector field, ∇ is the del operator, and dS and dr are the surface element and line element, respectively.

## 4. How is Stokes' Theorem verified or proven?

Stokes' Theorem can be verified by using the formula to solve a specific problem and then comparing the result to the surface and line integrals calculated directly. This process can also be extended to more complex problems involving multiple surfaces and curves.

## 5. What are the limitations of Stokes' Theorem?

Stokes' Theorem is limited to a specific type of vector field known as a conservative field, which means that the line integral of the field along any closed curve is zero. It also only applies to surfaces that are oriented and have a smooth boundary.

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