Question

Use Stokes' Theorem to find the circulation of F=<2y, 5z, 5x> around the triangle obtained by tracing out the path (4,0,0) to (4,0,4), to (4,4,3) back to (4,0,0).

Answer 1

Can you define what circulation is?

Answer 2

i guess it's just going along the path from a point to another?

Answer 3

I guess they just want you to do: \[
\int_C \mathbf{F}\cdot d\mathbf{r}
\]

Answer 4

i calculated the curl(F) first which was 5, and i multiplied that by 6 which was the region of the triangle but got the wrong answer

Answer 5

Well I'm guessing circulation is given by: \[
\iint_S \nabla \times \mathbf{F}\cdot d\mathbf{S}
\]And by Stokes theorem: \[
\iint_S \nabla \times \mathbf{F}\cdot d\mathbf{S} = \int_C \mathbf{F}\cdot d\mathbf{r}
\]

Answer 6

So basically, just do the line integral. Can you do it?

Answer 7

but what is the limit of the integral? from 0 to 4?

Answer 8

You have 3 curves.

Answer 9

(4,0,0) to (4,0,4)
(4,0,4) to (4,4,3)
(4,4,3) to (4,0,0)

Answer 10

Do the line integral across each one and add them up.

Answer 11

Do you know how to do a line integral?

Answer 12

Do you know how to parametrize the curves?

Answer 13

yes! :) i think i get it :P i'm solving it now :) thanks!!!

Answer 14

Wait... do you suppose it would be easier to find the surface and do the curl...?

Answer 15

yes but i got the wrong answer

Answer 16

What was your curl?

Answer 17

5

Answer 18

Curl is a vector dude....

Answer 19

curl doted with the normal vector is 5

Answer 20

that's the integrand

Answer 21

What was your curl and normal vector?

Answer 22

5,-5,-2 and 1.0.0

Answer 23

Also, your surface parametrization was?

Answer 24

oh i got the right answer! thanks a lot!

Answer 25

Yeah but I feel like we cheated.