(adsbygoogle = window.adsbygoogle || []).push({}); Why can't I get Stoke's Theorem to Work!!!?

1. The problem statement, all variables and given/known data

This is very similar to the problem I put up last night actually part c of the same problem. I really have put an amazing amount of time into this problem (week and a half/ 5 hrs a day) and it countinues to stump me. At this point I'm spinning mental wheels and could use some help.

we have a 3-4-5 right triangle lying on the x-y plane with the length three side on the x-axis and the length 4 side on the y-axis. we define a field

[tex]\vec{u}=x^2\hat{x} + xy\hat{y}[/tex]

and I am trying to evaluate the integral:

[tex]\oint{\vec{u}d\vec{l}}[/tex]

2. Relevant equations

I need to solve both:

[tex]\oint{\vec{u}d\vec{l}}[/tex]

and

[tex]\oint{(\nabla{}x\vec{u}) . \hat{n} dA}[/tex]

please excuse my clumsy notation, this should be del cross the vector u

3. The attempt at a solution

The surface integral after doing del cross u I end up with

[tex]\frac {1}{2} \oint{ y \hat{k} dxdy} [/tex]

evaluated from 0 - 3 on the x-axis and 0 - 4 on the y-axis

Doing this I get 12.

the 1/2 I added in to this equation is largely a guess on my part, and could be correct or not, just seemed to make some sense for a triagle who's area is 1/2 base*height

Looking at the line integral

I placed curve 1 along the y axis from 4 - 0 then paramaterized x = 0 and dx = 0

so for curve 1 is zero

Curve 2, I paramaterized y = 0, dy = 0 that left

[tex]\int{x^2 dx}[/tex] evaluated from 0 - 3 gives me 9

Curve 3 along the hypotinous (I can't spell ... sorry) paramaterized y = 4 - 4/3 * x dy = -3/4 * dx

plugging these new values into y and dy and integrating from 3 - 0 gives me -1

So for the whole curve I have 8

So somewhere I missed something because last I checked 8 =! 12, or if you ignore my guess of 1/2 in the surface integral 8 =! 24

so I'm only off by a factor of 3 somewhere, though it could be coincidence that's a pretty nice number to be off by, makes me think I'm just missing a term somewhere.

If I had to guess I would say somehow [tex] \hat{n} [/tex] is somehow equal to 1/3, or perhaps 1/6 to include my guess about 1/2 in the surface integral

Thank you for reading this long rambling thing, and thanks for any help

-Mo

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# Homework Help: Why can't I get Stoke's Theorem to Work ?

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