Stoke's Theorem

Jul 2007
69
1
In the interest of disclosure, this is a homework problem for which I will
receive a grade.

First the problem,

Let S be a closed surface given by x^4+y^4+z^4=1.

Evaluate
\(\displaystyle
\iint\limits_S {(\nabla \times F)} \bullet nd\sigma
\)

Where F=<2xy^2cosz,x+yze^z,xyz> and n is the outer normal.

Now my dilemma:
Using Stokes Theorem I believe I can evaluate as a line integral C or as a surface integral S.
My problem is with the parameterization.
The surface(x^4+y^4+z^4=1) looks similar to a cube but with rounded corners.
If I evaluate as a line integral then I am unsure of what parameters to use since this ( at least to me) is not a line I am used to defining. Unlike lets say, a circle, which can be parameterized as x=rcos, y=rsin.

Alternatively, if i try to evaluate as a surface integral then I think that to parameterize the position vector it would look something like
r=<x,y,(1-x^4-y^4)^1/4>.
This method begins to get complicated when I proceed to the next step and try to evaluate the cross product of dr/dx x dr/dy.
So, finally, I guess I question is-do I evaluate as a line or surface?
Also, how do parameterize this problem?
Any help is greatly appreciated.

Kid
 
Apr 2010
384
153
Canada
In the interest of disclosure, this is a homework problem for which I will
receive a grade.

First the problem,

Let S be a closed surface given by x^4+y^4+z^4=1.

Evaluate
\(\displaystyle
\iint\limits_S {(\nabla \times F)} \bullet nd\sigma
\)

Where F=<2xy^2cosz,x+yze^z,xyz> and n is the outer normal.

Now my dilemma:
Using Stokes Theorem I believe I can evaluate as a line integral C or as a surface integral S.
My problem is with the parameterization.
The surface(x^4+y^4+z^4=1) looks similar to a cube but with rounded corners.
If I evaluate as a line integral then I am unsure of what parameters to use since this ( at least to me) is not a line I am used to defining. Unlike lets say, a circle, which can be parameterized as x=rcos, y=rsin.

Alternatively, if i try to evaluate as a surface integral then I think that to parameterize the position vector it would look something like
r=<x,y,(1-x^4-y^4)^1/4>.
This method begins to get complicated when I proceed to the next step and try to evaluate the cross product of dr/dx x dr/dy.
So, finally, I guess I question is-do I evaluate as a line or surface?
Also, how do parameterize this problem?
Any help is greatly appreciated.

Kid
I'm fairly certain this would be best evaluated in its present form. In other words, I wouldn't do the parametric route.

We are asked for,

\(\displaystyle \iint_S curl F \cdot \hat N dS \)

We area given off the hop

\(\displaystyle F = (2xy^2cosz) \hat i + (x+yze^z) \hat j + (xyz) \hat k \)

So we can easily find Curl F in this form. Now what about N and dS? Well...

\(\displaystyle \hat N dS = +/- ( - \frac{ dg}{ dx} \hat i - \frac{ dg}{ dy} \hat j + \hat k ) \)

But I believe we can simpliy this to (after applying strokes twice)

\(\displaystyle \hat N dS = \hat k dA \)

We then get,

\(\displaystyle \iint_S curl F \cdot \hat N dS = \iint_D curl F \cdot \hat k dA\)

This should simplify to something simple (remember to evaluate curl F at z=0 ).

The domain might be a problem but I believe we can do this in cartesian just fine.