Question

Help on stokes theorem.
I have the two surfaces
z=-2x
z=x^2+y^2

What is dS and what is the n(normal vector)?

Answer 1

\[\int\limits_{}CurlF*ndS^{}\]
Are the normalvector the partial derivatives of the intersection equation?

[Zx,Zy,1] where the equation is z-z=0 x^2+y^2+2x

If so i get the normal vector to be [2x+2,2y,1] And what is dS?

Answer 2

Sorry should been a double integral thought

Answer 3

I think you should use 'surface independence of stokes theorem',

the intersection of the two equations will give you the 'shadow of the path on thxe x-y plane'

\[-2x=x^2 + y^2\\0=x^2+2x +1+y^2 -1 (add~subtract~1)\\0=(x+1)^2+y^2-1\\1=(x+1)^2+y^2\]

This is a circle,(this is the shadow of the bounding curve on the x-y plane) you can use this to find the parametric equation of the actual curve in space,

for normal vector and 'ds', since stokes theorem is surface indepndent, just use the simpler surface, i.e z=-2x

Answer 4

z=-2x
=> 2x+0y+z=0
normal vector = <2,0,1>

It's been a long time, and I have lost my grip on vector calculus, I'll just tag @ganeshie8 to verify

Answer 5

*just to clarify: the curve i'm talking about is the 'bounding curve',
stokes theorem will evaluate the line integral along this curve

Answer 6

I have done a parametrization of the curve and I know how to solve the line integral, I´m just preparing for my finals 13th december and from previous it seems like stokes is a well repeated question.

But you are right, stokes is surface independent so I could solve for the normal of the easier surface, thanks for clarifying that :) and for dS? it seems like there is several forms for stokes.
but im trying to solve the one where you dot CurlF with n and integrate over dS

Answer 7

i guess one way to find dS is to use this formula

\[\hat{n}dS= \pm <-f_x, ~-f_y~,~1> dxdy\]

where f(x,y) =-2x

note that this directly gives \(\hat{n}dS\) so you don't really add on the normal vector

Answer 8

where dxdy=dA so i can also use drdt? Thanks

Answer 9

once you do the dot product of ndS with curlF, the only thing left to do is to evaluate the integral... you can do that whichever way you like