# Stokes theorem application

1. Dec 4, 2011

### bmxicle

1. The problem statement, all variables and given/known data
Let $$F(x, y, z) = \left ( e^{-y^2} + y^{1+x^2} +cos(z), -z, y \right)$$
Let s Be the portion of the paraboloid $$y^2+z^2=4(x+1)$$ for $$0 \leq x \leq 3$$
and the portion of the sphere $$x^2 + y^2 +z^2 = 4$$ for $$x \leq 0$$

Find $$\iint\limits_s curl(\vec{F}) d \vec{s}$$
2. Relevant equations
I plotted the surfaces in matlab, including what i think should be the boundary.

3. The attempt at a solution
$$\vec{G} = curl(\vec{F}) = \left (2, -sin(z), h(x, y) \right)$$ where h(x, y) is some derivative i don't want to do.
$$div(G) = 0$$

So If we close the surface into a solid by adding the circle z^2 + y^2 = 16 so we have:

$$\iint\limits_s \vec{G} \cdot d\vec{s} + \iint\limits_{z^2 + y^2 = 16} \vec{G} \cdot d\vec{s} = \iiint\limits_E div(\vec{G}) = 0$$
$$\iint\limits_{z^2 + y^2 = 16} \vec{G} \cdot d\vec{s} = \iint\limits_{z^2 + y^2 = 16} \vec{G} \cdot \vec{n}ds$$ but we know that n is just the unit vector (1, 0, 0) so we have:

$$\iint\limits_{z^2 + y^2 = 16} \vec{G} \cdot d\vec{s} = \iint\limits_{z^2 + y^2 = 16} 2 ds = 16\pi*2 = 32\pi$$

Thus this gives the original integral as $$-32\pi$$

I have the solution to the question and they don't match so I'm wondering where my reasoning is going wrong.

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