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## Homework Statement

Calculate

[tex] \int_{S} \vec{F} \cdot d\vec{S} [/tex] where

[tex] \vec{F} = z \hat{z} - \frac{x\hat{x} + y \hat{y} }{ x^2 + y^2 } [/tex]

And S is part of the Ellipsoid [tex] x^2 + y^2 + 2z^2 = 4 , z > 0 [/tex] and the normal directed such that

[tex] \vec{n} \cdot \hat{z} > 0 [/tex]

## Homework Equations

All the above. There are the Gauss integrals

[tex] \int_{S} \vec{F} \cdot d\vec{S} = \int_{V} \vec{\nabla} \cdot \vec{F} dV [/tex]

## The Attempt at a Solution

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I tried solving the problem by introducing

[tex] \rho \hat{\rho} = x \hat{x} + y \hat{y} [/tex]

Thus expressing the vectorfield in cylindrical coordinates. I also drew the figure which resulted in a half sphere with origo at (0,0.0), top at [tex] (0,0,\sqrt{2}) [/tex] and the rest in XY has 2.

In order for me to solve the problem with Gauss I had to enclose the bottom with another surface.

Because the field is now

[tex] \vec{F} = z\hat{z} - \frac{1}{\rho} \hat{\rho} [/tex]. One can easily see that it is undefined when [tex] \rho \rightarrow 0 [/tex] So I will have to introduce another figure, perhaps a cylinder through origo and the subract this part from the total integral. I don't know how to do this or if I can.

I can also integrate the problem by dividing up the force field but I don't know how to do then. I'm also not sure if it is wise to implement cylindrical coordinates when the figure is clearly more spherical. Makes using the surface integral impossible since it encompasses a cylinder surface and not sphere.

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