- #1

Somefantastik

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## Homework Statement

S is surface of upper half of sphere (rad 3) at (0,0,0). Find the formula for n (in Cartesian, spherical, and cylindrical coord syst), then evaluate

[tex] \int\int\textbf{F}\cdot\textbf{n}dA [/tex]

Where

**F**(x,y,z) =

**k**

## Homework Equations

## The Attempt at a Solution

in a previous problem, if I let z = f(x,y) be the surface and take

[tex] \psi(x,y,z) = f(x,y) - z [/tex]

Since [tex] \nabla\psi [/tex] is orthogonal to the surface on which [tex] \psi [/tex] is constant, then I can use this to get the formula for the normal to the surface.

Problem is, I'm stuck trying to figure out what the formula for the surface is to start with.

A sphere with radius 3 is just x

^{2}+y

^{2}+z

^{2}= 3

so

[tex] z = 3 - \sqrt{x^{2}+y^{2}} [/tex]

where 0< x,y < 3

now

[tex] \textbf{n} = \frac{\nabla\psi}{\left\|\nabla\psi\right\|} [/tex]

Am I on the right track? The unit normal vector field gets pretty ugly after this step :-/