I'm trying to solve the flux integral of (using big letters for vectors). The vector field is the constant vector ${\bf A} = (1,2,3)$, andthe surface $S$ is $x^2 + y^2 + z^2 = 1$, with $z\geq 0$.
I want to calculate it as a flux integral, not using for example gauss. I'm trying to substitute $z$ for $\sqrt{1-x^2 -y^2}$, doing the cross product and then using polar coordinates but I can't get anywhere with that...
Thanks in advance.
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$\begingroup$By definition, the outward flux of ${\bf A}=(1,2,3)$ through the surface $S$ is given by $$\Phi:=\iint_{S} {\bf A}\cdot {\bf n}\ dS=$$ where $S$ is the upper unit hemi-sphere, ${\bf n}=(x,y,z)$ and $dS=\sqrt{1+f_x^2+f_y^2}dxdy$ with $z=f(x,y)=\sqrt{1-x^2-y^2}.$ Then $$\sqrt{1+f_x^2+f_y^2}=\frac{1}{\sqrt{1-x^2-y^2}}$$ and $$\Phi=\iint_{S}\frac{x+2y+3\sqrt{1-x^2-y^2}}{\sqrt{1-x^2-y^2}}\ dx dy.$$ Now you may use the polar coordinates $x=\rho\cos\theta$ and $y=\rho\sin\theta$ with $dxdy=\rho d\rho d\theta$.
Note that there is also a shortcut which uses a symmetric argument.
Can you take it from here?
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