Given a set of solids, compute the inward flux

[DottZakapa]

Homework Statement

Let $_\Omega \left\{ (x,y,z)\in R^3 : - \sqrt{3-y^2-z^2} \leq x \leq z+2 ,y^2+z^2 \leq 3 \right\}$
and consider the function
$f(x,y,z)=y^2x+z^2x$

Represent the domain $\Omega$
compute the vector field $F=\nabla f$
compute the inward flux.

Relevant Equations

flux integration

Let $_\Omega \left\{ (x,y,z)\in R^3 : - \sqrt{3-y^2-z^2} \leq x \leq z+2 ,y^2+z^2 \leq 3 \right\}$
and consider the function
$f(x,y,z)=y^2x+z^2x$

Represent the domain $\Omega$
compute the vector field $F=\nabla f$
compute the inward flux.

So I've found that one is a cylinder of radius $\sqrt 3$
the second figure is a sphere with radius $\sqrt 3$
then there is a plane passing through z=2 and x=2
the sphere is inside the cylinder, and concerning the sphere i consider just the part above the x axes, can't see how the plane intersect the two

now,
i'm having problems on finding the intersections in order to find also the boundaries of integration, in truth I'm always struggling at this step.
How do I have to proceed?

 

[mitochan]

So I've found that one is a cylinder of radius √3\sqrt 3
the second figure is a sphere with radius √3\sqrt 3
then there is a plane passing through z=2 and x=2

You see the sphere is contained in the cylinder. So the domain is the sphere cut by the plane passing the points (-2,y,0) and (0,y,2). Death Star like shape.

 

[member 428835]

Someone check me, but is it easiest if you adopt the cylindrical coordinate system as $x = x, y = r \sin \theta, z = r \cos \theta$? Then you have $_\Omega \left\{ (x,y,z)\in R^3 : - \sqrt{3-r^2} \leq x \leq r \cos \theta+2 ,r \leq 3 \right\}$ and $f(x,y,z)=r^2x$.

Then computing $\nabla f$ is straightforward (it's cylindrical coordinates) and the integration bounds would be as I've shown for $x$, $\theta \in [0,2\pi]$ and $r \leq 3$ (shown above).

If it's confusing, without loss of generality let $x = z$, which then implies $- \sqrt{3-r^2} \leq z \leq r\cos \theta+2$, $f = r^2 z$, and of course $\theta \in [0,2\pi]$, $r \in [0,3]$.

 

[DottZakapa]

Thank you all , problem solved and exam passed.
Long life to physics forum