calculate the flux of a vector field through a surface

Question

How to calculate the flux of a vector field through a surface in mathematica? I've this field:

F = (x, x^2 * y, y^2 * z) 

and this surface:

S = { (x,y,z) ∈ R^3 | 2 * Sqrt[x^2+y^2] <= z <= 1 + x^2 + y^2}

So, I'm trying:

region = ImplicitRegion[2 * Sqrt[x^2+y^2] <= z <= 1 + x^2 + y^2, {x, y, z}];

Integrate[#, {x,y,z} ∈ region]& /@ ({x, x^2 * y, y^2 * z} . {x, y, z})

I expect Pi/30 as a result, but it comes out "Infinity"...

Answer 1

The problem is the region is infinite. We need to restrict the radius like so:

region = ImplicitRegion[2*Sqrt[x^2+y^2] < z < x^2+y^2+1 && x^2+y^2 <= 1, {x,y,z}]

This region describes the volume in between a height 2 inverted cone of base radius 1, and a curved cap surface:

Plot3D[{2 Sqrt[x^2 + y^2], x^2 + y^2 + 1}, {x, -1, 1}, {y, -1, 1}, 
 PlotStyle -> {Opacity[.5]}, 
 RegionFunction -> Function[{x, y, z}, x^2 + y^2 < 1]]

We can then apply the divergence theorem to calculate the flux:

f = {x, x^2 y, y^2 z};
Integrate[Div[f, {x, y, z}], {x, y, z} ∈ region]
(* result Pi/5 *)

... or we can calculate the surface integral directly:

f = {x, x^2 y, y^2 z};
coneSurface = {x, y, 2 Sqrt[x^2 + y^2]};
capSurface = {x, y, x^2 + y^2 + 1};
(* multiply by -1 because the normal is pointing into the cone but we need it pointing out *)
coneNormal = -1*ResourceFunction["UnitNormal"][coneSurface, {x, y}];
capNormal = ResourceFunction["UnitNormal"][capSurface, {x, y}];
coneRegion = ImplicitRegion[x^2 + y^2 <= 1 && z == 2 Sqrt[x^2 + y^2], {x,y,z}];
capRegion = ImplicitRegion[x^2 + y^2 <= 1 && z == x^2 + y^2 + 1, {x,y,z}];
Integrate[coneNormal.f, {x,y,z} ∈ coneRegion] + Integrate[capNormal.f, {x,y,z} ∈ capRegion]

(* result: Pi/5 *)

The book result of $\pi/30$ is not correct. The reason why is because $\nabla\cdot F$ is $1+x^2+y^2$ not $x^2+y^2$, as the $F$ has a non-constant $x$ component.

If we make $F_x$ constant, e.g f = {0, x^2 y, y^2 z};, then both approaches above give $\pi/30$.