How to calculate the surface integral of a vector field?

Question

Suppose the oriented surface is described as, the outside of an upper hemisphere $S:x^2+y^2+z^2=1$ inside the cylinder $x^2-x+y^2=0$

The vector field is : ${\vec F}=<x^2,y^2,z^2>$

How to calculate the surface integral of the vector field: $$\iint\limits_{S^+} \vec F\cdot \vec n {\rm d}S $$

Is it the same thing to:

$$\iint\limits_{S^+}x^2{\rm d}y{\rm d}z+y^2{\rm d}x{\rm d}z+z^2{\rm d}x{\rm d}y$$

There is another post here with an answer by@MichaelE2 for the cases when the surface is easily described in parametric form. How to handle this case?

Answer 1

You can try using ImplicitRegion and Integrate:

region = ImplicitRegion[x^2 + y^2 + z^2 == 1 && x^2 - x + y^2 <= 0 && z >= 0, {x, y, z}];

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

(* 38/105 + (5 π)/32 *)

Note that the normal vector is just {x, y, z} for a unit sphere. I also map Integrate over the summands to make things easier for Integrate.

Answer 2

With the new-in-13.3 SurfaceIntegrate:

preg = 
  ParametricRegion[{u, v, Sqrt[1 - u^2 - v^2]}, 
                   {{u, 0, 1}, {v, -Sqrt[u - u^2], Sqrt[u - u^2]}}];

SurfaceIntegrate[{x^2, y^2, z^2}, {x, y, z} ∈ preg]
(* 38/105 + (5 π)/32 *)

Sadly the following doesn't work for the moment:

SurfaceIntegrate[{x^2, y^2, z^2}, {x, y, z} ∈ 
  ImplicitRegion[x^2 + y^2 + z^2 == 1 && x^2 - x + y^2 <= 0 && z >= 0, {x, y, z}]]
(* Input returned. *)

Answer 3

Just to illustrate use of MichaelE2's DoubleContourIntegral:

DoubleContourIntegral[{x^2, y^2, 
  z^2}, {({x, y, z} -> {u, v, Sqrt[1 - u^2 - v^2]}), {u, 0, 
   1}, {v, -Sqrt[u - u^2], Sqrt[u - u^2]}}]

yields: 38/105 + (5 π)/32

Note the limits of integration.