The idea of flux in three dimensions is very similar to the one for the flux in two dimensions, the only difference being flux in 3D is through a surface enclosing a region of space.

The approach is the same, break it up and add it all up. This time though, we will be walking across a surface in a region of space, so we have to involve a surface integral.

We first will want to compute the unit normal vector but in three dimensions, recalling that for a surface we walk along two parameters, a parameter space defined for the parameters which then define a step along the surface . We remember that these are the partial derivatives with respect to and of the vector.

We will want the unit normal vector. We first find the normal vector by the magnitude of the cross product of these, then divide it by its magnitude to get the unit normal vector.
We can invert the sign to make the flux inward (negative)

Knowing the flux integral, for our function , we will then want to define the flux in 3D, which is the flux integral but with the surface integral accounted for, and we get