[Math] Finding an outward pointing normal on the unit sphere,

Question

I am trying to apply the divergence theorem, but I need to find an outward pointing normal vector on the unit sphere. The answer gives $\hat n= (x_1,x_2,x_3)$.

Is the person who wrote up the solution just saying that any ordered triple on the sphere is an outward pointing normal – and this is all we need for the nds part of the integral?

Thanks,

Answer 1

The outward pointing normal, for a sphere, is in the same direction as the radius. That is because both are perpendicular to the tangent plane.
The position vector has unit length because it is a unit sphere. So the normal, which is the unit vector in the same direction as the position vector, must equal the position vector.