Function for a sphere

Question

I believe that there is something fundamentally wrong with my understanding of functions but I can't pin point what it is, so I would greatly appreciate any guidance.

Consider a unit sphere, centered at the origin, with equation: $$ x^2+y^2+z^2=1\,. $$

Now we can re-arrange this and arrive at a function $z(x,y)=\pm\sqrt{1-x^2-y^2}$, which we can graph and this is the graph of a surface of a sphere, correct? Is it correct to call $z$ a function for the surface of the unit sphere? I have been searching "function for a sphere" online and this doesn't seem to be a term which makes me think I am fundamentally misunderstanding something.

Answer 1

Sphere is a level set of the function $F(x,y,z)=x^2+y^2+z^2$. For example, $F(x,y,z)=1$ corresponds to the sphere of the radius $\sqrt{1}$. Being a 2-dimensional object one can locally re-parametrize it with two independent coordinates, for example in the top semi-sphere, simply by $x, y$, from which $z=\sqrt{1-x^2-y^2}.$

Answer 2

You are confusing the graph of a function with the function itself. It is easier to see it in less dimensions:

The unit circle is a subset of plane consisting of points $\left(x,y\right)$ such that $x^2+y^2=1$. If you isolate $y$, you get two functions: $$y=+\sqrt{1-x^2}$$ and $$y=-\sqrt{1-x^2}$$ The graph of the first function is a subset of the plane, equal to the upper part of the unit circle, and the graph of the second function is the lower part of the unit circle.

For the unit sphere, we also have two functions; one for the upper part of the sphere, containing the north pole, and another for the lower part, containing the south pole. If you unite the graph of both functions, you will get the whole unit sphere.