f is obviously smooth on R^3. So what you need to show is, when f is restricted to S^2, its image is in S^2.
That is, to show that [(x^2-y^2)^2 + (2xy)^2 + (2z)^2]/(1+z^2)^2 = 1, given that x^2+y^2+z^2=1.
Which is easy to show.
show that
defines a smooth function where so I am guessing it is a unit sphere.
How do we proceed with such problem?
f is a smooth function if all partial derivatives of all possible orders are defined in all points of the domain of f right?
I started with computing these partial derivatives
and these partial derivatives are all defined at any point from the domain of f
so I compute second order partial derivatives
and these 3 are also defined for all the points of the domain.
if I continue differentiating
and
which is defined for all points
What should I do next? can I conclude based on the calculations that f is certainly smooth?
Thanks in advance
so showing all the computations above were not necessary if I am supposed to restrict myself to right?
But I still don't get how to show how does this imply that f will be smooth on the sphere???
=
This shows that indeed maps to .
Then is smooth because "the restriction of a smooth function on a smooth sub-manifold is smooth".
To prove the above statement, let is the smooth embedding of in ,
is a smooth map on , then is smooth on .