I'm trying to get an intuition for smooth manifolds, and in particular the smoothness of transition functions. I haven't done that much calculus on $\mathbb{R}^n$ before, and would like to practice proving that some functions $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ are smooth or not smooth. I believe this is just a case of checking all partial derivatives $\frac{\partial f_i}{\partial x_j}$ exist to arbitrary order. Is this correct?

To facilitate practising this, does anyone know of a good selection of examples I could try? All the textbooks on differential geometry seem to assume I'm already completely comfortable with this, so provide no such exercises! Could someone point me at a good resource online or a good book? Alternatively could someone provide some interesting problems/examples in an answer?

Many thanks in advance.

  • $\begingroup$ Perhaps you should try working some relevant problems in Rudin or another undergraduate analysis textbook. $\endgroup$ – Neal Oct 9 '12 at 13:50
  • $\begingroup$ Rudin doesn't seem to have explicit examples of the form I am looking for! $\endgroup$ – Edward Hughes Oct 9 '12 at 16:47

You could try analyse the height function of some surface in $\mathbb{R}^{3}$.

For example take coordinates $(x,y,z)$ in $\mathbb{R}^{3}$. Now consider the unit sphere $S$ in $\mathbb{R}^{3}$ with center in $(0,0,1)$. Define the height function $f:S\rightarrow\mathbb{R}$ with respect to the plane $xy$ by $$f(x,y,z)=z$$

Is this function differentiable?

Try with the torus and so on.

  • $\begingroup$ Thanks - this a good idea. $\endgroup$ – Edward Hughes Oct 9 '12 at 16:47

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