If $M$ and $N$ are smooth manifolds, the jacobian matrix of the derivative of a smooth function $F: M \to N$ in a point $p \in M$, taking charts $(U,\varphi)$ around $p$ and $(V,\psi)$ around $F(p)$ with $F(U) \subseteq V,$ is given by
$$ [J_F(p)]^i_{\,j} = \frac{\partial(\psi \circ F \circ \varphi^{-1})^i}{\partial x^j}(\varphi(p)).$$
Now, I'm currently working on a problem that requires the derivative of a $C^k$ function where $k$ can be finite, from a book of foliation theory which assumes previous knowledge on manifolds, and though I studied manifolds before, all objects were smooth from the get go (Tu and Lee), so I'm not sure how to proceed. At first I blindly calculated the same way as I would a smooth function, which gave me the desired result, but I'm not sure if such operation is actually valid.
So, is the derivative of a $C^k$ function over smooth manifold computed the exact same way as a smooth function? I couldn't find literature on this, $C^k$ functions are briefly mentioned and computations are made with smooth functions everywhere I looked.
It feels like they would be the same, but since tangent spaces over $C^k$ manifolds are a bit different than their smooth counterparts (constructed by derivations) so I can't be sure. Any clarifications or sources about this are appreciated, thanks.
Best Answer
There are several approaches to defining tangent spaces. There is derivations on germs of smooth functions, or the “ algebraists “ tangent bundle. There are equivalence classes of paths through the point, where you restrict the function to the path and differentiate to get a derivation, the “physicists” tangent space, and you can just abstractly attach a vector space at each point in a coordinate chart and then identify tangent vectors in different charts using the correct transformation rules, the “bundle theorists “ tangent space. In the case of $C^{\infty}$-manifolds they are equivalent and yield the same answers for the Jacobian. In the $C^k$-case the definition via germs becomes more subtle. It is no longer a particularly useful viewpoint. However the other two definitions work, and you can compute as you did before. The only subtlety is that every time you differentiate, a function is less smooth.
If you look in older books on the subject you can find a detailed discussion of the different approaches that I mentioned and their comparisons. Boothby, and Hirsch come to mind.
The routine of turning geometry into linear algebra has become just that, a routine. Most advances in Differential Geometry now occur at the level of analysis. However the ideas of differential geometry are driving number theory. The analogy with geometry occurs at the level of sheaves. The germ approach to tangent vectors fits this, so it is now emphasized in introductory courses, at the cost of making it harder to transition to doing geometry.