I tried to understand the proof of Sard's theorem suggested in John M. Lee’s “Introduction to Smooth Manifolds”.
It said that $C_k$ is defined as $$ C_k := \{x \in U \vert \forall 1 \le i \le k\, \text{$i$-th partial derivatives of $f$ is $0$} \} $$ for smooth function, $f : M \to N$, where $M, N$ is smooth manifold.
I've assumed that $i$-th partial derivatives denote each components of $$ \dfrac{\partial f}{\partial x^i} $$ since Appendix C in same book follows this terminology.
However, when I read all parts of the proof, I'm strongly convinced that it means $$ \dfrac{\partial^i}{\partial x^I}f(\text{I is a sequence of indices, $|I| = i$}) $$, i.e, $i$-th order partial derivatives, rather than the above.
So, which one is correct? Is it just confusion in notations?
p.s.) Here is another question. If we accept $i$-th partial derivatives as $i$-th component partial derivates (rather than $i$-th order partial derivatives) in the proof, could we elaborate the proof of Sard's theorem in the case of $f \in C^\infty$ to the case of $f \in C^1$?
You're right -- it should have said "$i$th-order partial derivatives." I've added this to my online correction list.