I am trying to determine the earliest source for the definition of smooth ($C^\infty$) manifold with boundary. Milnor and Stasheff (1958) give a definition, but a scrutiny of that definition shows it to be incomplete (because they fail to define what smoothness means in that context).

What is the earliest known reference that is rigorous by modern standards?

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    $\begingroup$ Maybe p. 64 of Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, 17-86 (1954). ZBL0057.15502? $\endgroup$ – Francois Ziegler Feb 26 '19 at 23:17
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    $\begingroup$ You may want to check papers of H. Whitney from late 1930s and early 1940s. He certainly worked with manifolds with boundary (which he called partial manifolds), and he was comparting $C^r$ structures for different $r$ and proving Whitney embedding theorems, so he was aware of the issues. $\endgroup$ – Igor Belegradek Feb 27 '19 at 2:08
  • $\begingroup$ @FrancoisZiegler My French is not great, but I couldn't find it in that reference. Do you know where it might be in the paper? $\endgroup$ – John Klein Feb 27 '19 at 2:19
  • $\begingroup$ @IgorBelegradek I did. I couldn't find it. I'll check again. $\endgroup$ – John Klein Feb 27 '19 at 2:21
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    $\begingroup$ Right, his b) says a bit elliptically that at boundary points “there are charts” mapping $M$ to a closed half-space and restricting on the manifolds $M-V$ (resp. $V$) to charts with values in the open half-space (resp. the boundary hyperplane) — leaving talk about equivalent atlases of such charts to the reader. Maybe his Quelques propriétés des variétés-bords (1952) had more details? $\endgroup$ – Francois Ziegler Feb 27 '19 at 4:46

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