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Differential of a smooth function. Immersions, submersions, and embeddings.
A reference for this material is Chapters 3 and 4 of John M. Lee. Introduction to
smooth manifolds. Second edition. Grad. Texts in Math., 218. Springer, New York,
2013. xvi+708pp. ISBN: 978-1-4419-9981-8.
Differential of a map Let \(M\), \(N\) be smooth manifolds and \(f : M \to N\) a smooth map. The derivative
or differential of \(f\) is the map \(f_{\ast } : TM \to TN\) given by
\[ f_{\ast } L (h) : = L (h \circ f ) \]
for \(L \in TM\) and \(h \in C^{\infty }(N)\).
Differential is well-defined and linear For \(p\in M\), the map \(f_{\ast } \) sends \(T_pM\) linearly to \(T_{f(p)}N\). We call this
map \(d_pf : = f_{\ast } \vert _{T_pM}\).
Solution:
Let \(L \in T_pM\), and \(h_1, h_2 \in C^{\infty } (N)\). Then
\begin{align*} f_{\ast } L (h_1h_2) & = L ( (h_1 h_2 ) \circ f ) \\ & = L ( (h_1 \circ f) (h_2 \circ f) )\\ & = L( h_1 \circ f ) ( h_ 2 \circ f ) (p) + ( h_ 1 \circ f ) (p) L( h_2 \circ f ) \\ & = [ f_{\ast } L(h_1)] h_2 (f(p)) + h_1 (f(p)) [f_{\ast } L (h_2)]. \end{align*}
This shows that \(f_{\ast } L\) is a derivation at \(f(p)\).
Differential in terms of curves For a curve \(\gamma : (- \varepsilon , \varepsilon ) \to M\) one has
Smooth manifolds