Let \(F\colon M \to N\) denote a smooth map between smooth manifolds. In \(\mathbb{R}^n\) the total differential is a linear map on \(\mathbb{R}^n\). On manifolds it does not make sense to define a linear map. But on the tangent space it does.
We define the total derivative \(dF_p\colon T_pM\to T_{F(p)}N\) via
\begin{equation*} dF_p(v)(f)=v(f\circ F), \end{equation*}where \(v\in T_pM\) and \(f\in C^\infty(M)\).
Remarks
- The total derivative is also called differential, pushforward, tangent map or simply derivative of \(F\).
- The total derivative is a linear map.
- Alternatively, if tangent vectors are equivalence classes , then \[ dF_p\colon\, T_pM\to T_{F(p)}N, \quad [c]\mapsto [F\circ c]. \]
- Identities.
- Let \(F\colon M\to N\) and \(G\colon N\to P\) be smooth. Then the following chain rule holds \[ d(G\circ F)_p = dG_{F(p)}\circ dF_p. \]
- By the chain rule, if \(F\colon M\to N\) is diffeomorph , then \(dF_p\) is an isomorphism, and \[ (dF_p)^{-1}=d(F^{-1})_{F(p)}. \]
- \(d(\id_M )_p = \id_{T_pM}\colon T_pM\to T_pM\).