早上在证明黎曼流形上的极值原理的时候意识到了这件事，于是把完整过程写一下以作记录。

Proposition: For a smooth map $f:M\to N$, the representation matrix of its derivative $df\in\Gamma(M,T^*M\otimes f^*TN)$ is orthogonal.

Proof: $df=f^j_idx^i\otimes\frac{\partial}{\partial y^j}$, where $f_i^j=\frac{\partial f^j}{\partial x^i}$.

So $(df)^{-1}=f^i_j\frac{\partial}{\partial x^i}\otimes dy^j$.

And $df\circ(df)^{-1}=f^j_idx^i\otimes\frac{\partial}{\partial y^j}\circ f^k_l\frac{\partial}{\partial x^k}\otimes dy^l=f_i^jf_j^i=\mathrm{Id}$.

Hence ${f_i^j}$ is orthogonal.