> [!theorem] > > Let $U \subset \real^m$ and $V \subset \real^n$ be [[Open Set|open]], and $F: U \to V$ be a $C^p$-[[Space of Continuously Differentiable Functions|morphism]]. Let $(x^1, \cdots, x^m)$ be the standard coordinates in $U$, and $(y^1, \cdots, y^n)$ be the standard coordinates in $V$. Then the [[Differential|differential]] $dF_p: T_p\real^n \to T_{F(p)}\real^m$ has coordinate representation > $ > \frac{\partial}{\partial x^i}\bigg\vert_p \mapsto \frac{\partial F^j}{\partial x^i}(p) \frac{\partial}{\partial y^j}\bigg\vert_{F(p)} > $ > so $dF_p$ has the coordinate representation > $ > dF_p = \begin{bmatrix} > \frac{\partial F^1}{\partial x^1}(p) & \cdots & \frac{\partial F^1}{\partial x^n}(p) \\ > \vdots & \ddots & \vdots \\ > \frac{\partial F^m}{\partial x^1}(p) & \cdots & \frac{\partial F^m}{\partial x^n}(p) > \end{bmatrix} > $ > > *Proof*. Let $f \in C^p(V)$, then by the chain rule, > $ > \begin{align*} > dF_p\paren{\ppip} \cdot f &= \ppip \cdot (f \circ F) \\ > &= \braks{\ppi (f \circ F)}(p) \\ > &= \frac{\partial f}{\partial y^j}(F(p)) \cdot \frac{\partial F^j}{\partial x^i}(p) \\ > &= \braks{\frac{\partial F^j}{\partial x^i}(p) \cdot \ppj \bigg\vert_{F(p)}} \cdot f > \end{align*} > $ > [!theorem] > > Let $X$ and $Y$ be [[n-Manifold|n-manifolds]], $F: X \to Y$ be a [[Manifold Morphism|morphism]], and $p \in X$. Let $(U, \varphi) \in X$ and $(V, \psi) \in Y$ be charts at $p$ and $F(p)$, respectively, and $\widehat F = F_{U, V}$, then $dF_p$ has coordinate representation $d\widehat F_{\hat p}$ with > $ > \ppip \mapsto \frac{\partial \widehat F^j}{\partial x^i}(\hat p) \cdot \ppj\bigg\vert_{F(p)} > $ > *Proof*. Since $\widehat F = \psi \circ F \circ \varphi^{-1}$, > $ > \begin{align*} > dF_p\paren{\ppip} &= d\psi^{-1}_{\widehat {F(p)}} \circ d\widehat F_{\hat p} \circ d\varphi_p \cdot \paren{\ppip} \\ > &= d\psi^{-1}_{\widehat {F(p)}} \cdot \braks{\frac{\partial \widehat F^j}{\partial x^i}(\hat p) \cdot \ppj\bigg\vert_{\widehat {F(p)}}} \\ > &= \frac{\partial \widehat F^j}{\partial x^i}(\hat p) \cdot \ppj\bigg\vert_{F(p)} > \end{align*} > $