习题 An Introduction to Manifold - Loring W. Tu, Problem 8.1

给定 $F : \mathbb{R}^2 \to \mathbb{R}^3$ 为映射

$$ \bbox[5pt]{ (u,v,w) = F(x,y) = (x,y,xy) } $$

给定点 $p = (x,y) \in \mathbb{R}^2$ ，计算 $\displaystyle F_{\ast,p}\left(\left.\frac{\partial}{\partial{x}}\right|_p\right)$ 表示为 $\displaystyle \left.\frac{\partial}{\partial{u}}\right|_{F(p)} , \left.\frac{\partial}{\partial{v}}\right|_{F(p)} , \left.\frac{\partial}{\partial{w}}\right|_{F(p)}$ 的线性组合

Proof.
记

$$ \bbox[5pt]{ F_{\ast,p}\left(\left.\frac{\partial}{\partial{x}}\right|_p\right) = a\left.\frac{\partial}{\partial{u}}\right|_{F(p)} + b\left.\frac{\partial}{\partial{v}}\right|_{F(p)} + c\left.\frac{\partial}{\partial{w}}\right|_{F(p)} } $$

则

$$ \bbox[5pt]{ \begin{aligned} a = & F_{\ast,p}\left(\left.\frac{\partial}{\partial{x}}\right|_p\right)u \\ \mathtip{\color{skyblue}{=}}{定义: F_{\ast,p}} & \frac{\partial}{\partial{x}}(u \circ F(p))\big|_p \\ = & \frac{\partial}{\partial{x}}(x)\big|_p \\ = & 1 \\\\ b = & F_{\ast,p}\left(\left.\frac{\partial}{\partial{x}}\right|_p\right)v \\ \mathtip{\color{skyblue}{=}}{定义: F_{\ast,p}} & \frac{\partial}{\partial{x}}(v \circ F(p))\big|_p \\ = & \frac{\partial}{\partial{x}}(y)\big|_p \\ = & 0 \\\\ c = & F_{\ast,p}\left(\left.\frac{\partial}{\partial{x}}\right|_p\right)w \\ \mathtip{\color{skyblue}{=}}{定义: F_{\ast,p}} & \frac{\partial}{\partial{x}}(w \circ F(p))\big|_p \\ = & \frac{\partial}{\partial{x}}(xy)\big|_p \\ = & y \end{aligned} } $$

即

$$ \bbox[5pt]{ F_{\ast,p}\left(\left.\frac{\partial}{\partial{x}}\right|_p\right) = \left.\frac{\partial}{\partial{u}}\right|_{F(p)} + y\left.\frac{\partial}{\partial{w}}\right|_{F(p)} } $$