# 音乐同构

## 正式定义

${\displaystyle {\widehat {g}}_{x}:T_{x}M\longrightarrow T_{x}^{*}M}$

${\displaystyle {\widehat {g}}_{x}(X_{x})=\langle X_{x},\cdot \rangle \in T_{x}^{*}M\ ,}$

${\displaystyle {\widehat {g}}_{x}(X_{x})(Y_{x})=\langle X_{x},Y_{x}\rangle \ .}$

${\displaystyle {\widehat {g}}:TM\longrightarrow T^{*}M\ ,}$

${\displaystyle {\widehat {g}}:\alpha ^{i}{\frac {\partial }{\partial x^{i}}}\mapsto \alpha ^{i}g_{ij}d\,x^{j}\ .}$

${\displaystyle {\widehat {g}}^{-1}:\xi =\alpha _{i}d\,x^{i}\mapsto \alpha _{i}g^{ij}{\frac {\partial }{\partial x^{j}}}\ .}$

## 名称由来

${\displaystyle (\sum _{i}X^{i}{\frac {\partial }{\partial x^{i}}})^{\flat }=\sum _{ij}g_{ij}X^{i}dx^{j}:=\sum _{j}X_{j}dx^{j}\ ,}$

${\displaystyle (\sum _{i}\omega _{i}dx^{i})^{\sharp }=\sum _{ij}g^{ij}\omega _{i}{\frac {\partial }{\partial x^{j}}}\ ,}$

## 梯度、散度与旋度

{\displaystyle {\begin{aligned}\nabla f&=\left({\mathbf {d} }f\right)^{\sharp }\\\nabla \cdot F&=\star {\mathbf {d} }\star (F^{\flat })\\\nabla \times F&=\left[\star {\mathbf {d} }(F^{\flat })\right]^{\sharp }\end{aligned}}}

${\displaystyle \mathbf {v} \times \mathbf {w} =\left[\star \left(\mathbf {v} ^{\flat }\wedge \mathbf {w} ^{\flat }\right)\right]^{\sharp }.}$