# 第一基本形式

${\displaystyle \mathrm {I} (v,w)=\langle v,w\rangle .\,}$

${\displaystyle X(u,v)}$是一个参数曲面，则两个切向量的内积为

{\displaystyle {\begin{aligned}&{}\quad \mathrm {I} (aX_{u}+bX_{v},cX_{u}+dX_{v})\\&=ac\langle X_{u},X_{u}\rangle +(ad+bc)\langle X_{u},X_{v}\rangle +bd\langle X_{v},X_{v}\rangle \\&=Eac+F(ad+bc)+Gbd,\end{aligned}}}

${\displaystyle \mathrm {I} (v,w)=v^{T}{\begin{pmatrix}E&F\\F&G\end{pmatrix}}w.}$

## 进一步的记号

${\displaystyle \mathrm {I} (v)=\langle v,v\rangle =|v|^{2}.\,}$

${\displaystyle \left(g_{ij}\right)={\begin{pmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{pmatrix}}={\begin{pmatrix}E&F\\F&G\end{pmatrix}}}$

${\displaystyle g_{ij}=X_{i}\cdot X_{j}}$

i, j = 1, 2。具体例子可见下一节。

## 變數變換

${\displaystyle {\begin{pmatrix}E&F\\F&G\end{pmatrix}}={\begin{pmatrix}{\tilde {u}}_{u}&{\tilde {v}}_{u}\\{\tilde {u}}_{v}&{\tilde {v}}_{v}\end{pmatrix}}{\begin{pmatrix}{\tilde {E}}&{\tilde {F}}\\{\tilde {F}}&{\tilde {G}}\end{pmatrix}}{\begin{pmatrix}{\tilde {u}}_{u}&{\tilde {u}}_{v}\\{\tilde {v}}_{u}&{\tilde {v}}_{v}\end{pmatrix}}}$，其中${\displaystyle {\begin{pmatrix}{\tilde {u}}_{u}&{\tilde {u}}_{v}\\{\tilde {v}}_{u}&{\tilde {v}}_{v}\end{pmatrix}}={\frac {\partial ({\tilde {u}},{\tilde {v}})}{\partial (u,v)}}}$，所以說可以有

${\displaystyle EG-F^{2}=({\tilde {E}}{\tilde {G}}-{\tilde {F}}^{2})|{\frac {\partial (u,v)}{\partial (u,v)}}|^{2}}$

## 计算长度与面积

${\displaystyle ds^{2}=Edu^{2}+2Fdudv+Gdv^{2}\,}$.

${\displaystyle dA=|X_{u}\times X_{v}|\ du\,dv}$ 给出的经典面积元素可以用第一基本形式的系数利用拉格朗日恒等式写出，

${\displaystyle dA=|X_{u}\times X_{v}|\ du\,dv={\sqrt {\langle X_{u},X_{u}\rangle \langle X_{v},X_{v}\rangle -\langle X_{u},X_{v}\rangle ^{2}}}\ du\,dv={\sqrt {EG-F^{2}}}\,du\,dv.}$

### 例子

R3 中单位球面可如下参数化

${\displaystyle X(u,v)={\begin{pmatrix}\cos u\sin v\\\sin u\sin v\\\cos v\end{pmatrix}},\ (u,v)\in [0,2\pi )\times [0,\pi ).}$

${\displaystyle X(u,v)}$ 分别对 uv 微分得出

${\displaystyle X_{u}={\begin{pmatrix}-\sin u\sin v\\\cos u\sin v\\0\end{pmatrix}},\ X_{v}={\begin{pmatrix}\cos u\cos v\\\sin u\cos v\\-\sin v\end{pmatrix}}.}$

${\displaystyle E=X_{u}\cdot X_{u}=\sin ^{2}v}$
${\displaystyle F=X_{u}\cdot X_{v}=0}$
${\displaystyle G=X_{v}\cdot X_{v}=1}$

#### 球面上曲线的长度

${\displaystyle \int _{0}^{2\pi }{\sqrt {E\left({\frac {du}{dt}}\right)^{2}+2F{\frac {du}{dt}}{\frac {dv}{dt}}+G\left({\frac {dv}{dt}}\right)^{2}}}\,dt=\int _{0}^{2\pi }\sin v\,dt=2\pi \sin v=2\pi .}$

#### 球面上区域的面积

${\displaystyle \int _{0}^{\pi }\int _{0}^{2\pi }{\sqrt {EG-F^{2}}}\ du\,dv=\int _{0}^{\pi }\int _{0}^{2\pi }\sin v\,du\,dv=2\pi \left[-\cos v\right]_{0}^{\pi }=4\pi .}$

## 高斯曲率

${\displaystyle K={\frac {\det II}{\det I}}={\frac {LN-M^{2}}{EG-F^{2}}},}$