# 射流

## 欧氏空间之间的函数的射流

### 例：一维情况

${\displaystyle f:{\mathbb {R} }\rightarrow {\mathbb {R} }}$ 是实值函数，在点${\displaystyle x_{0}}$ 的领域U有至少k+1导数。那么根据泰勒定理，

${\displaystyle f(x)=f(x_{0})+f'(x_{0})(x-x_{0})+\cdots +{\frac {f^{(k)}(x_{0})}{k!}}(x-x_{0})^{k}+{\frac {R_{k+1}(x)}{(k+1)!}}(x-x_{0})^{k+1}}$

${\displaystyle |R_{k+1}(x)|\leq \sup _{x\in U}|f^{(k+1)}(x)|.}$

${\displaystyle (J_{x_{0}}^{k}f)(z)=f(x_{0})+f'(x_{0})z+\cdots +{\frac {f^{(k)}(x_{0})}{k!}}z^{k}.}$

### 例：从欧氏空间到另一个欧氏空间的映射

${\displaystyle f(x)=f(x_{0})+(Df(x_{0}))\cdot (x-x_{0})+{\frac {1}{2}}(D^{2}f(x_{0}))\cdot (x-x_{0})^{\otimes 2}+\cdots +{\frac {D^{k}f(x_{0})}{k!}}\cdot (x-x_{0})^{\otimes k}+{\frac {R_{k+1}(x)}{(k+1)!}}\cdot (x-x_{0})^{\otimes (k+1)}}$

${\displaystyle (J_{x_{0}}^{k}f)(z)=f(x_{0})+(Df(x_{0}))\cdot z+{\frac {1}{2}}(D^{2}f(x_{0}))\cdot z^{\otimes 2}+\cdots +{\frac {D^{k}f(x_{0})}{k!}}\cdot z^{\otimes k}}$

### 例：射流的代数属性

${\displaystyle f,g:{\mathbb {R} }^{n}\rightarrow {\mathbb {R} }}$ 是一对实值函数，则我们可以定义它们的射流的积为

${\displaystyle J_{x_{0}}^{k}f\cdot J_{x_{0}}^{k}g=J^{k}(f\cdot g)}$ .

• 在一维，令${\displaystyle f(x)=\log(1-x)}$ ${\displaystyle g(x)=\sin \,x}$ 。则
${\displaystyle (J_{0}^{3}f)(x)=-x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{3}}}$
${\displaystyle (J_{0}^{3}g)(x)=x-{\frac {x^{3}}{6}}}$

${\displaystyle (J_{0}^{3}f)\circ (J_{0}^{3}g)=-\left(x-{\frac {x^{3}}{6}}\right)-{\frac {1}{2}}\left(x-{\frac {x^{3}}{6}}\right)^{2}-{\frac {1}{3}}\left(x-{\frac {x^{3}}{6}}\right)^{3}\ \ ({\hbox{mod}}\ x^{4})}$
${\displaystyle =-x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{6}}}$

## 欧氏空间一点的射流：严格定义

### 解析定义

${\displaystyle C^{\infty }({\mathbb {R} }^{n},{\mathbb {R} }^{m})}$ 为光滑函数${\displaystyle f:{\mathbb {R} }^{n}\rightarrow {\mathbb {R} }^{m}}$ 的向量空间。令k为非负整数，并令p${\displaystyle {\mathbb {R} }^{n}}$ 的一点。我们在该空间定义一个等价关系${\displaystyle E_{p}^{k}}$ ，也就是令两个函数fg等价如果fgp有相同的值，并且所有它们的偏导数等价到k阶，若fgp数值相同，并且它们直到p阶的偏导数全部相同。

k阶射流空间${\displaystyle C^{\infty }({\mathbb {R} }^{n},{\mathbb {R} }^{m})}$ 在点p定义为${\displaystyle E_{p}^{k}}$ 的等价类集合，并记为${\displaystyle J_{p}^{k}({\mathbb {R} }^{n},{\mathbb {R} }^{m})}$

### 代数几何定义

${\displaystyle C^{\infty }({\mathbb {R} }_{p}^{n},{\mathbb {R} }^{m})}$ 光滑函数${\displaystyle f:{\mathbb {R} }^{n}\rightarrow {\mathbb {R} }^{m}}$ ${\displaystyle {\mathbb {R} }^{n}}$ 中的点p向量空间。令${\displaystyle {\mathfrak {m}}_{p}}$ 为在p为零的函数的理想。(这是局部环 ${\displaystyle C^{\infty }({\mathbb {R} }_{p}^{n},{\mathbb {R} }^{m})}$ 极大理想。)则理想${\displaystyle {\mathfrak {m}}_{p}^{k+1}}$ 由所有在点p直到k阶导数全部为零的函数的芽组成。现在我们可以定义p点的射流空间

${\displaystyle J_{p}^{k}({\mathbb {R} }^{n},{\mathbb {R} }^{m})=C^{\infty }({\mathbb {R} }_{p}^{n},{\mathbb {R} }^{m})/{\mathfrak {m}}_{p}^{k+1}}$

${\displaystyle f:{\mathbb {R} }^{n}\rightarrow {\mathbb {R} }^{m}}$ 为光滑函数，我们可以定义fpk阶射流为${\displaystyle J_{p}^{k}({\mathbb {R} }^{n},{\mathbb {R} }^{m})}$ 的如下元素

${\displaystyle J_{p}^{k}f=f\ ({\hbox{mod}}\ {\mathfrak {m}}_{p}^{k+1})}$

### 从一点到一点的射流空间

${\displaystyle J_{p}^{k}({\mathbb {R} }^{n},{\mathbb {R} }^{m})_{q}=\left\{J^{k}f\in J_{p}^{k}({\mathbb {R} }^{n},{\mathbb {R} }^{m})|f(p)=q\right\}}$

## 两个流形间的函数的射流

MN是两个光滑流形，我们如何定义函数${\displaystyle f:M\rightarrow N}$ 的射流?也许可以通过MN上的局部坐标来定义。这个方法的缺点是流形不能在这种方式下以等变的形式来定义。射流不像张量那样变换。实际上，两个流形间的函数的射流属于一个射流丛

### 从实直线到流形的函数的射流

${\displaystyle \varphi (Q)=\psi (x_{1}(Q),\dots ,x_{n}(Q))}$

${\displaystyle \varphi \circ f=\psi (x_{1}\circ f,\dots ,x_{n}\circ f)}$
${\displaystyle \varphi \circ g=\psi (x_{1}\circ g,\dots ,x_{n}\circ g)}$

${\displaystyle \left.{\frac {d}{dt}}\left(\psi \circ f\right)(t)\right|_{t=0}=\sum _{i=1}^{n}\left.{\frac {d}{dt}}(x_{i}\circ f)(t)\right|_{t=0}\ (D_{i}\psi )\circ f(0)}$

${\displaystyle I=J_{0}^{k}I=J_{0}^{k}(\rho \circ \rho ^{-1})=J_{0}^{k}(\rho )\circ J_{0}^{k}(\rho ^{-1})}$

• 如前所示，通过p的一条曲线的1阶射流就是一个切向量。在p的一个切向量就是一个一阶微分算子，它作用于p点的光滑实值函数。在局部坐标中，每个切向量有如下形式
${\displaystyle v=\sum _{i}v^{i}{\frac {\partial }{\partial x^{i}}}}$

${\displaystyle \varphi \circ f:{\mathbb {R} }\rightarrow {\mathbb {R} }}$

${\displaystyle J_{0}^{1}(\varphi \circ f)(t)=tv^{i}{\frac {\partial f}{\partial x^{i}}}(p)}$ .

• 过一点的曲线的二阶射流。

${\displaystyle x^{i}(t)=t{\frac {dx^{i}}{dt}}(0)+{\frac {t^{2}}{2}}{\frac {d^{2}x^{i}}{dt^{2}}}.}$

${\displaystyle {\frac {d}{dt}}y^{i}(x(t))={\frac {\partial y^{i}}{\partial x^{j}}}(x(t)){\frac {dx^{j}}{dt}}(t)}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}y^{i}(x(t))={\frac {\partial ^{2}y^{i}}{\partial x^{j}\partial x^{k}}}(x(t)){\frac {dx^{j}}{dt}}(t){\frac {dx^{k}}{dt}}(t)+{\frac {\partial y^{i}}{\partial x^{j}}}(x(t)){\frac {d^{2}x^{j}}{dt^{2}}}(t)}$

${\displaystyle {\dot {y}}^{i}={\frac {\partial y^{i}}{\partial x^{j}}}(0){\dot {x}}^{j}}$
${\displaystyle {\ddot {y}}^{i}={\frac {\partial ^{2}y^{i}}{\partial x^{j}\partial x^{k}}}(0){\dot {x}}^{j}{\dot {x}}^{k}+{\frac {\partial y^{i}}{\partial x^{j}}}(0){\ddot {x}}^{k}.}$

### 从流形到流形的函数的射流

MN为两个光滑流形。令pM一点。考虑由定义在p的某个邻域中的光滑映射${\displaystyle f:M\rightarrow N}$ 组成的空间${\displaystyle C_{p}^{\infty }(M,N)}$ 。在${\displaystyle C_{p}^{\infty }(M,N)}$ 上定义一个等价关系${\displaystyle E_{p}^{k}}$ 如下。两个映射fg称为等价的，若对于每条穿过p的曲线γ（按此处常规，这表示一个使得${\displaystyle \gamma (0)=p}$ 的映射${\displaystyle \gamma :{\mathbb {R} }\rightarrow M}$ ，）我们在p的某个领域上有${\displaystyle J_{0}^{k}(f\circ \gamma )=J_{0}^{k}(g\circ \gamma )}$

${\displaystyle J_{p}^{k}(M,N)}$ 的射流空间则定义为${\displaystyle C_{p}^{\infty }(M,N)}$ 以等价关系${\displaystyle E_{p}^{k}}$ 为模的等价类的集合。注意，因为目标空间N不需要有代数结构，${\displaystyle J_{p}^{k}(M,N)}$ 也可以没有这样的结构。也就是说，这和欧氏空间的情形实际上形成鲜明的对比。

${\displaystyle f:M\rightarrow N}$ 是定义在p附近的光滑函数，则我们定义fpk阶射流${\displaystyle J_{p}^{k}f}$ f${\displaystyle E_{p}^{k}}$ 为模所属的等价类。

## 截面的射流

E为流形M上的有限维光滑向量丛，其投影为${\displaystyle \pi :E\rightarrow M}$ 。则E的截面为满足${\displaystyle \pi \circ s}$ M上的恒等自同构的光滑函数${\displaystyle s:M\rightarrow E}$ 。截面sp的一个邻域上的射流就是从ME的光滑函数在点p的射流。

• 例：切丛的一阶射流丛。

${\displaystyle v=v^{i}(x)\partial /\partial x^{i}}$
M中点p的一个邻域。v的一阶射流可以通过取向量场的系数的一阶泰勒多项式得到：
${\displaystyle v^{i}(x)=v^{i}(0)+x^{j}{\frac {\partial v^{i}}{\partial x^{j}}}(0)=v^{i}+v_{j}^{i}x^{j}}$

${\displaystyle v=w^{k}(y)\partial /\partial y^{k}=v^{i}(x)\partial /\partial x^{i},}$

${\displaystyle w^{k}(y)=v^{i}(x){\frac {\partial y^{k}}{\partial x^{i}}}(x).}$

${\displaystyle w^{k}(0)+y^{j}{\frac {\partial w^{k}}{\partial y^{j}}}(0)=\left(v^{i}(0)+x^{j}{\frac {\partial v^{i}}{\partial x^{j}}}\right){\frac {\partial y^{k}}{\partial x^{i}}}(x)}$

${\displaystyle w^{k}={\frac {\partial y^{k}}{\partial x^{i}}}(0)v^{i}}$
${\displaystyle w_{j}^{k}=v^{i}{\frac {\partial ^{2}y^{k}}{\partial x^{i}\partial x^{j}}}+v_{j}^{i}{\frac {\partial y^{k}}{\partial x^{i}}}.}$

## 参考

• Ehresmann, C., "Introduction a la théorie des structures infinitésimales et des pseudo-groupes de mathcal{L}." Geometrie Differentielle, Colloq. Inter. du Centre Nat. de la Recherche Scientifique, Strasbourg, 1953, 97-127.
• Kolár(, I., Michor, P., Slovák, J., Natural operations in differential geometry. Springer-Verlag: Berlin Heidelberg, 1993. ISBN 3-540-56235-4, ISBN 0-387-56235-4.
• Saunders, D.J., "The Geometry of Jet Bundles", Cambridge University Press, 1989, ISBN 0-521-36948-7
• Bocharov, A.V. [et al.], "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, ISBN 0-8218-0958
• Olver, P.J., "Equivalence, Invariants and Symmetry", Cambridge University Press, 1995, ISBN 0-521-47811-1