order of contact

Suppose that $A$ and $B$ are smooth curves in $\\mathbb{R}^{n}$ which pass througha common point $P$ . We say that $A$ and $B$ have zeroth order contact if theirtangents at $P$ are distinct.

Suppose that $A$ and $B$ are tangent at $P$ . We may then set up a coordinatesystem in which $P$ is the origin and the $x_{1}$ axis is tangent to both curves.By the implicit function theorem, there will be a neighborhood of $P$ such that $A$ can be described parametrically as $x_{i}=f_{i}(x_{1})$ with $i=2,\\ldots,n$ and $B$ can be described parametrically as $x_{i}=g_{i}(x_{1})$ with $i=2,\\ldots,n$ . We then define the order of contact of $A$ and $B$ at $P$ to be the largest integer $m$ such that all partial derivatives of $f_{i}$ and $g_{i}$ of order not greater than $m$ at $P$ are equal.

单词	OrderOfContact
释义	order of contact Suppose that $A$ and $B$ are smooth curves in $\\mathbb{R}^{n}$ which pass througha common point $P$ . We say that $A$ and $B$ have zeroth order contact if theirtangents at $P$ are distinct. Suppose that $A$ and $B$ are tangent at $P$ . We may then set up a coordinatesystem in which $P$ is the origin and the $x_{1}$ axis is tangent to both curves.By the implicit function theorem, there will be a neighborhood of $P$ such that $A$ can be described parametrically as $x_{i}=f_{i}(x_{1})$ with $i=2,\\ldots,n$ and $B$ can be described parametrically as $x_{i}=g_{i}(x_{1})$ with $i=2,\\ldots,n$ . We then define the order of contact of $A$ and $B$ at $P$ to be the largest integer $m$ such that all partial derivatives of $f_{i}$ and $g_{i}$ of order not greater than $m$ at $P$ are equal.
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