# order of contact

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

Suppose that $A$ and $B$ are tangent at $P$. We may then set up a coordinate
system^{} 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 $i=2,\mathrm{\dots},n$
and $B$ can be described parametrically as ${x}_{i}={g}_{i}({x}_{1})$ with
$i=2,\mathrm{\dots},n$. 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.

Title | order of contact |
---|---|

Canonical name | OrderOfContact |

Date of creation | 2013-03-22 16:59:49 |

Last modified on | 2013-03-22 16:59:49 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 4 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 53A04 |

Synonym | order contact |