order of contact

Suppose that A and B are smooth curves in n which pass through a common point P. We say that A and B have zeroth order contact if their tangentsPlanetmathPlanetmathPlanetmath at P are distinct.

Suppose that A and B are tangent at P. We may then set up a coordinate systemMathworldPlanetmath in which P is the origin and the x1 axis is tangent to both curves. By the implicit function theoremMathworldPlanetmath, there will be a neighborhoodMathworldPlanetmath of P such that A can be described parametrically as xi=fi(x1) with i=2,,n and B can be described parametrically as xi=gi(x1) with i=2,,n. We then define the order of contact of A and B at P to be the largest integer m such that all partial derivativesMathworldPlanetmath of fi and gi 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