Contact (mathematics)
In mathematics, two functions have a contact of order k if they have the same value at a point P and also the same derivatives there, up to order k. This is an equivalence relation, whose equivalence classes are generally called jets. The point of osculation is also called the double cusp.
One speaks also of curves and geometric objects having k-th order contact at a point: this is also called osculation (i.e. kissing), generalising the property of being tangent. See for example osculating circle and osculating orbit.
Contact forms are particular differential forms of degree 1 on odd-dimensional manifolds; see contact geometry. Contact transformations are related changes of co-ordinates, of importance in classical mechanics. See also Legendre transformation.
Contact between manifolds is often studied in singularity theory, where the type of contact are classified, these include the A series (A0: crossing, A1: tangent, A2: osculating, ...) and the umbilic or D-series where there is a high degree of contact with the sphere.
Read more about Contact (mathematics): Contact Between Curves
Famous quotes containing the word contact:
“I stand in awe of my body, this matter to which I am bound has become so strange to me. I fear not spirits, ghosts, of which I am one,—that my body might,—but I fear bodies, I tremble to meet them. What is this Titan that has possession of me? Talk of mysteries! Think of our life in nature,—daily to be shown matter, to come in contact with it,—rocks, trees, wind on our cheeks! the solid earth! the actual world! the common sense! Contact! Contact! Who are we? where are we?”
—Henry David Thoreau (1817–1862)