Formal Definitions
There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via velocities of curves is quite straightforward given the above intuition, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.
Read more about this topic: Tangent Space
Famous quotes containing the words formal and/or definitions:
“It is in the nature of allegory, as opposed to symbolism, to beg the question of absolute reality. The allegorist avails himself of a formal correspondence between ideas and things, both of which he assumes as given; he need not inquire whether either sphere is real or whether, in the final analysis, reality consists in their interaction.”
—Charles, Jr. Feidelson, U.S. educator, critic. Symbolism and American Literature, ch. 1, University of Chicago Press (1953)
“The loosening, for some people, of rigid role definitions for men and women has shown that dads can be great at calming babiesif they take the time and make the effort to learn how. Its that time and effort that not only teaches the dad how to calm the babies, but also turns him into a parent, just as the time and effort the mother puts into the babies turns her into a parent.”
—Pamela Patrick Novotny (20th century)