Implicit Function Theorem
It can be shown that if R(x, y) is given by a smooth submanifold M in R2, and (a, b) is a point of this submanifold such that the tangent space there is not vertical (that is ), then M in some small enough neighbourhood of (a, b) is given by a parametrization (x, f(x)) where f is a smooth function. In less technical language, implicit functions exist and can be differentiated, unless the tangent to the supposed graph would be vertical. In the standard case where we are given an equation
the condition on R can be checked by means of partial derivatives.
Read more about this topic: Implicit Function
Famous quotes containing the words implicit, function and/or theorem:
“The vanity of men, a constant insult to women, is also the ground for the implicit feminine claim of superior sensitivity and morality.”
—Patricia Meyer Spacks (b. 1929)
“The art of living is to function in society without doing violence to one’s own needs or to the needs of others. The art of mothering is to teach the art of living to children.”
—Elaine Heffner (20th century)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)