MATH 234 Lecture Notes - Lecture 16: Implicit Function Theorem

• School: UW-Madison
• Department: Mathematics
• Course: MATH 234
• Professor: Betsy Stovall

Document Summary

Is there a function such that y = f(x) such that f(x, f(x)) > 0. 0 = ddx f(x, f(x)) = f x * 1 + f ydfdx dfdx = fx (x , f(cid:4666)x(cid:4667)) And the same for the 2nd case: fx(x, y) 0 the implicit function theorem with more variables. Answer is yes ( (xo, yo, zo) is a point on the zero set) Then, locally near (xo, yo, zo) the zero set f(x, y, z) = 0 is the graph of a function y = g(x, z) F(x, g(x, z), z) = 0 for any x and z. 0 = x f(x, g(x, z), z) = F x * 1 + f y g x + f z z x z x = 0. Let f(x, y) be a function of two variables, and also x = g(u, v) y = h(u, v)