Transcription of The Implicit Function Theorem - UCLA Mathematics
{{id}} {{{paragraph}}}
Example: bankruptcy
The Implicit Function Theorem - UCLA Mathematics
Math 32A Week 10 NotesNovember 29 and December 1, 2016 Austin ChristianThe Implicit Function TheoremSuppose we have a Function of two variables,F(x, y), and we re interested in its height-clevelcurve; that is, solutions to the equationF(x, y) =c. For instance, perhapsF(x, y) =x2+y2andc= 1, in which case the level curve we care about is the familiar unit circle. It wouldbe nice if choosing a value forxin the equationF(x, y) =cwould immediately determinethe value ofy that is, ifF(x, y) =cdeterminedyas a Function ofx. But we know thatthis isn t generally true. In the case of the unit circle, fixing a valuexleaves both 1 x2and 1 x2as possibilities for the value ofy. Graphically, this obstruction is representedby the fact thatx2+y2= 1 fails the familiar vertical line test, as can be seen in Figure 1: The level curvex2+y2= 1. The green segment represents a neighborhood of thered point on whichyis determined else that can be seen in Figure 1, though, is that our graph does pass thevertical line test locally.
assignment is makes z a continuous function of x and y. Colloquially, the upshot of the implicit function theorem is that for su ciently nice points on a surface, we can (locally) pretend this surface is the graph of a function. The primary use for the implicit function theorem in this course is for implicit di erentiation. You’ve
Loading..
Tags:
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: