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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 Implicit Function Theorem Suppose we have a function of two variables, F(x;y), and we’re interested in its height-c level ... The green segment represents a neighborhood of the red point on which y is determined by x. ... Here’s an example (due to Lincoln Chayes) to test our understand-ing.

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