First Example
If we define the function, then the equation cuts out the unit circle as the level set . There is no way to represent the unit circle as the graph of a function of one variable because for each choice of there are two choices of, namely .
However, it is possible to represent part of the circle as the graph of a function of one variable. If we let for, then the graph of provides the upper half of the circle. Similarly, if, then the graph of gives the lower half of the circle.
The purpose of the implicit function theorem is to tell us the existence of functions like and, even in situations where we cannot write down explicit formulas. It guarantees that and are differentiable, and it even works in situations where we do not have a formula for .
Read more about this topic: Implicit Function Theorem
Famous quotes containing the word example:
“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)