- #1

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##y^2=c-x^2## and then ##y=\sqrt |x|##

Thank you.

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- #1

- 761

- 11

##y^2=c-x^2## and then ##y=\sqrt |x|##

Thank you.

- #2

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In general, any function we get by taking a relation ##f(x,y) = g(x,y)## and solving for ##y## is called an implicit function for the relation at hand. But keep in mind that a relation may have more than one implicit function. The example you give i.e. ##x^2 + y^2 = c## has more than one implicit function. If you solve for ##y^2## as you did and you want to get ##y##, you need to take the square root of the right hand side and this leads to one positive square root (one implicit function) and one negative (a second implicit function) for any value you give to ##c##.

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Let's work in ##\Bbb R^2##, say. Suppose you had an equation ##F(x,y)=0##. Is there such a subset ## X\subseteq \Bbb R## such that for a fixed ##x\in X## there is a unique ##y## such that the equation is satisfied? If so, we say ##F## determines implicitly a mapping ##f:X\to\Bbb R## (satisfying ##F(x,f(x))=0, x\in X ##). We don't want to pick some funny weird subsets ##X##. We want it to be open so we could talk about differentiability (which is a very strong assumption - the course I took on optimal control theory Heavily relies on smoothness of the object function and the implicit function theorem becomes a powerful weapon - of course, it restricts the choice of ##F##, as well).

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