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Can I transform an implicit function of a cylinder to explicit form?

Lets say we have a cylinder

$$x^2 + y^2 - 1 = 0$$

and we want to have it expressed as function $f(x,y,z)$.

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What is wrong with $y = \pm \sqrt{1-x^2}$? $z$ can take any value. –  Henry Oct 17 '11 at 20:48
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You have already written it as $f(x,y,z)=0$, haven't you? –  lhf Oct 17 '11 at 21:35
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1 Answer

If what you want is an equation of the form $z=f(x,y)$, forget it: such a function would, by definition, give a unique value of $z$ for every pair of values $(x,y)$, but on your cylinder, a single pair $(x,y)$ may correspond to infinitely many values of $z$. That is, what would you want $f(1,0)$ to be? If what you want is $f(x,y,z)=0$, then, as lhf writes in the comments, you already have it. Think of it as $x^2+y^2+0z-1=0$, if it helps.

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or perhaps $1\times x^2+1\times y^2+0\times z^2 - 1 =0$ to emphasise that it is an even function of all three variables –  Henry Oct 17 '11 at 23:41
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