For $a,b,c >0$ let,
$E(a,b,c) = \{ (x,y,z) \in \mathbb {R^3} : \frac {x^2}{a^2} +\frac {y^2}{b^2} + \frac {z^2}{c^2} =1 \}$
How do i show that E(a,b,c) is a $C^{\infty}$ surface in $\mathbb {R^3}$
showing sphere like object is a $C^{\infty}$ surface.
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$\begingroup$
geometry
multivariable-calculus
surfaces
1 Answers
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Consider $f_{a,b,c}(x,y,z)={x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}$.
$f_{a,b,c}$ is a submersion on $R^3-\{0\}$ and $E(a,b,c)=f^{-1}_{a,b,c}(1)\subset R^3-\{0\}$.
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0so what i think your saying is that $f^{-1}$ is infinitely differentiable? – 2017-01-23
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0yes, it is a $C^{\infty}$-manifold since $f_{a,b,c}$ is $C^{\infty}$ this a standard way to show that a subset of $R^n$ is a $C^{\infty}$ manifold. – 2017-01-23