I have the ellipsoid
$x^2 + 2y^2 + z^2 = 4$.
It is mentioned that I can rewrite this equation as a level set
$f(x,y) = x^2 + 2y^2 + z^2$.
Of course I can look at Wikipedia where a level set is described as a
real-valued function $f$ of $n$ real variables is a set of the form $L_c(f) = \{(x_1, … , x_n) | f(x_1, … , x_n) = c\} $.
Fair enough. But are there any other, maybe more intuitive explanations, why I can rewrite the equation of the ellipsoid like that?
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$\begingroup$For a given $f:\mathbb{R}^n\to\mathbb{R}$, the level set $L_c(f)$ is defined, like you said, as $L_c(f)=\{(x_1, \dots, x_n):f(x_1, \dots, x_n)=c\}$. Intuitively, it means that we can look at the graph of $f$ and the plane $p(x_1, \dots, x_n)=c$ and see where they intersect. This can be visualised in the case of $n=3$: for example, consider $f(x, y)=x^2+y^2$, whose graph is a paraboloid. The level sets $L_c(f)$ for $c>0$ describe circles of radius $\sqrt{c}$.
In your example, the level set $L_4(f)$ for $f(x, y)=x^2+2y^2+z^2$, describes, by definition, the set of points which satisfies $x^2+2y^2+z^2=4$. This set is indeed an ellipsoid, since it takes the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ for $a=2, b=\sqrt{2}, c=2$ (verify!)
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