prove S is a surface

logistic_guy

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Apr 17, 2024
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here is the question

Prove that S:(x2+y2)2+3z2=1\displaystyle S: (x^2 + y^2)^2 + 3z^2 = 1 is a surface.

Definition

if i take gradient of the function and set it to zero i get x=y=z=0\displaystyle x = y = z = 0. do this proof it is surface? if yes i don't see how. if no, how to proof?
 
if i take gradient of the function and set it to zero i get x=y=z=0\displaystyle x = y = z = 0x=y=z=0. do this proof it is surface? if yes i don't see how. if no, how to proof?
The point you found is the only point with zero gradient, but that point does not belong to SS. I.e., the gradient is non-zero on SS. I believe this is sufficient for SS to be a smooth surface. Here is a sketch of a proof: for any given point pp on SS there exists a tangent plane (because the gradient is non-zero). For some small neighborhood USU\subset S of pp a projection of UU on the tangent plane will be a smooth map.
More rigorous proof would probably use the implicit function theorem.
 
The point you found is the only point with zero gradient, but that point does not belong to SS. I.e., the gradient is non-zero on SS. I believe this is sufficient for SS to be a smooth surface. Here is a sketch of a proof: for any given point pp on SS there exists a tangent plane (because the gradient is non-zero). For some small neighborhood USU\subset S of pp a projection of UU on the tangent plane will be a smooth map.
More rigorous proof would probably use the implicit function theorem.
thank blamocur. i think i'm understand now
 
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