prove S is a surface

[logistic_guy]

here is the question

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

Definition

definition of differentiability on a regular surface

all. I am studying the book "Differential Geometry of Curves and Surfaces" written by do Carmo, and there is one thing that confuses me so much: In his book, a regular surface refers to a subset of $\

math.stackexchange.com

if i take gradient of the function and set it to zero i get $\displaystyle x = y = z = 0$. do this proof it is surface? if yes i don't see how. if no, how to proof?

 

[blamocur]

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 $S$. I.e., the gradient is non-zero on $S$. I believe this is sufficient for $S$ to be a smooth surface. Here is a sketch of a proof: for any given point $p$ on $S$ there exists a tangent plane (because the gradient is non-zero). For some small neighborhood $U\subset S$ of $p$ a projection of $U$ on the tangent plane will be a smooth map.
More rigorous proof would probably use the implicit function theorem.

 

[logistic_guy]

The point you found is the only point with zero gradient, but that point does not belong to $S$. I.e., the gradient is non-zero on $S$. I believe this is sufficient for $S$ to be a smooth surface. Here is a sketch of a proof: for any given point $p$ on $S$ there exists a tangent plane (because the gradient is non-zero). For some small neighborhood $U\subset S$ of $p$ a projection of $U$ 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