some question about a problem in gtm218:the introduction to smooth manifolds
$begingroup$
let $F:R^2to R$ be defined by $F(x,y)=x^3+xy+y^3$.which level sets of $F$ are embedded submanifolds of $R^2$ ?
For each level set,prove either that it is or that it is not an embedded manifold.
My idea is showed as following:
Firstly,we assume $x^3+xy+y^3=c$ amd we can easily calculate that $DF=(3x^2+y,x+3y^2)$, so only when $y=0$ or $y=frac{-1}{3}$,$DF$ is not surjective.
so we just need to check whether $F^{-1}(c)$ is embedded manifold when $c=0$ and $c=frac{1}{27}$ .
And i can see that when $c=0$ $F^{-1}(c)$ is not embedded manifold,however when $c=frac{1}{27}$ $F^{-1}(c)$ is embedded manifold from their graph.
I want to know whether $x^3+xy+y^3=frac{1}{27}$ have some specialty?
differential-geometry smooth-manifolds
asked Dec 28 '18 at 15:36
calibertytzcalibertytz
1
$endgroup$
add a comment |
$begingroup$
let $F:R^2to R$ be defined by $F(x,y)=x^3+xy+y^3$.which level sets of $F$ are embedded submanifolds of $R^2$ ?
For each level set,prove either that it is or that it is not an embedded manifold.
My idea is showed as following:
Firstly,we assume $x^3+xy+y^3=c$ amd we can easily calculate that $DF=(3x^2+y,x+3y^2)$, so only when $y=0$ or $y=frac{-1}{3}$,$DF$ is not surjective.
so we just need to check whether $F^{-1}(c)$ is embedded manifold when $c=0$ and $c=frac{1}{27}$ .
And i can see that when $c=0$ $F^{-1}(c)$ is not embedded manifold,however when $c=frac{1}{27}$ $F^{-1}(c)$ is embedded manifold from their graph.
I want to know whether $x^3+xy+y^3=frac{1}{27}$ have some specialty?
differential-geometry smooth-manifolds
asked Dec 28 '18 at 15:36
calibertytzcalibertytz
1
$endgroup$
$begingroup$
Just to clarify, $DF$ is surjective if and only if nonzero, so if and only if $(x,y)neq (0,0),(-1/3,-1/3)$. Therefore, you have to check when one of those points satisfies $x^3+xy+y^3=c$, this happens when $c=0,1/27$, as you stated. I would also like to include a formal proof of the fact that $x^3+xy+y^3=0$ is not a manifold, as a sketch is not a proof.
$endgroup$
– C. Falcon
Dec 28 '18 at 15:56
add a comment |
$begingroup$
let $F:R^2to R$ be defined by $F(x,y)=x^3+xy+y^3$.which level sets of $F$ are embedded submanifolds of $R^2$ ?
For each level set,prove either that it is or that it is not an embedded manifold.
My idea is showed as following:
Firstly,we assume $x^3+xy+y^3=c$ amd we can easily calculate that $DF=(3x^2+y,x+3y^2)$, so only when $y=0$ or $y=frac{-1}{3}$,$DF$ is not surjective.
so we just need to check whether $F^{-1}(c)$ is embedded manifold when $c=0$ and $c=frac{1}{27}$ .
And i can see that when $c=0$ $F^{-1}(c)$ is not embedded manifold,however when $c=frac{1}{27}$ $F^{-1}(c)$ is embedded manifold from their graph.
I want to know whether $x^3+xy+y^3=frac{1}{27}$ have some specialty?
differential-geometry smooth-manifolds
asked Dec 28 '18 at 15:36
calibertytzcalibertytz
1
$endgroup$
let $F:R^2to R$ be defined by $F(x,y)=x^3+xy+y^3$.which level sets of $F$ are embedded submanifolds of $R^2$ ?
For each level set,prove either that it is or that it is not an embedded manifold.
My idea is showed as following:
Firstly,we assume $x^3+xy+y^3=c$ amd we can easily calculate that $DF=(3x^2+y,x+3y^2)$, so only when $y=0$ or $y=frac{-1}{3}$,$DF$ is not surjective.
so we just need to check whether $F^{-1}(c)$ is embedded manifold when $c=0$ and $c=frac{1}{27}$ .
And i can see that when $c=0$ $F^{-1}(c)$ is not embedded manifold,however when $c=frac{1}{27}$ $F^{-1}(c)$ is embedded manifold from their graph.
I want to know whether $x^3+xy+y^3=frac{1}{27}$ have some specialty?
differential-geometry smooth-manifolds
differential-geometry smooth-manifolds
asked Dec 28 '18 at 15:36
calibertytzcalibertytz
1
asked Dec 28 '18 at 15:36
calibertytzcalibertytz
1
asked Dec 28 '18 at 15:36
calibertytzcalibertytz
1
asked Dec 28 '18 at 15:36
calibertytzcalibertytz
1
asked Dec 28 '18 at 15:36
calibertytzcalibertytz
1
1
$begingroup$
Just to clarify, $DF$ is surjective if and only if nonzero, so if and only if $(x,y)neq (0,0),(-1/3,-1/3)$. Therefore, you have to check when one of those points satisfies $x^3+xy+y^3=c$, this happens when $c=0,1/27$, as you stated. I would also like to include a formal proof of the fact that $x^3+xy+y^3=0$ is not a manifold, as a sketch is not a proof.
$endgroup$
– C. Falcon
Dec 28 '18 at 15:56
add a comment |
$begingroup$
Just to clarify, $DF$ is surjective if and only if nonzero, so if and only if $(x,y)neq (0,0),(-1/3,-1/3)$. Therefore, you have to check when one of those points satisfies $x^3+xy+y^3=c$, this happens when $c=0,1/27$, as you stated. I would also like to include a formal proof of the fact that $x^3+xy+y^3=0$ is not a manifold, as a sketch is not a proof.
$endgroup$
– C. Falcon
Dec 28 '18 at 15:56
$begingroup$
Just to clarify, $DF$ is surjective if and only if nonzero, so if and only if $(x,y)neq (0,0),(-1/3,-1/3)$. Therefore, you have to check when one of those points satisfies $x^3+xy+y^3=c$, this happens when $c=0,1/27$, as you stated. I would also like to include a formal proof of the fact that $x^3+xy+y^3=0$ is not a manifold, as a sketch is not a proof.
$endgroup$
– C. Falcon
Dec 28 '18 at 15:56
$begingroup$
Just to clarify, $DF$ is surjective if and only if nonzero, so if and only if $(x,y)neq (0,0),(-1/3,-1/3)$. Therefore, you have to check when one of those points satisfies $x^3+xy+y^3=c$, this happens when $c=0,1/27$, as you stated. I would also like to include a formal proof of the fact that $x^3+xy+y^3=0$ is not a manifold, as a sketch is not a proof.
$endgroup$
– C. Falcon
Dec 28 '18 at 15:56
add a comment |
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$begingroup$
Just to clarify, $DF$ is surjective if and only if nonzero, so if and only if $(x,y)neq (0,0),(-1/3,-1/3)$. Therefore, you have to check when one of those points satisfies $x^3+xy+y^3=c$, this happens when $c=0,1/27$, as you stated. I would also like to include a formal proof of the fact that $x^3+xy+y^3=0$ is not a manifold, as a sketch is not a proof.
$endgroup$
– C. Falcon
Dec 28 '18 at 15:56