Find the tangent space of Ellipsoid $M = {(x,y,z)|frac{x^2}{a^2}+frac{y^2}{b^2}+frac{z^2}{c^2}=1}$
Find the tangent space of
$$M = {(x,y,z)|frac{x^2}{a^2}+frac{y^2}{b^2}+frac{z^2}{c^2}=1}$$
So I know the formula of tangent space for a manifold represnted by $F$ such that $F=0$: it is $ker (DF)$.
So I'll define - $F = frac{x^2}{a^2}+frac{y^2}{b^2}+frac{z^2}{c^2} - 1$ and of course $F=0$.
By definition, $DF = (frac{2x}{a^2},frac{2y}{b^2},frac{2z}{c^2})$ and we just need to find $ker (DF)$.
Besides $x=y=z=0$, the solutions are $(x,y,(-frac{x^2}{a^2} -frac{y^2}{b^2} )c^2)$, $(x,(-frac{x^2}{a^2} -frac{z^2}{c^2})b^2,z)$ and $(-frac{y^2}{b^2} -frac{z^2}{c^2} )a^2,y,z)$.
But what is the final tangent space that is spanned by these solutions?
calculus multivariable-calculus differential-geometry manifolds tangent-spaces
edited Dec 8 at 17:22
Nosrati
26.4k62353
asked Dec 5 at 22:23
ChikChak
810418
add a comment |
Find the tangent space of
$$M = {(x,y,z)|frac{x^2}{a^2}+frac{y^2}{b^2}+frac{z^2}{c^2}=1}$$
So I know the formula of tangent space for a manifold represnted by $F$ such that $F=0$: it is $ker (DF)$.
So I'll define - $F = frac{x^2}{a^2}+frac{y^2}{b^2}+frac{z^2}{c^2} - 1$ and of course $F=0$.
By definition, $DF = (frac{2x}{a^2},frac{2y}{b^2},frac{2z}{c^2})$ and we just need to find $ker (DF)$.
Besides $x=y=z=0$, the solutions are $(x,y,(-frac{x^2}{a^2} -frac{y^2}{b^2} )c^2)$, $(x,(-frac{x^2}{a^2} -frac{z^2}{c^2})b^2,z)$ and $(-frac{y^2}{b^2} -frac{z^2}{c^2} )a^2,y,z)$.
But what is the final tangent space that is spanned by these solutions?
calculus multivariable-calculus differential-geometry manifolds tangent-spaces
edited Dec 8 at 17:22
Nosrati
26.4k62353
asked Dec 5 at 22:23
ChikChak
810418
add a comment |
Find the tangent space of
$$M = {(x,y,z)|frac{x^2}{a^2}+frac{y^2}{b^2}+frac{z^2}{c^2}=1}$$
So I know the formula of tangent space for a manifold represnted by $F$ such that $F=0$: it is $ker (DF)$.
So I'll define - $F = frac{x^2}{a^2}+frac{y^2}{b^2}+frac{z^2}{c^2} - 1$ and of course $F=0$.
By definition, $DF = (frac{2x}{a^2},frac{2y}{b^2},frac{2z}{c^2})$ and we just need to find $ker (DF)$.
Besides $x=y=z=0$, the solutions are $(x,y,(-frac{x^2}{a^2} -frac{y^2}{b^2} )c^2)$, $(x,(-frac{x^2}{a^2} -frac{z^2}{c^2})b^2,z)$ and $(-frac{y^2}{b^2} -frac{z^2}{c^2} )a^2,y,z)$.
But what is the final tangent space that is spanned by these solutions?
calculus multivariable-calculus differential-geometry manifolds tangent-spaces
edited Dec 8 at 17:22
Nosrati
26.4k62353
asked Dec 5 at 22:23
ChikChak
810418
Find the tangent space of
$$M = {(x,y,z)|frac{x^2}{a^2}+frac{y^2}{b^2}+frac{z^2}{c^2}=1}$$
So I know the formula of tangent space for a manifold represnted by $F$ such that $F=0$: it is $ker (DF)$.
So I'll define - $F = frac{x^2}{a^2}+frac{y^2}{b^2}+frac{z^2}{c^2} - 1$ and of course $F=0$.
By definition, $DF = (frac{2x}{a^2},frac{2y}{b^2},frac{2z}{c^2})$ and we just need to find $ker (DF)$.
Besides $x=y=z=0$, the solutions are $(x,y,(-frac{x^2}{a^2} -frac{y^2}{b^2} )c^2)$, $(x,(-frac{x^2}{a^2} -frac{z^2}{c^2})b^2,z)$ and $(-frac{y^2}{b^2} -frac{z^2}{c^2} )a^2,y,z)$.
But what is the final tangent space that is spanned by these solutions?
calculus multivariable-calculus differential-geometry manifolds tangent-spaces
calculus multivariable-calculus differential-geometry manifolds tangent-spaces
edited Dec 8 at 17:22
Nosrati
26.4k62353
asked Dec 5 at 22:23
ChikChak
810418
edited Dec 8 at 17:22
Nosrati
26.4k62353
asked Dec 5 at 22:23
ChikChak
810418
edited Dec 8 at 17:22
Nosrati
26.4k62353
edited Dec 8 at 17:22
Nosrati
26.4k62353
edited Dec 8 at 17:22
Nosrati
26.4k62353
26.4k62353
asked Dec 5 at 22:23
ChikChak
810418
asked Dec 5 at 22:23
ChikChak
810418
asked Dec 5 at 22:23
ChikChak
810418
810418
add a comment |
add a comment |
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