“Critical point” - single-variable calculus v.s. differential geometry
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In differential geometry, given a smooth map $Phi: M rightarrow N$ between smooth manifolds $M$ and $N$, a point $p in M$ is said to be a critical point if $dPhi_p: T_pM rightarrow T_{Phi(p)}N$ is not surjective.
In single-variable calculus, one learns about differentiating a function and finding its "critical points" for local extrema or inflexions. Does the term "critical point" in single-variable calculus have the same meaning as the differential geometric definition? Or is the terminology just a coincidence?
If it is not the same, what is the intuition behind a "critical point" in the differential geometric definition? I don't really understand it.
For context, I am learning about Sard's Theorem.
calculus differential-geometry
asked Dec 2 '18 at 0:17
Frederic ChopinFrederic Chopin
321111
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add a comment |
$begingroup$
In differential geometry, given a smooth map $Phi: M rightarrow N$ between smooth manifolds $M$ and $N$, a point $p in M$ is said to be a critical point if $dPhi_p: T_pM rightarrow T_{Phi(p)}N$ is not surjective.
In single-variable calculus, one learns about differentiating a function and finding its "critical points" for local extrema or inflexions. Does the term "critical point" in single-variable calculus have the same meaning as the differential geometric definition? Or is the terminology just a coincidence?
If it is not the same, what is the intuition behind a "critical point" in the differential geometric definition? I don't really understand it.
For context, I am learning about Sard's Theorem.
calculus differential-geometry
asked Dec 2 '18 at 0:17
Frederic ChopinFrederic Chopin
321111
$endgroup$
add a comment |
$begingroup$
In differential geometry, given a smooth map $Phi: M rightarrow N$ between smooth manifolds $M$ and $N$, a point $p in M$ is said to be a critical point if $dPhi_p: T_pM rightarrow T_{Phi(p)}N$ is not surjective.
In single-variable calculus, one learns about differentiating a function and finding its "critical points" for local extrema or inflexions. Does the term "critical point" in single-variable calculus have the same meaning as the differential geometric definition? Or is the terminology just a coincidence?
If it is not the same, what is the intuition behind a "critical point" in the differential geometric definition? I don't really understand it.
For context, I am learning about Sard's Theorem.
calculus differential-geometry
asked Dec 2 '18 at 0:17
Frederic ChopinFrederic Chopin
321111
$endgroup$
In differential geometry, given a smooth map $Phi: M rightarrow N$ between smooth manifolds $M$ and $N$, a point $p in M$ is said to be a critical point if $dPhi_p: T_pM rightarrow T_{Phi(p)}N$ is not surjective.
In single-variable calculus, one learns about differentiating a function and finding its "critical points" for local extrema or inflexions. Does the term "critical point" in single-variable calculus have the same meaning as the differential geometric definition? Or is the terminology just a coincidence?
If it is not the same, what is the intuition behind a "critical point" in the differential geometric definition? I don't really understand it.
For context, I am learning about Sard's Theorem.
calculus differential-geometry
calculus differential-geometry
asked Dec 2 '18 at 0:17
Frederic ChopinFrederic Chopin
321111
asked Dec 2 '18 at 0:17
Frederic ChopinFrederic Chopin
321111
asked Dec 2 '18 at 0:17
Frederic ChopinFrederic Chopin
321111
asked Dec 2 '18 at 0:17
Frederic ChopinFrederic Chopin
321111
asked Dec 2 '18 at 0:17
Frederic ChopinFrederic Chopin
321111
321111
add a comment |
add a comment |
1 Answer
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In single variable, surjetivity of $df_x$ is equivalent to saying that $df_xneq 0$ therefore both definition are equivalent. $M=N=mathbb{R}$ and $df_x:T_xmathbb{R}=mathbb{R}rightarrow T_{f(x)}mathbb{R}=mathbb{R}$ defined by $df_x(u)=f'(x)u$ is a linear function which is surjective if and only if $f'(x)neq 0.$
answered Dec 2 '18 at 0:25
Tsemo AristideTsemo Aristide
57.2k11444
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1 Answer
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1 Answer
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oldest
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active
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$begingroup$
In single variable, surjetivity of $df_x$ is equivalent to saying that $df_xneq 0$ therefore both definition are equivalent. $M=N=mathbb{R}$ and $df_x:T_xmathbb{R}=mathbb{R}rightarrow T_{f(x)}mathbb{R}=mathbb{R}$ defined by $df_x(u)=f'(x)u$ is a linear function which is surjective if and only if $f'(x)neq 0.$
answered Dec 2 '18 at 0:25
Tsemo AristideTsemo Aristide
57.2k11444
$endgroup$
add a comment |
$begingroup$
In single variable, surjetivity of $df_x$ is equivalent to saying that $df_xneq 0$ therefore both definition are equivalent. $M=N=mathbb{R}$ and $df_x:T_xmathbb{R}=mathbb{R}rightarrow T_{f(x)}mathbb{R}=mathbb{R}$ defined by $df_x(u)=f'(x)u$ is a linear function which is surjective if and only if $f'(x)neq 0.$
answered Dec 2 '18 at 0:25
Tsemo AristideTsemo Aristide
57.2k11444
$endgroup$
add a comment |
$begingroup$
In single variable, surjetivity of $df_x$ is equivalent to saying that $df_xneq 0$ therefore both definition are equivalent. $M=N=mathbb{R}$ and $df_x:T_xmathbb{R}=mathbb{R}rightarrow T_{f(x)}mathbb{R}=mathbb{R}$ defined by $df_x(u)=f'(x)u$ is a linear function which is surjective if and only if $f'(x)neq 0.$
answered Dec 2 '18 at 0:25
Tsemo AristideTsemo Aristide
57.2k11444
$endgroup$
In single variable, surjetivity of $df_x$ is equivalent to saying that $df_xneq 0$ therefore both definition are equivalent. $M=N=mathbb{R}$ and $df_x:T_xmathbb{R}=mathbb{R}rightarrow T_{f(x)}mathbb{R}=mathbb{R}$ defined by $df_x(u)=f'(x)u$ is a linear function which is surjective if and only if $f'(x)neq 0.$
answered Dec 2 '18 at 0:25
Tsemo AristideTsemo Aristide
57.2k11444
answered Dec 2 '18 at 0:25
Tsemo AristideTsemo Aristide
57.2k11444
answered Dec 2 '18 at 0:25
Tsemo AristideTsemo Aristide
57.2k11444
answered Dec 2 '18 at 0:25
Tsemo AristideTsemo Aristide
57.2k11444
57.2k11444
add a comment |
add a comment |
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