Is restriction of $C^infty$ map to a subspace differentiable
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Suppose $U subset mathbb R^n$ is an open set. Let $f in C^infty(U)$. Let $S$ be a vector subspace of $mathbb R^n$ equipped with subspace topology. Let $g$ be the restriction of $f$ on $S cap U$, i.e., $g = f|_S cap U$. $g$ is certainly continuous. I am wondering whether it makes sense to talk about differentiability of $g$ if considered as a function over $S cap U$ and the ambient space as $S$. In this way, $S cap U$ is open in $S$ and it seems $g$ would as well be $C^infty$. But this does not feel completely right.
real-analysis
asked Sep 9 at 7:27
user9527
1,0971525
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up vote
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Suppose $U subset mathbb R^n$ is an open set. Let $f in C^infty(U)$. Let $S$ be a vector subspace of $mathbb R^n$ equipped with subspace topology. Let $g$ be the restriction of $f$ on $S cap U$, i.e., $g = f|_S cap U$. $g$ is certainly continuous. I am wondering whether it makes sense to talk about differentiability of $g$ if considered as a function over $S cap U$ and the ambient space as $S$. In this way, $S cap U$ is open in $S$ and it seems $g$ would as well be $C^infty$. But this does not feel completely right.
real-analysis
asked Sep 9 at 7:27
user9527
1,0971525
why does this not feel right?
– Thomas
Sep 9 at 7:28
(suggestion: start with differentiable instead of infinitely often continuously differentiable and write down the definition. Then investigate continuity and start an induction)
– Thomas
Sep 9 at 7:31
(Alternative suggestion: recall the relation between continuous differentiabllity and partial derivatives. Then choose a coordinate system of the ambient space which is adapted to the subspace $S$, i.e. such that the first $k$, say, coordinates span $S$).
– Thomas
Sep 9 at 7:34
You can define the directional derivative of $f:Ssubseteq Bbb R^nto Bbb R^k$ along $S$ at $pin S$ as the linear map $partial_Sf(p):Sto Bbb R^k$ such that $$lim_hto 0, h, p+hin S fracf(p+h)-f(p)-partial_Sf(p)(h)=0, $$if it exists. For $S = Bbb R v$ with $|v|=1$ we recover $partial f/partial v$.
– Ivo Terek
Sep 9 at 7:47
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose $U subset mathbb R^n$ is an open set. Let $f in C^infty(U)$. Let $S$ be a vector subspace of $mathbb R^n$ equipped with subspace topology. Let $g$ be the restriction of $f$ on $S cap U$, i.e., $g = f|_S cap U$. $g$ is certainly continuous. I am wondering whether it makes sense to talk about differentiability of $g$ if considered as a function over $S cap U$ and the ambient space as $S$. In this way, $S cap U$ is open in $S$ and it seems $g$ would as well be $C^infty$. But this does not feel completely right.
real-analysis
asked Sep 9 at 7:27
user9527
1,0971525
Suppose $U subset mathbb R^n$ is an open set. Let $f in C^infty(U)$. Let $S$ be a vector subspace of $mathbb R^n$ equipped with subspace topology. Let $g$ be the restriction of $f$ on $S cap U$, i.e., $g = f|_S cap U$. $g$ is certainly continuous. I am wondering whether it makes sense to talk about differentiability of $g$ if considered as a function over $S cap U$ and the ambient space as $S$. In this way, $S cap U$ is open in $S$ and it seems $g$ would as well be $C^infty$. But this does not feel completely right.
real-analysis
real-analysis
asked Sep 9 at 7:27
user9527
1,0971525
asked Sep 9 at 7:27
user9527
1,0971525
asked Sep 9 at 7:27
user9527
1,0971525
asked Sep 9 at 7:27
user9527
1,0971525
asked Sep 9 at 7:27
user9527
1,0971525
1,0971525
why does this not feel right?
– Thomas
Sep 9 at 7:28
(suggestion: start with differentiable instead of infinitely often continuously differentiable and write down the definition. Then investigate continuity and start an induction)
– Thomas
Sep 9 at 7:31
(Alternative suggestion: recall the relation between continuous differentiabllity and partial derivatives. Then choose a coordinate system of the ambient space which is adapted to the subspace $S$, i.e. such that the first $k$, say, coordinates span $S$).
– Thomas
Sep 9 at 7:34
You can define the directional derivative of $f:Ssubseteq Bbb R^nto Bbb R^k$ along $S$ at $pin S$ as the linear map $partial_Sf(p):Sto Bbb R^k$ such that $$lim_hto 0, h, p+hin S fracf(p+h)-f(p)-partial_Sf(p)(h)=0, $$if it exists. For $S = Bbb R v$ with $|v|=1$ we recover $partial f/partial v$.
– Ivo Terek
Sep 9 at 7:47
add a comment |Â
why does this not feel right?
– Thomas
Sep 9 at 7:28
(suggestion: start with differentiable instead of infinitely often continuously differentiable and write down the definition. Then investigate continuity and start an induction)
– Thomas
Sep 9 at 7:31
(Alternative suggestion: recall the relation between continuous differentiabllity and partial derivatives. Then choose a coordinate system of the ambient space which is adapted to the subspace $S$, i.e. such that the first $k$, say, coordinates span $S$).
– Thomas
Sep 9 at 7:34
You can define the directional derivative of $f:Ssubseteq Bbb R^nto Bbb R^k$ along $S$ at $pin S$ as the linear map $partial_Sf(p):Sto Bbb R^k$ such that $$lim_hto 0, h, p+hin S fracf(p+h)-f(p)-partial_Sf(p)(h)=0, $$if it exists. For $S = Bbb R v$ with $|v|=1$ we recover $partial f/partial v$.
– Ivo Terek
Sep 9 at 7:47
why does this not feel right?
– Thomas
Sep 9 at 7:28
why does this not feel right?
– Thomas
Sep 9 at 7:28
(suggestion: start with differentiable instead of infinitely often continuously differentiable and write down the definition. Then investigate continuity and start an induction)
– Thomas
Sep 9 at 7:31
(suggestion: start with differentiable instead of infinitely often continuously differentiable and write down the definition. Then investigate continuity and start an induction)
– Thomas
Sep 9 at 7:31
(Alternative suggestion: recall the relation between continuous differentiabllity and partial derivatives. Then choose a coordinate system of the ambient space which is adapted to the subspace $S$, i.e. such that the first $k$, say, coordinates span $S$).
– Thomas
Sep 9 at 7:34
(Alternative suggestion: recall the relation between continuous differentiabllity and partial derivatives. Then choose a coordinate system of the ambient space which is adapted to the subspace $S$, i.e. such that the first $k$, say, coordinates span $S$).
– Thomas
Sep 9 at 7:34
You can define the directional derivative of $f:Ssubseteq Bbb R^nto Bbb R^k$ along $S$ at $pin S$ as the linear map $partial_Sf(p):Sto Bbb R^k$ such that $$lim_hto 0, h, p+hin S fracf(p+h)-f(p)-partial_Sf(p)(h)=0, $$if it exists. For $S = Bbb R v$ with $|v|=1$ we recover $partial f/partial v$.
– Ivo Terek
Sep 9 at 7:47
You can define the directional derivative of $f:Ssubseteq Bbb R^nto Bbb R^k$ along $S$ at $pin S$ as the linear map $partial_Sf(p):Sto Bbb R^k$ such that $$lim_hto 0, h, p+hin S fracf(p+h)-f(p)-partial_Sf(p)(h)=0, $$if it exists. For $S = Bbb R v$ with $|v|=1$ we recover $partial f/partial v$.
– Ivo Terek
Sep 9 at 7:47
add a comment |Â
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why does this not feel right?
– Thomas
Sep 9 at 7:28
(suggestion: start with differentiable instead of infinitely often continuously differentiable and write down the definition. Then investigate continuity and start an induction)
– Thomas
Sep 9 at 7:31
(Alternative suggestion: recall the relation between continuous differentiabllity and partial derivatives. Then choose a coordinate system of the ambient space which is adapted to the subspace $S$, i.e. such that the first $k$, say, coordinates span $S$).
– Thomas
Sep 9 at 7:34
You can define the directional derivative of $f:Ssubseteq Bbb R^nto Bbb R^k$ along $S$ at $pin S$ as the linear map $partial_Sf(p):Sto Bbb R^k$ such that $$lim_hto 0, h, p+hin S fracf(p+h)-f(p)-partial_Sf(p)(h)=0, $$if it exists. For $S = Bbb R v$ with $|v|=1$ we recover $partial f/partial v$.
– Ivo Terek
Sep 9 at 7:47