Proof of $k$ differentiable function
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Let $f: E subset mathbb{R}^n to mathbb{R}^m$ such that $f$ is $k-1$ differentiable around $a$. Prove that $f$ is $k$ differentiable at $a$ given that $$ f(a+h) -sum limits_{j=0}^{k} T_j (h,cdots,h) = o(|h|^k)$$ such that $T_j$ is a linear transformation for all $j=0, cdots ,k$.
real-analysis multivariable-calculus
edited Nov 16 at 16:47
Masacroso
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asked Nov 16 at 16:42
Ahmad
2,4501625
add a comment |
up vote
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down vote
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Let $f: E subset mathbb{R}^n to mathbb{R}^m$ such that $f$ is $k-1$ differentiable around $a$. Prove that $f$ is $k$ differentiable at $a$ given that $$ f(a+h) -sum limits_{j=0}^{k} T_j (h,cdots,h) = o(|h|^k)$$ such that $T_j$ is a linear transformation for all $j=0, cdots ,k$.
real-analysis multivariable-calculus
edited Nov 16 at 16:47
Masacroso
12.2k41746
asked Nov 16 at 16:42
Ahmad
2,4501625
you must add to your question what had you tried, otherwise it will be unlikely that you get an answer
– Masacroso
Nov 16 at 16:47
What do you mean by linear? Multilinear? (with what multiplcity?) What is the sense of evaluating it on $(h,ldots,h)$?
– Michał Miśkiewicz
Nov 16 at 19:49
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $f: E subset mathbb{R}^n to mathbb{R}^m$ such that $f$ is $k-1$ differentiable around $a$. Prove that $f$ is $k$ differentiable at $a$ given that $$ f(a+h) -sum limits_{j=0}^{k} T_j (h,cdots,h) = o(|h|^k)$$ such that $T_j$ is a linear transformation for all $j=0, cdots ,k$.
real-analysis multivariable-calculus
edited Nov 16 at 16:47
Masacroso
12.2k41746
asked Nov 16 at 16:42
Ahmad
2,4501625
Let $f: E subset mathbb{R}^n to mathbb{R}^m$ such that $f$ is $k-1$ differentiable around $a$. Prove that $f$ is $k$ differentiable at $a$ given that $$ f(a+h) -sum limits_{j=0}^{k} T_j (h,cdots,h) = o(|h|^k)$$ such that $T_j$ is a linear transformation for all $j=0, cdots ,k$.
real-analysis multivariable-calculus
real-analysis multivariable-calculus
edited Nov 16 at 16:47
Masacroso
12.2k41746
asked Nov 16 at 16:42
Ahmad
2,4501625
edited Nov 16 at 16:47
Masacroso
12.2k41746
asked Nov 16 at 16:42
Ahmad
2,4501625
edited Nov 16 at 16:47
Masacroso
12.2k41746
edited Nov 16 at 16:47
Masacroso
12.2k41746
edited Nov 16 at 16:47
Masacroso
12.2k41746
12.2k41746
asked Nov 16 at 16:42
Ahmad
2,4501625
asked Nov 16 at 16:42
Ahmad
2,4501625
asked Nov 16 at 16:42
Ahmad
2,4501625
2,4501625
you must add to your question what had you tried, otherwise it will be unlikely that you get an answer
– Masacroso
Nov 16 at 16:47
What do you mean by linear? Multilinear? (with what multiplcity?) What is the sense of evaluating it on $(h,ldots,h)$?
– Michał Miśkiewicz
Nov 16 at 19:49
add a comment |
you must add to your question what had you tried, otherwise it will be unlikely that you get an answer
– Masacroso
Nov 16 at 16:47
What do you mean by linear? Multilinear? (with what multiplcity?) What is the sense of evaluating it on $(h,ldots,h)$?
– Michał Miśkiewicz
Nov 16 at 19:49
you must add to your question what had you tried, otherwise it will be unlikely that you get an answer
– Masacroso
Nov 16 at 16:47
you must add to your question what had you tried, otherwise it will be unlikely that you get an answer
– Masacroso
Nov 16 at 16:47
What do you mean by linear? Multilinear? (with what multiplcity?) What is the sense of evaluating it on $(h,ldots,h)$?
– Michał Miśkiewicz
Nov 16 at 19:49
What do you mean by linear? Multilinear? (with what multiplcity?) What is the sense of evaluating it on $(h,ldots,h)$?
– Michał Miśkiewicz
Nov 16 at 19:49
add a comment |
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you must add to your question what had you tried, otherwise it will be unlikely that you get an answer
– Masacroso
Nov 16 at 16:47
What do you mean by linear? Multilinear? (with what multiplcity?) What is the sense of evaluating it on $(h,ldots,h)$?
– Michał Miśkiewicz
Nov 16 at 19:49