Fdtxjr

How do I create uniquely male characters?

Pristine Bit Checking

Why was the "bread communication" in the arena of Catching Fire left out in the movie?

Manga about a female worker who got dragged into another world together with this high school girl and she was just told she's not needed anymore

LWC and complex parameters

New order #4: World

Is there a name of the flying bionic bird?

Typesetting a double Over Dot on top of a symbol

What is the meaning of "of trouble" in the following sentence?

What do the Banks children have against barley water?

Extreme, but not acceptable situation and I can't start the work tomorrow morning

Can the Produce Flame cantrip be used to grapple, or as an unarmed strike, in the right circumstances?

How to make payment on the internet without leaving a money trail?

Is it legal to have the "// (c) 2019 John Smith" header in all files when there are hundreds of contributors?

Are white and non-white police officers equally likely to kill black suspects?

Why is the design of haulage companies so “special”?

What to wear for invited talk in Canada

What is GPS' 19 year rollover and does it present a cybersecurity issue?

How is it possible for user's password to be changed after storage was encrypted? (on OS X, Android)

How to deal with fear of taking dependencies

Crop image to path created in TikZ?

How to move the player while also allowing forces to affect it

Can I legally use front facing blue light in the UK?

How can I fix this gap between bookcases I made?

Lipschitz continuous and Jacobian matrix

Recovering vector-valued function from its Jacobian MatrixJacobian matrix of an inverse diffeomorphismWhen is a given matrix-valued function the Jacobian of something?Determinant of the Jacobian of a short mapJacobian Matrix - unknown functionNon-vanishing Jacobian determinant is bounded below?Is the Spectral Norm of the Jacobian of an M-Lipschitz Function bounded by M?Jacobian with right inverseJacobian matrix of $|cdot|_2$Is the derivative of a Lipschitz continuous gradient function is a continuous vector function?

2

1

$begingroup$

Consider a function $f:mathbbR^nlongrightarrowmathbbR^m$ with partial derivatives everywhere so that the Jacobian matrix is well-defined. Let $L>0$ be a real number. Is it true that:
$$|f(x)-f(y)|leq L|x-y|,forall x,y Longleftrightarrow |J_f(x)|_2leq L,forall x$$
where $|cdot|$ denotes the euclidean vector norm and $|cdot|_2$ the spectral matrix norm.

multivariable-calculus lipschitz-functions jacobian

share|cite|improve this question

asked Mar 22 at 16:03

user404332user404332

748

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Consider the inequality $|f(x) - f(y)| leq L|x-y|$. We can rewrite it as $fracx-y leq L$. Is there any equation/formula from calculus you recall that you can relate to the left-hand side of this inequality? In particular, can you think of a formula that involves a derivative?
  $endgroup$
  – kkc
  Mar 22 at 19:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The result is indeed trivial for $n=m=1$ by using the definition of derivative for one implication and by using the mean value theorem for the other one. The question is about extending this result to multidimension.
  $endgroup$
  – user404332
  Mar 25 at 8:33
  

  
  
  
  

add a comment | 

2

1

$begingroup$

Consider a function $f:mathbbR^nlongrightarrowmathbbR^m$ with partial derivatives everywhere so that the Jacobian matrix is well-defined. Let $L>0$ be a real number. Is it true that:
$$|f(x)-f(y)|leq L|x-y|,forall x,y Longleftrightarrow |J_f(x)|_2leq L,forall x$$
where $|cdot|$ denotes the euclidean vector norm and $|cdot|_2$ the spectral matrix norm.

multivariable-calculus lipschitz-functions jacobian

share|cite|improve this question

asked Mar 22 at 16:03

user404332user404332

748

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Consider the inequality $|f(x) - f(y)| leq L|x-y|$. We can rewrite it as $fracx-y leq L$. Is there any equation/formula from calculus you recall that you can relate to the left-hand side of this inequality? In particular, can you think of a formula that involves a derivative?
  $endgroup$
  – kkc
  Mar 22 at 19:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The result is indeed trivial for $n=m=1$ by using the definition of derivative for one implication and by using the mean value theorem for the other one. The question is about extending this result to multidimension.
  $endgroup$
  – user404332
  Mar 25 at 8:33
  

  
  
  
  

add a comment | 

$begingroup$

Consider a function $f:mathbbR^nlongrightarrowmathbbR^m$ with partial derivatives everywhere so that the Jacobian matrix is well-defined. Let $L>0$ be a real number. Is it true that:
$$|f(x)-f(y)|leq L|x-y|,forall x,y Longleftrightarrow |J_f(x)|_2leq L,forall x$$
where $|cdot|$ denotes the euclidean vector norm and $|cdot|_2$ the spectral matrix norm.

multivariable-calculus lipschitz-functions jacobian

share|cite|improve this question

asked Mar 22 at 16:03

user404332user404332

748

$endgroup$

share|cite|improve this question

asked Mar 22 at 16:03

user404332user404332

748

share|cite|improve this question

asked Mar 22 at 16:03

user404332user404332

748

asked Mar 22 at 16:03

user404332user404332

748

asked Mar 22 at 16:03

user404332user404332

748