Fdtxjr

Sum letters are not two different

Find 108 by using 3,4,6

Drawing without replacement: why is the order of draw irrelevant?

Why should I vote and accept answers?

Why do early math courses focus on the cross sections of a cone and not on other 3D objects?

A term for a woman complaining about things/begging in a cute/childish way

How would a mousetrap for use in space work?

How much damage would a cupful of neutron star matter do to the Earth?

Why do we bend a book to keep it straight?

Most bit efficient text communication method?

What order were files/directories outputted in dir?

Converted a Scalar function to a TVF function for parallel execution-Still running in Serial mode

Why is Nikon 1.4g better when Nikon 1.8g is sharper?

What are the diatonic extended chords of C major?

What is the appropriate index architecture when forced to implement IsDeleted (soft deletes)?

Performance gap between vector<bool> and array

What was the first language to use conditional keywords?

Is CEO the "profession" with the most psychopaths?

What is the meaning of 'breadth' in breadth first search?

Why does it sometimes sound good to play a grace note as a lead in to a note in a melody?

How do I find out the mythology and history of my Fortress?

What is the difference between globalisation and imperialism?

Can the Great Weapon Master feat's damage bonus and accuracy penalty apply to attacks from the Spiritual Weapon spell?

Should I use a zero-interest credit card for a large one-time purchase?

Jacobian of the inverse function

0

$begingroup$

Let $F=$$y_i(x_1,x_2,..,x_n)$$_i=1^n$:$R^nrightarrow $$R^n$ be a map with continous nonvanishing partial derivatives then its jacobian at a point $p$ is defined by the matrix $M_i,j(F)=partial y_i/partial x_jlvert_p $. Now by the inverse function theorem $F$ has an inverse $F^-1=$$x_i(y_1,y_2,..,y_n)$ in a neigbourhood of $F(p)$ and so its jacobian must be defined by $M_i,j(F^-1)=partial x_i/partial y_jlvert_F(p) $. However from the univariate version of the inverse function theorem $partial x_i/partial y_jlvert_F(p)=(partial y_i/partial x_jlvert_p)^-1$

CONCLUSION:

Therefore one can conclude that jacobian of the inverse function $F^-1$ at the point $F(p)$ is the same as the matrix of the jacobian of $F$ at $p$ but with its terms reciprocated.

By the multivariable case of inverse function theorem my conclusion is wrong infact it follows from the chain rule that $M_i,j(F^-1)=M_i,j(F)^-1$.

Therefore my conclusion is wrong however I dont understand what is wrong behind my reasoning.

Thanks in advance

multivariable-calculus

share|cite|improve this question

edited Mar 27 at 18:37

TheGeometer

asked Mar 27 at 18:00

TheGeometerTheGeometer

942619

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  "with its terms reciprocated" Utterly false. The Jacobian matrix of $F^-1$ is the inverse of the Jacobian matrix of $F$.
  $endgroup$
  – Martín-Blas Pérez Pinilla
  Mar 27 at 18:18
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Try a simple example such as $F:(x,y)mapsto(x+y,y)$ to see how this fails.
  $endgroup$
  – amd
  Mar 27 at 18:22
  
  

  
  
  
  

add a comment | 

0

$begingroup$

Let $F=$$y_i(x_1,x_2,..,x_n)$$_i=1^n$:$R^nrightarrow $$R^n$ be a map with continous nonvanishing partial derivatives then its jacobian at a point $p$ is defined by the matrix $M_i,j(F)=partial y_i/partial x_jlvert_p $. Now by the inverse function theorem $F$ has an inverse $F^-1=$$x_i(y_1,y_2,..,y_n)$ in a neigbourhood of $F(p)$ and so its jacobian must be defined by $M_i,j(F^-1)=partial x_i/partial y_jlvert_F(p) $. However from the univariate version of the inverse function theorem $partial x_i/partial y_jlvert_F(p)=(partial y_i/partial x_jlvert_p)^-1$

CONCLUSION:

Therefore one can conclude that jacobian of the inverse function $F^-1$ at the point $F(p)$ is the same as the matrix of the jacobian of $F$ at $p$ but with its terms reciprocated.

By the multivariable case of inverse function theorem my conclusion is wrong infact it follows from the chain rule that $M_i,j(F^-1)=M_i,j(F)^-1$.

Therefore my conclusion is wrong however I dont understand what is wrong behind my reasoning.

Thanks in advance

multivariable-calculus

share|cite|improve this question

edited Mar 27 at 18:37

TheGeometer

asked Mar 27 at 18:00

TheGeometerTheGeometer

942619

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  "with its terms reciprocated" Utterly false. The Jacobian matrix of $F^-1$ is the inverse of the Jacobian matrix of $F$.
  $endgroup$
  – Martín-Blas Pérez Pinilla
  Mar 27 at 18:18
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Try a simple example such as $F:(x,y)mapsto(x+y,y)$ to see how this fails.
  $endgroup$
  – amd
  Mar 27 at 18:22
  
  

  
  
  
  

add a comment | 

$begingroup$

Let $F=$$y_i(x_1,x_2,..,x_n)$$_i=1^n$:$R^nrightarrow $$R^n$ be a map with continous nonvanishing partial derivatives then its jacobian at a point $p$ is defined by the matrix $M_i,j(F)=partial y_i/partial x_jlvert_p $. Now by the inverse function theorem $F$ has an inverse $F^-1=$$x_i(y_1,y_2,..,y_n)$ in a neigbourhood of $F(p)$ and so its jacobian must be defined by $M_i,j(F^-1)=partial x_i/partial y_jlvert_F(p) $. However from the univariate version of the inverse function theorem $partial x_i/partial y_jlvert_F(p)=(partial y_i/partial x_jlvert_p)^-1$

CONCLUSION:

Therefore one can conclude that jacobian of the inverse function $F^-1$ at the point $F(p)$ is the same as the matrix of the jacobian of $F$ at $p$ but with its terms reciprocated.

By the multivariable case of inverse function theorem my conclusion is wrong infact it follows from the chain rule that $M_i,j(F^-1)=M_i,j(F)^-1$.

Therefore my conclusion is wrong however I dont understand what is wrong behind my reasoning.

Thanks in advance

multivariable-calculus

share|cite|improve this question

edited Mar 27 at 18:37

TheGeometer

asked Mar 27 at 18:00

TheGeometerTheGeometer

942619

$endgroup$

share|cite|improve this question

edited Mar 27 at 18:37

TheGeometer

asked Mar 27 at 18:00

TheGeometerTheGeometer

942619

share|cite|improve this question

edited Mar 27 at 18:37

TheGeometer

asked Mar 27 at 18:00

TheGeometerTheGeometer

942619

asked Mar 27 at 18:00

TheGeometerTheGeometer

942619

asked Mar 27 at 18:00

TheGeometerTheGeometer

942619