About the Jacobian determinant and the bijection

In summary, the conversation discusses the equivalence between the conditions of a matrix being locally inversible and its associated linear map being bijective, both of which are equivalent to the determinant of the matrix not being equal to zero. This is a general theorem known as the "implicit function theorem" which allows for solving for one variable in terms of the others if the Jacobian is not zero at a given point.
  • #1
simpleeyelid
12
0
Hello!

I am having problems with the inverse function theorem.

In some books it says to be locally inversible: first C1, 2nd Jacobian determinant different from 0

And I saw some books say to be locally inversible, it suffices to change the 2NDto "F'(a) is bijective"..

How could these two be equivalent.

Thank you for your kindness in advance,

Sincerely
 
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  • #2
F'(a) is a linear map and its being bijective is (by linear algebra) equivalent to det F'(a) not being equal to zero.
 
  • #3
Pere Callahan said:
F'(a) is a linear map and its being bijective is (by linear algebra) equivalent to det F'(a) not being equal to zero.

Thanks and, could you tell me the name of this proposition?
 
  • #4
simpleeyelid said:
Thanks and, could you tell me the name of this proposition?

I'm not sure if this proposition has a special name. You may look up in wikipedia the equivalent conditions for a matrix to be invertible (which means that the associated linear map is an vector space isomorphism, i.e. bijective.)

http://en.wikipedia.org/wiki/Matrix_inversion" [Broken]
 
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  • #5
Pere Callahan said:
I'm not sure if this proposition has a special name. You may look up in wikipedia the equivalent conditions for a matrix to be invertible (which means that the associated linear map is an vector space isomorphism, i.e. bijective.)

http://en.wikipedia.org/wiki/Matrix_inversion" [Broken]

MERCI beaucoup~~ I will check it..
 
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  • #6
simpleeyelid said:
Thanks and, could you tell me the name of this proposition?
The general theorem is the "implicit function theorem" which basically says if the Jacobian of f(x,y,z) is not 0 at (x0, y0, z0) then we can solve for anyone of the variables as a function of the other two in some neighborhood of (x0, y0, z0).
 

1. What is the Jacobian determinant?

The Jacobian determinant is a mathematical concept that is used in multivariable calculus to study the relationship between two sets of variables. It represents the ratio of the change in one set of variables to the change in another set of variables.

2. How is the Jacobian determinant related to a bijection?

A bijection is a type of function that has a one-to-one correspondence between two sets, meaning that each element in one set is paired with only one element in the other set. The Jacobian determinant is used to determine if a function is a bijection by checking if it is invertible and has a non-zero determinant.

3. What is the importance of the Jacobian determinant in mathematics?

The Jacobian determinant is important in mathematics because it is used to solve many problems in physics, engineering, and economics that involve multiple variables. It is also a fundamental concept in the study of differential equations and vector calculus.

4. How is the Jacobian determinant calculated?

The Jacobian determinant is calculated using the partial derivatives of a function. For a function with n variables, the Jacobian determinant is calculated by taking the determinant of the n by n matrix of partial derivatives.

5. Can the Jacobian determinant be negative?

Yes, the Jacobian determinant can be negative. In fact, a negative Jacobian determinant indicates that the orientation of the transformation is reversed, meaning that the transformed shape will be "flipped" compared to the original shape. This is important in applications such as vector fields and fluid dynamics.

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