Critical points of maps f:R^n->R^m , m>1

In summary, the conversation discusses the critical points of a map from R^n to R^m, where m is greater than 1. The speaker is having trouble determining the critical points and asks for hints. Another speaker clarifies that the Jacobian matrix being invertible is a sufficient condition for differentiability, but not necessary. The conversation also touches on the concept of differentiable functions and diffeomorphisms.
  • #1
Bacle
662
1
Critical points of maps f:R^n-->R^m , m>1

Hi, Everyone:

I am having trouble figuring out what the critical points would be for any such map

f:R^n -->R^m , m>1.

Problem is that I know sufficient conditions for differentiability (partials exist and

are continuous.) but not necessary ones. So our differential is given by an

mxn matrix. In the case of n=m, we just want the Jacobian matrix to be invertible,

but I am not clear on how to deal with this issue when n=/m.

Any Hints, Please.?

Thanks.
 
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  • #2


In the case m=n, why do you say that the jacobian must be invertible for the function to be differentiable? Consider the null function from R^n to itself. This is differentiable, but the jacobian is not invertible. I think you confuse the concepts of differentiable functions and diffeomorphisms (invertible differentiable functions). Or maybe I misunderstood your question.
 
  • #3


Thanks, Mugver:

What do you mean by the null function.?

Actually, the condition of the Jacobian being invertible is, I think, sufficient,

but not neccessary: I think we assume when we have a Jacobian, that

the partials Delf/Delx_i are continuous. Then we have that the partials

exist and they are continuous. Then f is differentiable.
 
  • #4


Bacle said:
What do you mean by the null function.?

I mean the function from R^m to R^n defined by f(x) = 0.

Bacle said:
Actually, the condition of the Jacobian being invertible is, I think, sufficient,

but not neccessary: I think we assume when we have a Jacobian, that

the partials Delf/Delx_i are continuous. Then we have that the partials

exist and they are continuous. Then f is differentiable.

The rank of the matrix formed by the derivatives of the function has nothing to do with the differentiability of the function. Maybe you should state more precisely your question... :)
 

Related to Critical points of maps f:R^n->R^m , m>1

1. What are critical points of a map f:R^n->R^m, m>1?

Critical points of a map are the points where the derivative of the map is equal to zero or undefined. These points are important because they can help us determine the behavior of the map and identify any extreme points such as local maxima or minima.

2. How do you find critical points of a map f:R^n->R^m, m>1?

To find critical points, we need to take the partial derivatives of the map with respect to each variable and set them equal to zero. Then, we can solve the resulting system of equations to find the critical points.

3. What is the significance of critical points in maps f:R^n->R^m, m>1?

Critical points help us understand the behavior of the map by identifying important points such as local maxima or minima. They can also help us determine the stability of the map and its behavior near these critical points.

4. Can a map f:R^n->R^m, m>1 have more than one critical point?

Yes, a map can have multiple critical points. The number of critical points depends on the dimension of the domain and the range of the map, as well as the complexity of the map itself.

5. How are critical points related to the Jacobian matrix of a map f:R^n->R^m, m>1?

The Jacobian matrix of a map is a matrix of partial derivatives that can help us identify the critical points of the map. Specifically, the critical points are the points where the determinant of the Jacobian matrix is equal to zero.

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