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

[Bacle]

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.

 

[Petr Mugver]

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.

 

[Bacle]

Thanks, Mugver:

What do you mean by the null function.?

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

but not necessary: 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.

 

[Petr Mugver]

What do you mean by the null function.?

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

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

but not necessary: 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... :)