Can the Inverse Function Theorem Be Applied to Complex Multivariable Functions?

[DuskStar]

I've had a read through some of the topics about this but I am struggling to understand how to apply it.

(1) Is it possible to apply the inverse function theorem to a function like f(x,y)=-xye(-(x^2+y^2)/2) or f(x,y)=2x^2+y^2-xy-7y

(2) I am confused about how to compute the jacobian matrix when differentating only gives two terms and my matrix needs four?

Any help would be great

 

[HallsofIvy]

What "inverse function theorem" are you referring to? The "inverse functions theorems" I know are from Rⁿ to Rⁿ. Here, your function is from R² to R. An "inverse" would be from R to R². I don't see any reasonable way to do that!

 

[DuskStar]

In that case I am completely confused. I was just given these examples and told to show whether they were invertible or not by the 'inverse function theorem'. Is it possible for someone to post any example where I could apply the inverse function theorem?

 

[HallsofIvy]

You originally said the question was " Is it possible to apply the inverse function theorem to a function like f(x,y)=-xye(-(x^2+y^2)/2) or f(x,y)=2x^2+y^2-xy-7y?"

Possibly the answer is "no"!

 

[mathwonk]

the inverse function theorem, as stated, applies only to functions between euclidean spaces of the same dimension, since the hypothesis, that the jacobian matrix be invertible, is otherwise false.

however it is a special case of the "rank theorem', which says, also in a special case, that if the rank of the jacobian matrix equals the dimension of the target, say n, (and the map is smooth), then in some smooth coordinate system, the map becomes projection on the last n coordinates.

so in this example of a map from R^2 to R, near any point where the partials are not both zero, the map can be expressed in some coordinate system as (x,y)-->y.

this theorem, also called the implicit function theorem, can be proved as a corollary of the inverse function theorem, by augmenting the given map as (f(x,y), x), or f(x,y),y), depending on which partial is non zero.

so in some sense the inverse function theorem can be applied to this situation. the result however gives that some auxiliary function is invertible, not the original f. for the original f it gives a structure theorem.