Questions tagged [inverse-function-theorem]

Recommended to be used when the Inverse function theorem is being employed in a question, and also for those users that need help understanding it.

Filter by
Sorted by
Tagged with
2
votes
1answer
277 views

Let $M$ be a smooth manifold and let $N$ be a manifold with boundary. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}M $.

Let $M$ be a smooth manifold and let $N$ be a manifold with boundary, both with the same dimension $n$. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}N $. I am trying to prove ...
4
votes
3answers
612 views

Show function $f(x,y)=(x^2-y^2,2xy)$ is $1$-$1$ by Inverse Function Theorem

I'm trying to prove the problem below - which comes from Munkres' "Analysis on Manifolds" book in the section on the Inverse function theorem. Since its in the chapter on the Inverse Function Theorem ...
0
votes
1answer
277 views

Inverse Operator Theorem, counter example

Let $X=Y=C([0,1])$ be the space of continuous functions $f:[0,1]\to \mathbb{R}$. The normed space $(X,\Vert \cdot \Vert_X) $ with $\Vert f\Vert_X:=\sup_{0\leq t\leq 1}\vert f(t)\vert$ is complete and ...
0
votes
1answer
219 views

How is the Inverse function theorem used to prove that the formulae in this question are the same?

I was informed in my last question that the Inverse function theorem: $$(f^{-1})^{\prime}(f(a))=\cfrac{1}{f^{\prime}(a)}\tag{I*}$$ was needed to show that $$\rho_x (x)=\rho_\alpha(\alpha)\left|\...
5
votes
2answers
3k views

Inverse Function Theorem and global inverses

We learnt the Inverse Function Theorem for multi-variable functions, and it only dealt with "local" inverses, not "global" inverses. Is my interpretation of a global inverse just that there exists an ...
8
votes
2answers
3k views

Differentiable bijection $f:\mathbb{R} \to \mathbb{R}$ with nonzero derivative whose inverse is not differentiable

I had an exam today, and I was asked about the inverse function theorem, and the exact conditions and statement (as stated in Mathematical Analysis by VA Zorich): Let $X, Y \subset \mathbb{R}$ be open ...
1
vote
1answer
175 views

If $f\colon \mathbb{R} \to \mathbb{R}$ is one-to-one and differentiable at $a$ with $f'(a) \ne 0$, must $f^{-1}$ be differentiable at $f(a)$?

If $f\colon \mathbb{R} \to \mathbb{R}$ is one-to-one and differentiable at $a$ with $f'(a) \ne 0$, must $f^{-1}$ be differentiable at $f(a)$? I'd like proofs and/or counterexamples, or citation of ...
4
votes
1answer
2k views

Inverse Function Theorem for Manifolds with Boundary as the Domain

In Lee's Smooth Manifolds, it is written that the inverse function theorem can fail for manifolds with boundary. As hint, it is given the inclusion of half space into euclidean space $\iota:\mathbb{H}^...
6
votes
1answer
105 views

Is a differentiable (but not $C^1$) function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with invertible derivative everywhere an open map?

Is a differentiable (but not $C^1$) function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with invertible derivative everywhere an open map? I know that if we assume the function is $C^1$, then this is ...
3
votes
2answers
400 views

Why do we want TWO open sets from the inverse function theorem?

I have been analyzing Rudin's proof of the Inverse Function Theorem closely over the last two days, and trying to understand what the purpose of every assumption made is. The first assumption that he ...
0
votes
1answer
182 views

An application of the Inverse function theorem

I have recently come across two formula's that I am unfamiliar with and would like to know if they are both aspects of the same thing: $$\color{purple}{\cfrac{1}{f^{\prime}(a)}=f(a)(f^{-1})^{\prime}}\...
3
votes
2answers
479 views

inverse function theorem and matrix square root

Define $\mathbf{f}(A) = A^2$, for $A \in \mathbb{R}^{n \times n}$. (a) Applying the Inverse Function Theorem, show that every matrix $B$ in a neighbourhood of $I$ has (atleast) 2 square roots $A$, ...
3
votes
1answer
624 views

Conditions for Inverse Function Theorem

Suppose I have some function $f$: $\mathbb{R}_+\rightarrow\mathbb{R}_+$ that is bijective, continuous and strictly increasing. Moreover, it is (at least) twice continuously differentiable everywhere....
1
vote
0answers
90 views

How do I set up Implicit Function Theorem to verify this function is a $C^{r}$ diffeomorphism?

Suppose I have a $C^{r}$ diffeomorphism $F: \mathbb{R}^{n} \mapsto \mathbb{R}^{n}$ that I write as $x_{k+1} = F(x_{k})$, with $x \in \mathbb{R}^{n}$. Note that here $x_{k+1}$ is the image of $x_{k}$ ...
1
vote
1answer
34 views

$f(u,v)=uv+u^2v$ can be written as $u^2+v^2$ by the change of coordinate in some neighborhood of $(0,0)$.

Proposition: Let $f:\mathbb{C}^2\to \mathbb{C}$ such that $f(0,0)=0, Df(0,0)=0$ and the Hessian of $f$ at $(0,0)$ is of rank $2$. Then there exists coordinates in an appropriately small neighborhood ...
1
vote
2answers
100 views

Un-proving $1=-1$ in the development of the implicit mapping theorem. Edwards: Advanced Calculus of Several Variables.

The follow conundrum arose while attempting to translate to tensor notation the development of the implicit mapping theorem in C.H. Edwards, Jr.'s Advanced Calculus of Several Variables. I refer to ...
1
vote
1answer
842 views

How can I use inverse/implicit function theorem to find a function and its inverse?

My task is this: Let $f:\mathbb{R}^3 \to \mathbb{R}$ be the function $f(x,y,z) = xy^2e^z + z$. Show that there exists a function $g(x,y)$ defined around $\textbf{x} =(-1, 2)$ s.t. $g(\textbf{x}) = ...
0
votes
0answers
51 views

If $\phi$ is a homeomorphism and both $\phi$ and $\phi^{-1}$ extend to differentiable functions, is ${\rm D}\phi(x)$ invertible?

Let $k,d\in\mathbb N$, $\Omega\subseteq\mathbb R^d$, $U\subseteq\mathbb R^k$ and $\phi:\Omega\to U$ be a homeomorphism and $\psi:=\phi^{-1}$. Assume $$\phi=\left.\tilde\phi\right|_\Omega\tag1$$ for ...