# 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.

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### 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 ...
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 ...
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 ...
219 views

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### 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 $A$, ...
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### 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....
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### 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 $x_{k}$ ...
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### $f(u,v)=uv+u^2v$ 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 ...