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 ...
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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 ...
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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 ...
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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 ...
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Proof that the inverse of an analytic function is analytic which uses only real analysis.
I would like to prove the following result:
Let $f:R\to R$ be such that $f(x)=\sum\limits_{k=0}^\infty a_k(x-c)^k$ for all $x$ in some open set $O\subset R$.
Suppose that $f'(c)\ne 0$.
Then there ...
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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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Inverse function theorem on smooth manifolds
Following this thread, I'm trying to prove in detail this theorem. Could you please check if my proof is fine or contains logical mistakes?
Theorem: Let $X \subseteq \mathbb R^M$ and $Y \subseteq \...
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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 ...
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How do we need to apply the inverse function theorem here? (diffeomorphically mapping an open subset of a submanifold)
Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$, $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$, $T$ be a $C^1$-diffeomorphism from $\mathbb R^d$ and $N:=T(M)$.
It's easy to see that $...
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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|\...
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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 ...
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Spivak, Calculus on Manifolds, Problem 2-37 (b)
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(a) Let $f:\mathbb{R^2}\to\mathbb{R}$ be a continuously differentiable function. Show that $f$ is $not\mbox{ 1-1}$. $Hint:$ If, for example, $D_1f(x,y)\neq0$ for all $(x,y)$ in some open set $A$...
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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 ...
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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}^...
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Does the inverse function theorem require continuity as a hypothesis?
This question is about the inverse function theorem for real-valued functions.
Suppose $f$ is a one-to-one, that $a$ is in the domain of $f$, and that $f$ is defined on an open interval containing $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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Identity up to isomorphism treated as identity in proof
In the following corollary to the inverse mapping theorem by Serge Lang, Fundamentals of Differential Geometry, 1999, p.17-18, there are two things in the proof which I don't understand, the first ...
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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 ...
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Serge Lang's projection
This question is a follow-up to Identity up to isomorphism treated as identity in proof. I thought that with all the kind help given there, now I would be able to work out the sketch of a proof given ...
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Surface in $\mathbb{R}^3$ implicitely defined by a $C^1$-function is locally similar to $xy$-plane
Let $F:\mathbb{R}^3\to \mathbb{R}$ be a $C^1$-function and suppose that $(dF)(x,y,z)\not=0$ wherever $F(x,y,z)=0$. Call $$O = \{(x,y,z)\mid F(x,y,z)=0\}.$$
Then, for every point $p\in O$, there is an ...
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Understanding the proof of the Implicit Mapping Theorem
I am following Advanced Calculus of Several Variables by C.H. Edwards, Jr. I failed to build the logic of the theorem III-$3.4$ stated below,
Theorem $3.4$: Let the mapping $G: \mathscr{R}^{m+n} \...
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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 ...
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Solving $y=x\tan(\theta),f(x,y)=0\implies f(x,x\,\tan(\theta))=0$. Generalized abcissa and ordinate trigonometric functions.
I was inspired to return to an old problem I came up with after seeing
this question.
This was the problem of finding the analogs of other trigonometric functions which would parametrize a certain ...
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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 ...
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Proving a particular case of the complex analytic implicit function theorem using the complex analytic inverse function theorem
In the real analytic case the implicit and inverse function theorem are essentially equivalent, i.e. one can be deduced from the other (from what I know it is more usual to prove the inverse function ...
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Does this version of inverse function theorem hold for Banach space?
Let $E, F$ be Banach spaces over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$. Let $\mathcal L_{\text{is}} (E, F)$ be the set of all topological isomorphisms from $E$ to $F$. Then $\mathcal L_{\text{is}...
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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 ...
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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}) = ...
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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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Should the points in the fiber lie in different connected component?
Suppose I have two connected real smooth manifolds $M$ and $N$ of the same dimension $n$. Suppose that $f:M\rightarrow N$ is a differentiable map between them so that a regular value $q\in N$ exists. ...
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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}}\...
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Generic Intersection of Submanifolds
This is the geometric version of this linear algebra question.
Given submanifolds $N_1\cdots N_n$ of $M$ and $p\in M$, show that the map induced by inclusions
$f:\bigoplus_{i}T_pN_i\to(T_pM)^n/{\Delta}...
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How do I find the Inverse of $f(x)$?
Given
$f(x+1)+f(x-1)=x^2$
I have subtituted $(a=x+1)$ and $(a=x-1)$ and got
$$f(x)+f(x-2)=(x-1)^2 \text{ and } f(x+2)+f(x)=(x+1)^2$$
Combining those equations, I got
$$f(x+2)-f(x-2)=4x$$
I could not ...
