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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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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260 views

Explanation of “without loss of generality” in an application of Inverse Function Theorem.

Let $U$ be an open subset of $\mathbb R^{n+m}=\mathbb R^n\times \mathbb R^m$ and $g:U\to\mathbb R^m$ a $C^1$ function. Let $p=(x_0,y_0)\in U$ be a point such that $$g'(p):\mathbb R^{n+m}\to \mathbb R^...
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205 views

Example appliction of Nash-Moser inverse function theorem

I have basic knowledge of PDE and know how to use the standard (Banach space) inverse function theorem to solve $$ -\Delta u+ g(u)=f $$ when $g(0)=g'(0)=0$ and $f$ is small: Define $A(u):=-\Delta u+...
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30 views

Converse of inverse function theorem

Consider the following statement of the inverse function theorem: Let $f : W \to \mathbb{R}^n$ be a $C^1$ function from an open subset $W \subseteq \mathbb R^n$ and let $p \in W$. $f$ has a local $C^...
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152 views

Applications of Inverse Function Theorem

I am searching for applications of the Inverse Function Theorem for smooth maps: Inverse Function Theorem. Let $E\subset \mathbb{R}^n$ be an open subset and let $f\colon E \rightarrow \mathbb{R}^n$ ...
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656 views

Prove case of Implicit Function Theorem using Inverse Function Theorem

Use the Inverse Function Theorem to prove the following case of the Implicit Function Theorem. If $f$ is a real-valued $C^1$ function on an open subset $U\subseteq \mathbb{R}^2$ with $f(a,b) = 0$, $\...
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57 views

Inverse of Higher order partial derivatives: How can I get them?

In my current research, a curious problem arose. I will elaborate. I have a Volume function, which is given by: $$ V(u,v,w) = \sum_{i=1}^n N_i(u)M_i(v)L_i(w)\omega _iPi \text{,} $$ where $N_i(u)$, $...
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166 views

Implicit function theorem implies inverse function theorem proof

I believe this will be a very long answer if anyone tries to write the full proof or anything so I'll specify which specific parts I am having trouble with to save people's time. I want prove three ...
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97 views

Inverse function theorem via degree theory

In this MO question Terrence Tao writes the following about the inverse function theorem. The Brouwer fixed point theorem gives local surjectivity, and degree theory gives local injectivity if $\...
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239 views

Extension of Inverse Function Theorem to Manifold with Boundary

Recently i learned that we can't extend inverse function theorem in Euclidean space to manifold with boundary as domain. More precisely, if we have a smooth map $F : M \to N$ where $M$ is a manifold ...
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127 views

Common techniques/tricks for Implicit/Inverse Function Theorem Problems

I am currently in a graduate level real analysis course, and we have been working with the Inverse and Implicit Function Theorem. With these problems, I often find myself struggling to set the work up ...
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243 views

Generalization of the Inverse Function Theorem

This is exercise 10 on page 19 in Guillemin/Pollack Differential Topology. Let $f : X \to Y$ be a smooth map which is one-to-one on some compact submanifold $Z$ of $X$. Moreover, let $df_p$ be ...
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89 views

Historically, who realized the inverse function theorem could be proved with the use of contraction mappings?

Historically, who realized the inverse function theorem could be proved with the use of contraction mappings? And moreover, how was the connection between contraction mappings and the inverse function ...
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1answer
1k views

Finding the inverse of $f(x,y)=(e^{x}\cos(y),e^{x}\sin(y))$ around a neighborhood.

Show that $f\left(x,y\right)=\left(e^{x}\cos y,e^{x}\sin y\right)$ is one-to-one around any point of $\mathbb{R}^{2}$. For the points $\left(0,\pi\right)$ and $\left(-1,\frac{\pi}{2}\right)$ find such ...
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57 views

Unique Solution for Square-root function on Matrices

Prove that for every n by n matrix M sufficiently close to the identity matrix there exists a square-root matrix (solution of $A^2 = M$) and the solution is unique if $A$ is required to be ...
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20 views

A function that is not invertible but has an implicit form locally

I'm trying to find out the difference between the implicit function theorem and the inverse function theorem. One of the obstacles of my understanding, is that I can't find a function that it ...
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22 views

Confusions on functions on open intervals

Assume $f \in C^1(\mathbb{R}^2)$ such that $f(0,0)=0$, $f_x(0,0) \ne −1$, and $f_y(0,0) \ne 0$. Assume $g : \mathbb{R}^2 → \mathbb{R}$ such that $$g(x,y) = f(f(x,y),y).$$ Let $t$ be a function defined ...
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52 views

Inverse Map of a Set

Suppose I have $B=F^{-1}(A)$, shouldn’t $F^{-1}$ defined on all of $A$ for the inverse map to make sense? Or can we define $B=({{ u\in B|F(u)\in A }})$? My question is regarding a proof in Pressley’s ...
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30 views

Inverse function theorem - derivative has not full rank

I am a little bit confused... Suppose that the derivative of a function $f$ has not full rank at a point $a$. However, this does not imply that there doesn't exist an inverse function. We can only ...
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46 views

Is this proof about Inverse function Theorem okay?

I'm proving this: "Let $U$ open set in $\mathbb R^n$ and $f:U \to \mathbb R^n$. $f$ is $C^1$ in $U$ and $\det(D(f(x)) \neq 0$. Prove that $f(U)$ is and open set in $\mathbb R^n$" My proof is the ...
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38 views

Showing inverse of multivariate holomorphic map is holomorphic using real analysis techniques

I'm grading for a graduate real analysis course and looking through the text, trying to do some problems before the term starts so I can stay ahead of things. I found the following exercise in the ...
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161 views

Generalization of inverse function theorem to allow unequal dimensions?

My book is An Introduction to Manifolds by Loring W. Tu. The last sentence here says being an immersion or a submersion at p is equivalent to the maximality of rk $[\frac{\partial f^i}{\partial ...
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1answer
91 views

I’m missing a step in a proof of the Holomorphic Inverse Function Theorem.

I’m looking at the proof of Theorem $1.7$ (Inverse Function Theorem), chapter VI of Lang’s Complex Analysis (which can be found on page $182$ in the fourth edition of the book), and I’m stuck with one ...
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1answer
41 views

Showing that a set cannot be expressed as the local graph of a $C^1-$function over the $x-$axis or the $y-$axis near the origin.

Consider the function $h(x,y)=(x−y^2)(x−3y^2)$, $(x,y)\in \mathbb{R}^2$. Show that the set $\{(x,y) | h(x,y) = 0\}$ cannot be expressed as the local graph of a $C^1-$function over the $x-$axis or the ...
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152 views

real analytic inverse function theorem

I have a two part question about real analyticity and whether I have a an error in reasoning. Suppose I have a multivariate holomorphic mapping $f \colon U \to \mathbb{C}$, where $U$ is an open set in ...
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1answer
68 views

Higher order Inverse Function Theorem

Consider a smooth function $F : \Bbb{R}^n \to \Bbb{R}^m$ where $n \geq m$ are positive integers. Consider a curve $\alpha : [0,1 ) \to F^{-1}(0)$ such that $\alpha(0) = x \in F^{-1}(0)$. Then $F(\...
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1answer
44 views

Inverse function theorem to prove onto

Let $f: \mathbb R^2 \to \mathbb R^2$ be $C^1$, $D_f(x)$ is invertible everywhere, and $\lim_{|x| \to \infty}|f(x)| = \infty$ Show that $\min_{x \in \mathbb R^2}|f(x)-a|$ exists, and that $f$ is onto. ...
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44 views

Given a function $f:\mathbb{R}^2 \to \mathbb{R}^2$, to find derivative of its inverse at a given point

Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a function given by $$f\left(x,y\right)=\left(x^2-y^2,2xy\right),\,\, \left(x,y\right) \in \mathbb{R}^2$$ Question : Compute $Df^{-1}\left(0,1\right)$. I have ...
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62 views

Piecewise smooth function , Jacobian and locally invertible?

Say $f$ is a piecewise smooth function that is $f(x,y;\chi)=f_{1}(x,y;\chi)$ if $y\leq g(x;\chi)$ and $f(x,y;\chi)=f_{2}(x,y;\chi)$ if $y \geq g(x;\chi)$ where $g$ is a $C^2$ function and $\chi$ a ...
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30 views

Inverse Function Theorem Problem Feedback

I'm working on the following problem: Let $f(x) = x^4 - x^3 + x$, for $x \geq 0$. Show that (i) $f$ is strictly increasing on $[0,\infty)$, (ii) $f$ is a one-to-one and onto function on $[0,\infty)$,...
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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 $x_{k}$ ...
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Function satisfying $(Df(x)h,h) \geq \alpha(h,h), \forall x,h \in \mathbb{R}^n$ has an inverse around every point?

$f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ class $C^1$ such that: there exists $\alpha >0$: $$(Df(x)h,h) \geq \alpha(h,h), \forall x,h \in \mathbb{R}^n$$ (where $(x,x)$ is the standard scalar ...
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22 views

Inverting a map locally and interpretation of line integral

Consider the mapping $y(x)$ given by $ \left\{ \begin{array}{ll} y_1=x_1+x_1x_2x_3\\ y_2=x_2 + x_1x_2\\ y_3=x_1^2 - 2x_1x_3 +...
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1answer
25 views

Inverse relation graphical shape

I have a few points among two quantities which are inversely related. The points are $(0,20), (1,19)$ and $(5,15)$. I have drawn these points in Matlab but it gives me an inverted triangle which isn't ...
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How does inverse function theorem show how interior or boundary points map?

So I've read many references that make proofs about interior points mapping to interior points and boundary points mapping to boundary points and they all seem to cite that the inverse function ...
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101 views

Jacobian Criterion from $\mathbb R^n$ to $\mathbb P^n(\mathbb R)$

$\newcommand\R{\mathbb{R}} \renewcommand\P{\mathbb{P}}$ $\newcommand\pd[2]{\frac{\partial #1}{\partial #2}}$ Let $f:\R^n\rightarrow \P^n(\R)$ be smooth, then why can one apply the inverse function ...
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1answer
986 views

Jacobian and local invertibility of function

Following is a question in the text book: Show by an example that the condition that the Jacobian vanishes at a point is not necessary for a function to be locally invertible at that point..... and ...
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22 views

Implicit function for quadrants of real Banach space

Definition 1 Let $X$ be a real Banach space and $X^+\subset X$. Then, $X^+$ is called a quadrant of $X$ iff there exists a nonempty finite linearly independent subset $\{\lambda_1,...,\lambda_n\}$ of ...
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29 views

Variant of local inversion theorem in special case

Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be defined by $F(x,y)=(x+2y+x^2\ ,\ y-x^3+y^2)$. Then show that for $p_0=(4,1)$ and $p_1=(1,1)$ there exists $\delta>0$ such that for every $\vec y\in B(p_0,\...
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176 views

Inverse function theorem and integrals.

Consider this integral: $$I = \int g\left(f(x)\right)dx.$$ Assuming all regularity conditions, by inverse function theorem, $$\frac{df(x)}{dx}=\frac{1}{\left[f^{-1}\right]'\left(f(x)\right)}$$ and ...
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134 views

inverse function theorem problem

I am interested in finding a formula for the inverse of the function $$f(x,y) = (x^2 + y^2, xy)$$ which works on the set $S = \{(x,y) \in \mathbb{R}^2 : -x < y < x\}$. I determined that the ...
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Is this function from $\mathbb{R}^3 $ to $\mathbb{R}^4$ locally invertible at $(x,y,z)$?

Is this function from $\mathbb{R}^3 $ to $\mathbb{R}^4$ invertible at $(x,y,z)$? $k(x,y,z) = (x+y+z, e^x \cos z, e^x \sin z, \cos z)$ at $(x,y,z)$ I have only studied the Inverse Function Theorem ...
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Map of an inverse function

This passage is taken from a lecture where we went over the inverse function theorem, it is an intro to dynamical systems course whihc is why I think they want us to be paying attention to the set we'...
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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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16 views

Diffeomorphisms between a nonopen set and an open set of Banach spaces

Let $E_i$ be a $\mathbb R$-Banach space, $U_2\subseteq E_2$ be open and $f$ be a $C^1$-diffeomorphism from $B_1$ onto $U_2$. What do we need to assume in order to conclude that $B_1$ is open? By ...
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32 views

If $\\f:\ U\to\ R^N$ is a submersion, Prove that $g$ is $C^k$

If $\\f:\ U\to\ R^N$ is a submersion of class $C^k$ and $g:f(U)\to\ R^M$ is such that $g\circ f : U\to\ R^M$ is $C^k$ then $g$is $C^k$ In my attempt I know that $D_{f(p_0)}$ is onto, $p_0 \in U$ and ...
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32 views

Is the parameterization of a regular surface a local diffeomorphism?

Let $\left\{ U_{\alpha },\varphi _{\alpha }\right\}$ be a local chart of a regular surface S $\subset \mathbb{R}^{3}$. That is: $\varphi_{\alpha}:U_{\alpha}\subseteq\mathbb{R}^2\rightarrow \varphi_{\...
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19 views

convergence to level set

Say we have a function $f:\Bbb R^N\to\Bbb R$ and a scalar $J$, we define $X := \{x | f(x) = J \}$ and assume $X\ne\emptyset$. We have a sequence $\{x_n\}$ s. t. $\{J(x_n)\}\to J$. Under what ...
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16 views

What is solution of $f'(f(x))=\exp(f'^{-1}(x))$ with $ f'^{-1}$ is a compositional inverse of $f'$?

,Assume $f$ a bijective and differentiable function on its domain , I want to find solutions of this functional such that $f\colon\mathbb{R} \to \mathbb{R}$; $f'(f(x))=\exp(f'^{-1}(x))$ such that $f'...
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45 views

Inverse Function Theorem to solve functions

Prove that for $(x, y) \in \Bbb R^2$ \begin{cases} x+y+\sin(xy)=2c \\ \sin(x^2 + y) = c^2 \\ \end{cases} can give a solution for all $c ∈ \Bbb R$ close ...