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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148 views

Can Integration by Parts be used to define the explicit inverse or integral-inverse of a function defined by an integral?

Suppose you have a function $f(x)$ defined as an integral, such that $$f(x)=\int g(x)dx$$ like say the natural logarithm defined as $$\ln(x)= \int_{1}^{x} \frac{1}{t}dt.$$ Unfortunately, the inverse ...
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58 views

Existence of neighbourhoods and diffeomorphism to form a projection

Let $f \in C^1(\mathbb{R^n})$ be a given function with $f(0)=0$ and $ ∂_1f(0) \neq 0$. Show that there exist neighbourhoods $U$ and $V$ of $x=0 \in \mathbb{R^n}$ and a diffeomorphism $\phi: U \...
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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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57 views

Existence of a Diffeomorphism with $f(\phi(x)) = x_1$

Let $f \in C^1(\mathbb{R}^n)$ a function with $f(0) = 0$ and $\partial_1f(0) \neq 0$ Show that neighbourhoods $U, V$ of $x = 0$ as well as a diffeomorphism $\phi: U \to V$ exist so that $f(\phi(x)) ...
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60 views

Inverse function theorem and finding the open set that allows $f$ to have a local inverse

The inverse function theorem says that there exists an open set $U$ such that there is a local inverse of $f$ around a point $a$. However, how would we actually find an open set containing this point $...
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101 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 ...
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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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1answer
292 views

Inverse Function Theorem proof: f is injective

I am trying to prove the Inverse Function Theorem from the Implicit Function Theorem for Banach spaces. My attempt so far is as follows: Let $f:\mathbf{X}\to \mathbf{Y}$ be a $\mathcal{C}^k$ function ...
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1answer
284 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 ...
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102 views

Prove that a differentiable function with a special propriety is a Isometry

Hi folks! I'm trying to answer this one exercise: Let $f:\mathbb R^m\to\mathbb R^m$ be a $C^1$ function such that, for all $x\in\mathbb R^m$ $|f'(x)\cdot v|=||v||$ (where $||\cdot||$ is the ...
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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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75 views

What is this notation regarding the Inverse Function Theorem?

A definition in my notes state: If $Df(a)$ is invertible (as a matrix), then $f$ is invertible on an open set $U$ containing $a$. So given that $f(x,y) = (a,b)$ and there exists a $C^1$ local ...
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Prove $\det[D f (x)] \neq 0$

Let $U, V ⊆ \mathbb R^n$ be open sets and $f : U → V$ a differentiable bijection with a differentiable inverse. Show that $\det[D f (x)] \neq 0$ for any $x ∈ U$. This shows that the converse of the ...
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68 views

Prove a bijection and $D_{(2,0)}(f \circ g^{-1})$ as a matrix. Inverse functions

Let $g:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ and $f:\mathbb{R}^2 \rightarrow \mathbb{R}^3$ be the functions defined by: $$g(x,y):=(2ye^{2x},xe^y)$$ $$f(x,y):=(3x-y^2,2x+y,xy+y^3)$$ $a)$Show that ...
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173 views

How to find the inverse of a function in the neighborhood of a point at which the derivative is invertible

According to Rudin, The inverse function theorem states, roughly speaking, that a continuously differentiable mapping $f$ is invertible in a neighborhood of any point $x$ at which the linear ...
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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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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
71 views

$f$ diffeomorphism $\Leftrightarrow f'(x)\cdot f'(y)>0$

Let $I\subset \mathbb R$ be an open interval and $f:I\to \mathbb R$ be a differentiable function. I would like to prove the following equivalence: $$f:I\to\mathbb R\ \text{a diffeomorphism over $f(...
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What is the compositional inverse of nul map or $f(x)=0 $?

Let $f$ denote a function and $f^{-1}$ the compositional inverse of $f$ from: $\mathbb{R}\to \mathbb{R}$, for example $log$ is the compositional inverse of $exp$ function ,Really I w'd like to ask if $...
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1answer
129 views

Extension of Inverse Function Theorem from $\mathbb{R}$ to $\mathbb{R^n}$

Consider the Inverse Function Theorem in $\mathbb{R}$: Let $O ∈\mathbb{R}$ be open for $f:O → \mathbb{R}$. If $f$ is continuously differentiable, and for a $x_0 ∈O$, $f'\left(x_0\right) \ne 0$. Then ...
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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
71 views

Showing that a differential is nonzero

Problem Statement: Let $g:U\rightarrow \mathbb{R}^{n}$ be a $C^{1}$ map ($U\subset \mathbb{R}^{n}$ open), with $dg(\mathbf{x})\in GL(\mathbb{R}^{n})$ for all $\mathbf{x}\in U$. Suppose $\mathbf{y}_{0}\...
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1answer
988 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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1answer
215 views

Multivariate Newton's method - how to apply the chain rule to $g(x)=x-(D_xf)^{-1}f(x)$

Let $f:\mathbb R ^n\to \mathbb R ^n$. Following page 6 here, I'm trying to show the multivariable iteration function $g(x)=x-(D_xf)^{-1}f(x)$ (the analogue of the 1d version for Newton-Raphson) ...
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1answer
67 views

logarithm of the square of a nonsingular matrix

Let $C\in\mathbb{R}^{n\times n}$ be a matrix such $det(C)\neq0$ b. I need to prove that exists $B\in\mathbb{R}^{n\times n}$ such $$e^B=C^2$$ This is a question from a curse of ODE, we have seen ...
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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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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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1answer
681 views

Jacobian determinant of a function $f$ if it has a differentiable inverse.

This question is from the book "Differential Topology First Steps" by Andrew Wallace. The question is as follows: Let $f$ be a differentiable map of an open set in Euclidean n-space into n-space and ...
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1answer
97 views

Find conditions on $x$ and $y$ which guarantee that one can locally solve the following for $u(x, y)$ and $v(x, y)$

My understanding of this question is that I need to show that the following equations can be solved where $u$ and $v$ can be written as a function of $x$ and $y$. $xu^2+yv^2=9$ $xv^2-yu^2=7$ I ...
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2answers
483 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 $A$, ...
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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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1answer
163 views

Use the inverse function theorem to find $y = G(u,v)$ given $u = f(x,y), v = g(x,y)$

Consider the functions $u = f(x,y) = x^2-y^2$ and $v = g(x,y) = 2xy$. What does the inverse function theorem tell you about defining $x = F(u,v)$ and $y = G(u,v)$? Find an expression for $\frac{\delta ...
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1answer
280 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 ...
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1answer
110 views

Question related to differentiable functions on Banach spaces

There is an interesting exercise on my Analysis book that I have not been able to solve: Let $\mathbb{E,F}$ be Banach spaces, $f:\mathbb{E}\to\mathbb{F}$ of type $\mathcal{C}^k$, $k\geq1$. Asume $f'(...
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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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1answer
283 views

Approximating the implicit function

Last week in our Calculus II course we learned about the implicit function theorem, to prove this theorem we used the inverse function theorem which we proved using using the Banach fixed-point ...
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1answer
127 views

assumptions for existence of envelope of a family of curves

Given a family of curves $F(x, y, c) = 0$ for $c$ in a range of real numbers, the envelope $E$ is the curve tangent to every member of the family. I see that it is defined by the solution of $F(x, y,...
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1answer
843 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}) = ...
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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 ...
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1answer
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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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Showing a function is inversible

Hi I'm struggling with this question. Let $f : ℝ → ℝ$ be defined by $f(x)=\frac{5}{\sqrt {x^3+4}}$. Show that this function is invertable.
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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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1answer
93 views

Showing that a function is injective

I am trying to show that the following function is injective in some neighborhood of $(0, 0)$: $f:\mathbb R^2 \rightarrow \mathbb R^2$ given by $$f(x, y)=(\sin(x^3)\cosh(y), \cos(x^3)\sinh(y))$$ I ...
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1answer
143 views

How to prove a weak version of Inverse Function Theorem using Extreme Value Theorem?

Suppose that $f:(a,b) \rightarrow \mathbb{R}$ is continuously differentiable with $f'(x)>0$ for all $x \in (a,b)$. Let $g$ be the inverse function of $f$. How can I prove that $g$ is differentiable ...
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1answer
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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3 questions over the function $f(x_0,x_1,x_2)=\frac{1}{1+x_0+x_1+x_2}(x_0+x_1,x_1+x_2,x_2+x_0).$

Let $D=\{(x_0,x_1,x_2)\in\mathbb{R}^3|\ 1+x_0+x_1+x_2 =0\}$ the plane in $\mathbb{R}^3$ passing for the points $(-1,0,0)$, $(0,-1,0)$, $(0,0,-1)$ and let $E=\mathbb{R}^3-D$ its complement. Note that $...
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1answer
264 views

Prove that $f(A)$ is an open set and $f^{-1}:f(A)\to A$ is differentiable.

Let $A\subset \mathbb{R}^n$ an open set, and $f:A\to \mathbb{R}^n$ a one to one and continuously differentiable function so that $\det f'(x)\ne 0$ for all $x\in A$ Prove that $f(A)$ is an open set and ...
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1answer
89 views

How to prove this assertion? Do I need to use Inverse function Theorem?

Let $f:R^2\to R$ be a continuously differentiable function. Show that there exist a continuous one-one function $g:[0,1]\to R^2$ such that $f\circ g:[0,1]\to R $ is constant.
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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}}\...
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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|\...