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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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Wrong proof of Inverse function theorem

Is there anything wrong in my attempt to prove the following section of Inverse function theorem by following: The section I try to prove: Let $f:U\subseteq \mathbb{R}^n\to \mathbb{R}^m$ be ...
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$f : I \to \mathbb{R}$ differentiable at $x_0 \in I$ but $f^{-1}$ discontinuous at $f(x_0)$ [duplicate]

I need to find a function $f: I \to \mathbb{R}$, where $I$ is an interval (can be open, closed, bounded or unbounded - it doesn't matter), so that $f: I \to f(I)$ is bijective, $f$ is differentiable ...
Michael Novak's user avatar
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Application of inverse function theorem.

I am working on this problem. Let $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a function given by $f(x) = x + g(x)$, where $g : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is $C^1$. Suppose that for ...
user123456's user avatar
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Find the inverse of $f(x,y,z)=(x+y+z^2,x-y+z,2x+y-z)$ [duplicate]

This question is linked to Prove that the given function is invertible Find the inverse of $f(x,y,z)=(x+y+z^2,x-y+z,2x+y-z)$. I am looking for an approach to find the inverse of such a 3d function, ...
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Levenberg-Marquardt algorithm and inverse Jacobian/Hessian

Let’s say I have a function $f(p,q):R^{n+m}→R$, with $p∈R^n$ and $q∈R^m$. I have a set of $q_{i=1,…,k},y_{i=1,…,k}$ and I want to find $p$ so I use the Levenberg-Marquardt algorithm to resolve the ...
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Prove that if $Df(x)=\eta (x)Dg(x)$, then g is locally a composition of f and some function $\varphi$

I have this problem where I can't figure out the final step: $f,g:U\subseteq\mathbb{R}^n\to\mathbb{R}$ are $\mathscr{C}^1$ functions and $Df(x)\neq 0$ for all $x\in U$. Suppose that there is some ...
Andrés Vásquez's user avatar
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If $\psi(u,v) = (u, u^2 + v^2, v)$ and $\alpha(t)=(ct - b, 5t^2 - ct + c, t + a)$ is a curve with $\alpha$ in $\text{img}(\psi)$, then $a + b + c$ is

If $\psi(u,v) = (u, u^2 + v^2, v)$ with $(u,v) \in \mathbb{R}^2$ and $\alpha(t) = (ct - b, 5t^2 - ct + c, t + a)$ is a curve with $\alpha \subset \text{img}(\psi)$, then $a + b + c$ is .... Is anyone ...
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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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making sense of calculations in the proof that the image of an embedding is an embedded submanifold

I am having trouble understanding the proof of Theorem If $F : M \to N$ is an embedding, then $F (M)$ is an embedded submanifold of $N$ Proof. Let $p \in M$. Since $F$ is an immersion, by Theorem 1 (...
some_math_guy's user avatar
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2 answers
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How do I remove the specific assumptions in the proof of the rank theorem in the special case of injective differential?

I have the following proof in my notes. It is a particular case of the more general rank theorem. Theorem (Rank theorem for injective differential). Suppose $M$ is a smooth manifold of dimension $m$, ...
some_math_guy's user avatar
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Is the determinant of the Jacobian matrix of $g$ at $f(2,1)$ correct?

Let $f(x,y)=(x^2-y^2,2xy),$ where $x>0,y>0.$ Let $g$ be the inverse of $f$ in a neighbourhood of $f(2,1)$. Then the determinant of the Jacobian matrix of $g$ at $f(2,1)$ is equal to.................
GATE's user avatar
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Inverse Function Theorem For Function Transforming Polar Coordinates Into Cartesian Coordinates. [closed]

Let $U=(0,\infty)\times (0,2\pi]$ Define a function $F:U \rightarrow \mathbb R^2$ by $$F(r,\theta)= (r\cos(\theta),r\sin(\theta)).$$ Here all $4$ partial derivatives of component functions exist and ...
Meet Patel's user avatar
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Can we solve $u=(x^4+y^4)/x,v=\sin x+\cos y$ for $(x,y)$ near $(\pi/2,\pi/2)$?

Let $U=\{(x,y)\in\mathbb{R}^2\colon x\neq 0\}$. For each $(x,y)\in U$, let $$u(x,y)=\frac{x^4+y^4}{x}\quad\text{ and }\quad v(x,y)=\sin(x)+\cos(y).$$ Let $f=(u,v)$ on $U$ and $c=(\pi/2, \pi/2).$ Can ...
Brody's user avatar
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1 answer
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How to get inverse function of $x-\sin(ax)$

$$f(x)=\frac{x}{2}-\frac{\sin(200πx)}{400π}$$ I'm not a mathematician. I'm making a heater which uses this formula to calculate the watt of given x. The problem is I need the inverse of this function ...
Taner Caslaman's user avatar
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Use the inverse function theorem to prove that the image of a neighbourhood is a neighbourhood of the image.

Let $a$ be in a point in $\mathbb{R}^m$. $f$ is a partial function from $\mathbb{R}^m$ to $\mathbb{R}^n$, where $m\geq n$, that is differentiable in a neighbourhood $V$ of $a$ and continuously ...
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Does being bijective and having a full rank jacobian Matrix imply being a Diffeomorphism?

I was thinking it to be true, as the inverse function theorem gives us contious differentiablity of the inverse locally at any point. With continous differntiablity being a local attribute of the ...
watertrainer's user avatar
1 vote
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Question about the inverse function and the implicit function theorems.

The theorems in question are and A consecuence of (9.24) is and a consecuence of (9.28) is the following highlighted text I'm having trouble at how he arrived a these conclusions
Manuel Ocaña's user avatar
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If $f(1)=3$, $f'(1)=2$, $f''(1)=4$, then $(f^{-1})''(3) =$ [duplicate]

I tried using the inverse function theorem, but the answer I got through that was $-1$. The answer key says the answer is $-1/2$. Where am I going wrong?
Sanchita's user avatar
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1 answer
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A doubt in a problem regarding Inverse function theorem

Considering the function $f(x)= \begin{cases} x+ x^2 \sin \frac{1}{x} \mbox{ if } x \neq 0 \\ 0 \mbox{ if } x=0 \end{cases},$ show that the hypothesis: $f'$ is continuous can not be removed from ...
Learning's user avatar
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Extension of Inverse Function theorem to k-times continuously differentiable functions.

In an exercise, I am asked to show that If in the (single variable) Inverse Function Theorem $f$ has $k$ continuous derivatives, then the inverse function $g$ also has $k$ continuous derivatives. ...
cosmic_crafter's user avatar
2 votes
2 answers
256 views

Solve nonlinear system of equations and show it has infinite solutions

So we have this system of nonlinear equations \begin{align*} \sin(x+u) - e^y + 1 = 0\\ x^2 + y + e^u = 1 \end{align*} and we want to show that it has infinitely many solutions $(x,y,u)$. I tried ...
Math Wrath's user avatar
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Intuition behind local coordinates reformulation of Inverse Function Theorem

I'm reading from Guillemin and Pollack's Differential Topology, and I don't understand one of the reformulations of the Inverse Function Theorem as well as I'd like to. The Inverse Function Theorem as ...
Keshav Balwant Deoskar's user avatar
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Confused about a step of the proof of the inverse function theorem for differentiable, non $C^1$ mappings.

Here is the proof of the inverse function theorem for differentiable, non $C^1$ mappings, supposing that $Df$ has maximum rank https://terrytao.wordpress.com/2011/09/12/the-inverse-function-theorem-...
Lorenzo Vanni's user avatar
2 votes
2 answers
1k views

Multi-variate cross-partial derivative of a inverse function

I have an invertible mapping $y=f(x,\theta)$ where $x,y\in\mathbb{R}^K$ with a scalar parameter $\theta\in\mathbb R$. Consider its inverse $x=g(y,\theta)$. I'm interested in the matrix $\partial^2g/\...
Kirill Borusyak's user avatar
1 vote
1 answer
35 views

For $F:R^{n} \to R^{m}, m>n$, with local inverse $G$ at $x$ such that $G(F(x)) = x$, is $DG$ a left inverse of $DF$? Why/Why not?

Assuming both functions are differentiable in some neighboorhood, under which conditions, if any, is the Jacobian of $G$ the left inverse of the Jacobian of $F$, ie $(JG)(JF) = Id^{nxn}$? Since we're ...
gaaaaaaaaaah's user avatar
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pdf transformation of periodic scalar random variables related by a derivative

Two random scalar variables, x and y, are related by the following expression: $y(t) = f[x(t)] = a\frac{dx(t)}{dt}$, where $x(t)$ is periodic and $a$ is a constant. How can I calculate the pdf $g_{Y}[...
unkown's user avatar
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Proposition 0.2 Introduction to PDE's - Folland, G.

I'm reading the following proposition of Introducion to PDE's of G. Folland but I don't understand the following part: For each $x\in S$ its Jacobian matrix at $(x,0)$ is nonsingular since $\nu$ is ...
matdlara's user avatar
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4 votes
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Inverse Function Theorem applied to two open sets

Considering the system of equations: $x^2\cos(xy)=a$ ; $e^y=b$ Show that if $V$ is a significantly small neighbourhood of $(1,1)$, the system above has at least two solutions for each $(a,b)$ $\in$ $...
J P's user avatar
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0 answers
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Variant of inverse function theorem

I am trying to work out if the following variant of the inverse function theorem holds: Conjecture: Suppose $f$ is a $C^0$ function, and $f'(0)$ exists and is non-singular. Then there exists a ...
JDoe2's user avatar
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Help with proof of Inverse-Function Theorem in $ \mathbb R ^ n $

Let $U$ be an open set in $\mathbb{R^n}$, and let $f: U \to \mathbb{R^n}$ be a $C^1$ function such that, for all $x$ $\in$ $U$, $Df(x)$ is an isomorphism. My goal is to show if $f$ is one-to-one, ...
J P's user avatar
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Locally invertible function is onto if its preimage on a compact subset is compact.

Question Suppose $f:\mathbb R^n\to \mathbb R^n$ is a $C^1$ function and $Df(x)$ is invertible for all $x\in \mathbb R^n$. Then $f$ is onto if $f^{-1}(K)$ is compact for all compact set $K\subseteq \...
SuperSupao's user avatar
1 vote
1 answer
60 views

Last step in the proof of $ G(F):=\{(p, F(p); p \in M\}$ is an embedded submanifold of $M \times N$ of dimension $m$, $F:M\rightarrow N$ smooth

Consider the definition of embedded submanifold as follows :Let $M$ be a smooth manifold of dimension $m$ and $S\subseteq M$. $S$ is an embedded manofold of dimension $k$ if for every $p \in S$, ...
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Inverse function theorem Proof in Rudin Book (Openness of V)

In Proving the Inverse function theorem in Mathematical Analysis Book of Rudin, when he want to show that V is open, Why he simply does not use that under continuous function (Mapping), the open set U ...
Mehdi Mowlavi's user avatar
1 vote
0 answers
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On the PDF of a function of a random variable [duplicate]

Suppose I have a random variable $X$ and I know its PDF $p_X$. The goal is to find the corresponding $p_Z$ of the random variable $Z=f(X)$. For now, $f$ is any Borel measurable function. I'm trying to ...
haiku's user avatar
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Is it possible to find the inverse of the following function explicitly?

I am interested in the following function $$ f(x) = \sqrt{3x \left(1-\frac{x}{\theta} \right) \ln{\left[a \left(1-\frac{\theta}{x} \right) \right]}}, $$ with $ 0<a \leq 1$ and $\theta<x$. ...
Ario Derek's user avatar
1 vote
1 answer
41 views

Manifolds - Inverse Function Theorem Form?

For context, I am reading Appendix F is Milnor's book on Sullivan's No Wandering Theorem and this is where he uses the following manifolds result: Let $F:X \rightarrow Y$ be a smooth function between ...
OllyT777's user avatar
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1 vote
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Proof that locally defining function implies submanifold of codimension $m-n$.

Setup: We note the following theorem: Let $U \subset \mathbb{R}^m$ and let $f \in C^{\infty}(U,\mathbb{R}^n)$. Now if $n \leq m$ and $Df|_x$ is surjective, there is a neighbourhood $V \subset \mathbb{...
Ben123's user avatar
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2 votes
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Show that $F(t) = (\cos t, \sin t)$ is locally one-to-one but not globally one-to-one.

I am studying Implicit Function Theorem and Inverse Function Theorem. The problem I want to ask is: Show that $F(t) = (\cos t, \sin t)$ is locally one-to-one but not globally one-to-one. I have two ...
Beerus's user avatar
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Is every $C^k$ embedding the restriction of a diffeomorphism on a subspace?

Let $D$ be an open subset of $\mathbb{R}^d$ and $\phi:D\rightarrow \mathbb{R}^n$ a $C^k$ embedding. Is $\phi$ the restriction of some diffeomorphism (on $\mathbb{R}^n$) on the subspace $\mathbb{R}^d \...
Plemath's user avatar
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2 votes
0 answers
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Proof verification: $dF_p$ nonsingular, then $F(p)\in \textrm{Int}N$

I am trying to solve Problem 4-2 on Lee's Introduction to Smooth Manifold. The problem states the following: Suppose $M$ is a smooth manifold (without boundary), $N$ is a smooth manifold with ...
Kaira's user avatar
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1 vote
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Proving a set is a regular surface using inverse function theorem

I was recently introduced to a particular definition of a regular surface which is: A (regular) surface is a subset $S \subset \mathbb{R}^{3}$ with the property that around every point $p \in S$ we ...
JLGL's user avatar
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3 votes
1 answer
186 views

Guillemin & Pollack Exercise 1.8.14 (Generalized inverse function theorem)

I have a question regarding Exercise 1.8.14 in Guillemin & Pollack. (I think this question also applies to the answers here.) Here's the exercise: Inverse Function Theorem Revisited. Use a ...
Hrhm's user avatar
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2 answers
118 views

Jacobian, inverse function theorem and continuously differentiable functions

Question: Let $f \colon \Omega \to \mathbb{R}^n$ be such that $f$ is continuously differentiable where $\Omega$ is a bounded connected set in $\mathbb{R}^n$. For each $t \in \mathbb{R}$ define $f_t(x) ...
L-JS's user avatar
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1 vote
1 answer
79 views

Inverse implicit function theorem

Let $f:\mathbb{R}^{2}\to \mathbb{R}$ be a twice continuously differentiable function such that $f(0,0)=0$ and $f_{y}(0,0)\ne 0$. By the implicit function theorem, there exists an $\epsilon >0$ and ...
maths and chess's user avatar
1 vote
0 answers
89 views

Continuity of derivative in Inverse Function theorem in Deimling's book

I am having a question on the following proof of the Inverse function Theorem in Deimling's book. It is stated that if $G$ is continuously differentiable, then $G_{|U}^{-1}$ is also continuosly ...
Petar's user avatar
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1 answer
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Inverse function theorem from Deimling's book

I am looking at the following proof of the Inverse function theorem in "Nonlinear Functional Analysis" by Deimling. Here in the proof, he uses the following argument which seems extremely ...
Petar's user avatar
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Implicit function theorem: am I using it correctly?

Suppose that $F\left ( x,y,u,v \right )$ and $G\left ( x,y,u,v \right )$ have continuous first partial derivatives. Suppose that the equations $F\left ( x,y,u,v \right )=0$ and $G\left ( x,y,u,v \...
mlrofcloud's user avatar
3 votes
4 answers
147 views

How do I learn to stop worrying and love the substitution $y'' = y' (dy'/dy)$

The following is a solution of the differential equation $y'' = y$ with initial values $y(0) = 3$, $y'(0) = 1$. Considering $y$ to be a function of $x$ and omitting some standard details: Let $z = y'$...
ronno's user avatar
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2 votes
0 answers
71 views

Am I applying the Implicit Function Theorem correctly?

I have solved a $3\times 3$ Non-linear system using numerical methods (post here), and now would like to argue for the uniqueness of my solution. My approach would be to make use of the Implicit ...
Weierstraß Ramirez's user avatar
3 votes
0 answers
167 views

Analytical computation of an inverse function

A function defined as $f(x,y)=0$ is said to be implicitly defined by the equation. The concept of inverting $f$ is closely related to the idea of solving the equation $y=f(x)$ for x as a function of y....
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