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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If there is a diffeomorphism from an open subset of $\mathbb R^d$ onto a submanifold $M$ of $\mathbb R^d$, then $M$ is open

Let $d\in\mathbb N$ $O\subseteq\mathbb R^d$ $k\in\{1,\ldots,d\}$ $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary $f:O\to M$ be a $C^1$-diffeomorphism How can we ...
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Relaxing continuous differentiability in the inverse function theorem

Let $f$ be a continuous injective function on a ball centered at the origin of $\mathbb{R}^n$ into $\mathbb{R}^n$. Suppose that $f$ is differentiable at the origin with non-zero Jacobian determinant. ...
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linear isomorphism preserves coordinates

I am working in differential geometry homework, and I really appreciate any help! My understanding that we can apply inverse function theorem but not sure how that's preserves coordinates. Thank you! ...
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How to find range of a function $f : \mathbb{R}^2\to \mathbb{R}^2 $

How to find range of a function $f : \mathbb{R}^2\to \mathbb{R}^2 $ defined by $f(x,y)=(x^2-y^2, 2xy)$ I am new to calculus of several variables and I have no idea on how to solve such questions. I ...
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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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Can we prove that polynomial functions can always be split into bijective restrictions?

Let us consider a polynomial function $P:[x_{1},x_{2}]\to[0,1]$ of degree $n\geq 2$. I would like to know if it is always possible to prove that it can be split into restrictions $\left.P\right|_{D_{k}...
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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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Inverse theorem proof question

The question pertains to the notes found at: https://ocw.mit.edu/courses/mathematics/18-101-analysis-ii-fall-2005/lecture-notes/lecture6.pdf. In the proof, $U_1$ is defined to be a $\delta$ open ...
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If $T_t$ is a diffeomorphism and $t\mapsto T_t(x)$ is differentiable, can we find a map $v$ with $v(t,T_t(x))=\frac{\partial T}{\partial t}(t,x)$?

Let $d\in\mathbb N$, $\tau>0$, $U\subseteq\mathbb R^d$ be open and $T_t$ be a $C^1$-diffeomorphism from $U$ onto an open subset of $\mathbb R^d$ for $t\in[0,\tau)$ with $T_0=\operatorname{id}_U$. ...
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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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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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How do I show If $f(B_r)$ is in a rectangle with side $Mr$ and $ \epsilon r$.

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a continuously differentiable function. Suppose that the Jacobian determinant $Df(0,0)$ is equal to zero. show that for $\epsilon >0$ there exist $...
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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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Show that there exists a neighborhood $U$ of $(0,1)$ such that the restriction $g:U \rightarrow g[U]$ is invertible

Let $g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be defined by $g(x,y)=(2ye^{2x},xe^y)$. Show that there exists a neighborhood $U$ of $(0,1)$ such that the restriction $g:U \rightarrow g[U]$ is ...
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Inverse function of $f(x)=e^x+x-1$.

Please find the inverse function of $f(x)=e^x+x-1$. I want to integrate the inverse functions of this but I am not able to find out any possible way to do so.
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Find affine function to approximate local inverse

Let $$ f(x,y) = (x^2 - y^2, 2xy) $$ then by the inverse function theorem $f$ is invertible locally at any point $(x,y) \neq (0,0)$ because $$ \det Df(x,y) = \det \begin{pmatrix} 2x & -2y \\ 2y &...
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Assumptions of the inverse mapping theorem

Inverse mapping theorem: Let $f: U \to \mathbb{R}^m$ be continuously differentiable, and $a \in U$. Suppose that $df_a$ is invertible, i.e. $det(J_f(a)) \neq 0$. Then $a$ has an open neighbourhood $V \...
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If $f:U\to\mathbb{R}^N$ is a $C^k$-submersion, $g:f(U)\to \mathbb{R}^M$ and $g\circ f :U\to\mathbb{R}^M$ is $C^k$ then $g$ is $C^k$

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

I have learned the inverse function theorem which ensures that a regular mapping (which has its inverse) is a (local) diffeomorphism. But I wonder whether a diffeomorphism is regular. I guess the ...
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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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Is monotonicity a necessary condition for the inverse function theorem?

A textbook I was reading, Introduction To Real Analysis By Robert G. Bartle (page 169) states that the inverse theorem is defined as: Let $I$ be an interval in $\mathbb{R}$ and let $f: I \...
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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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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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Unique Solution to 1st Order Autonomous ODE

Take the ODE $y'=F(y)$. Show it has a unique solution with initial condition $y(t_0) = y_0$ in a neighborhood of $t_0$ provided $F$ in continuous and $F(y_0) \neq 0$. I am trying to use the inverse ...
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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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Inverse Function Theorem for functions $f(x,y)$ and $\int\limits_0^1\frac{\partial f}{\partial x}(tx,y)dt$

I'm struggling with the following problem: Let $f\colon\mathbb{R}^2\to\mathbb{R}$ be a twice continuously differentiable function satisfying $$f(0,y)=0\mbox{ for all }y\in\mathbb{R}$$ (a) ...
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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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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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Determine the inverse g(x) of the function f(x)=1+1/x , stating its domain and range. Verify that f(g(x)) = g(f(x)) = x and that g’(f(x))= 1/(f’(x))

can anyone kindly show me how to do this question? Any help is appreciated. Thank you in advance.
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Application of Implicit Function Theorem to a function $\psi:U\subset\Bbb{R}^{2}\rightarrow \Bbb{R}^{4}$

Let $U$ be an open subset of $\Bbb{R}^{2}$ and \begin{align*}\psi:U&\rightarrow\Bbb{R}^{4}\\ x\mapsto &(\psi_1(x),\psi_2(x),\psi_3(x),\psi_4(x)) \end{align*} a $\mathcal C^1$ function. ...
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proving bi-Lipschitz function has an inverse that is lipschitz

How could I go about proving that a bi-Lipschitz function has an inverse that is a Lipschitz function. Definition 1: bi-Lipschitz function. Given metric spaces $(X,d_X)$, $(Y,d_Y)$, a function $f:X \...
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Assuming $A = Df(x_0)$ is invertible, prove that there exists $\mu > 0$ such that for all $x \in R^n$ $ ||Ax|| \geq \mu||x||$

My questions are Let $U \subset R^n$ be an open set, $f: U \rightarrow R^n$ be a $C^1(U)$ function and $x_0 \in U$. 1) Assuming $A = Df(x_0)$ is invertible, prove that there exists $\mu > 0$ such ...
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Set of points such that derivative is injective is an open set

Suppose that $A$ and $B$ are finite dimensional vector spaces. Let $U \subseteq A$ be open and $f:U \to B$ be $C^{\infty}$. Show that $\{a \in U : (Df)_a \text{ is injective}\}$ is open. I tried ...
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Showing that a function in $\Bbb{R}^{2}$ is a diffeomorphism.

Let $f:\Bbb{R}\rightarrow\Bbb{R}$ a function of class $C^{1}$ such that $|f'(t)|\leq k < 1\, \forall\, t\in\Bbb{R}$. Define $\phi:\Bbb{R}^{2}\rightarrow\Bbb{R}^2$ by $$\phi(x,y)=(x+f(y),y+f(x)). $$...
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Spivak, Calculus on Manifolds, Problem 2-37 (b)

2-37 (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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Show that non-singular is necessary for the Inverse Function Theorem

I am attempting the following problem: Show that the condition that $dF(a)$ be non-singular is necessary in the inverse function theorem by showing that if a function $F$ from a neighborhood of $a$ ...
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Invertible $ T \in L(R^n, R^n) $ such that there is no $S \in L(R^n,R^n)$ with $e^S = T.$

I am having a hard time coming up with an example such that an invertible $ T \in L(R^n, R^n)$ such that there is no $S \in L(R^n,R^n)$ with $e^S = T.$
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Find the differential of an inverse function of $G(u,v) = (u^4 - u + uv + v^2,\cos u + \sin v)$

I am attempting to solve this problem: Show that the system of equations $$ \begin{align} x &= u^4 - u + uv + v^2, \\ y &= \cos u + \sin v \end{align} $$ can be solved for $(u,v)$ as a ...
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Is a bijective smooth map of the closed disk with invertible differential a diffeomorphism?

Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Let $f:D \to D$ be a smooth bijective map with everywhere invertible differential. Is $f$ a diffeomorphism of the closed disk? Here is what ...
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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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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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46 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 ...
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Cauchy-Riemann Equations + Locally Invertible implies non-zero derivative

Suppose $C^1$ $f:\mathbb{R}^2 \to \mathbb{R}^2$ satisfies the Cuachy-Riemann equations $\frac{\partial f_1}{\partial x} = \frac{\partial f_2}{\partial y}, \frac{\partial f_1}{\partial y} = -\frac{\...
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Proof of Inverse Function Theorem, class k

I was studying the Inverse Function Theorem, and I found this proof on the internet: http://virtualmath1.stanford.edu/~andras/174A-2.pdf In the proof, there is this line about $C^k$ functions: If ...
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Can a diffeomorphism have critical points?

I'm trying to understand the step highlighted in this demonstration: from Zorich, Mathematical Analysis I, sec. 8.6, pag 510. What I know is that if $f'(x_0)$ is invertible ($f:G\subset\mathbb{R}^m\...
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Solve inverse trigonometric equation $\sin\left(\operatorname{cot^{-1}}(x + 1)\right) = \cos\left(\tan^{-1}x\right)$

If $\sin\left(\operatorname{cot^{-1}}(x + 1)\right) = \cos\left(\tan^{-1}x\right)$, then find the value of $x$. Please solve this question by using $\cos\left(\dfrac\pi2 - \theta\right) = \sin\theta$ ...
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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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