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 $\phi$ 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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If $f(x)=\frac{3x-1}{x^3-27}$, compute $[f^{-1}]'(27)$

If $\displaystyle f(x)=\frac{3x-1}{x^3-27}$, compute $[f^{-1}]'(27)$. I tried to compute the value $x_0$ such that $f(x_0)=27$ and then I could apply the inverse function theorem, but I can't find the ...
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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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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 $\\f:\ U\to\ R^N$ 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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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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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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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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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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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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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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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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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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How to begin to prove that f is injective in R^n?

Given that $F:R^n\rightarrow R^n$ is $C^1$ on $R^n$ s.t. the jacobian matrix $Df(p)$ is invertible for all $p \in R^n$, prove that f must be injective. To show this in $R\rightarrow R$, I applied ...
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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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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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Verify procedure to determine density of $S = S_0 \exp(X)$ where $p_X(x) = \frac{\lambda}{2}\exp(-\lambda |x |)$

Determine the density of the random variable $S = S_0 \exp(X)$ where $p_X(x) = \frac{\lambda}{2}\exp(-\lambda |x |)$ i.e. $X$ is a Laplace distribution with parameter $\lambda$, and $S_0$ is a ...
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Does the inverse function theorem have an analogue for Boolean algebra?

This is a more focused version of a question (unresolved) about finding the inverse of a system of Boolean equations. For a vector space over the reals, the inverse function theorem says that the ...
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Exercise 4.4.5 Introduction to Real Analysis by Jiri Lebl / inverse function theorem

Exercise 4.4.5: (required 4.3) Show that if in the inverse function theorem $f$ has $k$ continuous derivatives, then the inverse function $g$ also has $k$ continuous derivatives. We know that $$g'(x) ...
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Prove that near the orign $0 \in \mathbb R^{4}$, its solutions from the graph of a continuously differentiable function $ G: R^{2} \rightarrow R^{2}$.

Consider the system of equations in the variables $u,v,s,t$: $(uv)^{4}+ (u+s)^{3}+t=0$ $sin(uv)+e^{v}+t^{2}-1=0$ Prove that near the origin $0 \in \mathbb R^{4}$, its solutions from the graph of a ...
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77 views

Inverse Function Theorem and open sets

Let $\gamma: \mathbb{R} \to \mathbb{R}^2$, $\gamma$ continuously differentiable with $\gamma'(t) \neq 0$, $\forall t \in \mathbb{R}$. Then, for all $t_1 \in \mathbb{R}$, there is a $\epsilon>0$ ...
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1answer
29 views

Proof of the inverse function theorem

I am studying the proof of the inverse function theorem (see picture) in multivariable analysis. I understand the proof, but I do not understand why we can assume without loss of generality that $x_0=...
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Question about proof of the inverse function theorem

I am studying this proof (see picture) of the implicit function theorem in multivariable analysis. I understand the proof, but I do not understand why we can assume without loss of generality that $...
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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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422 views

What are some real world applications of the Inverse Function Theorem?

I am teaching an AP calc AB course and just covered the Inverse Function Theorem. While I know some of the applications in higher math, I could not come up with any real world applications to use for ...
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Function from $\mathbb {R}$ to $\mathbb{R}^2$ - real analysis problem

Function $f(x)$ maps $(-a,a)$ to $\mathbb{R}^2$ and $f \in C^1$ (continously differentiable). Is it possible that image of every open interval $(-b,b)$ (for $b<a$ of course) contains neighborhood ...