# 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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### 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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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 $... 1answer 292 views ### Inverse Function Theorem proof: f is injective I am trying to prove the Inverse Function Theorem from the Implicit Function Theorem for Banach spaces. My attempt so far is as follows: Let$f:\mathbf{X}\to \mathbf{Y}$be a$\mathcal{C}^k$function ... 2answers 39 views ### Show that there exists a neighborhood$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]$is ... 2answers 41 views ### Inverse function of$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. 1answer 38 views ### 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 &... 1answer 41 views ### 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 \... 1answer 154 views ### 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)} ... 0answers 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 gis C^k In my attempt I know that D_{f(p_0)} is onto, p_0 \in U and ... 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 ... 1answer 20 views ### 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 ... 0answers 33 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_{\... 1answer 34 views ### 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 \... 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 ... 0answers 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 ... 1answer 31 views ### 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 ... 0answers 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 ... 1answer 291 views ### 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) ... 0answers 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'... 0answers 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 ... 1answer 27 views ### 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. 1answer 68 views ### 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. ... 1answer 108 views ### 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 \... 2answers 37 views ### 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 ... 2answers 41 views ### 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 ... 1answer 40 views ### 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)). $$... 1answer 218 views ### Spivak, Calculus on Manifolds, Problem 2-37 (b) 2-37 (a) Let f:\mathbb{R^2}\to\mathbb{R} be a continuously differentiable function. Show that f is not\mbox{ 1-1}. Hint: If, for example, D_1f(x,y)\neq0 for all (x,y) in some open set A... 1answer 18 views ### Show that non-singular is necessary for the Inverse Function Theorem I am attempting the following problem: Show that the condition that dF(a) be non-singular is necessary in the inverse function theorem by showing that if a function F from a neighborhood of a ... 1answer 19 views ### Invertible T \in L(R^n, R^n) such that there is no S \in L(R^n,R^n) with e^S = T. I am having a hard time coming up with an example such that an invertible T \in L(R^n, R^n) such that there is no S \in L(R^n,R^n) with e^S = T. 1answer 32 views ### Find the differential of an inverse function of G(u,v) = (u^4 - u + uv + v^2,\cos u + \sin v) I am attempting to solve this problem: Show that the system of equations$$ \begin{align} x &= u^4 - u + uv + v^2, \\ y &= \cos u + \sin v \end{align} $$can be solved for (u,v) as a ... 2answers 60 views ### Is a bijective smooth map of the closed disk with invertible differential a diffeomorphism? Let D \subseteq \mathbb{R}^2 be the closed unit disk. Let f:D \to D be a smooth bijective map with everywhere invertible differential. Is f a diffeomorphism of the closed disk? Here is what ... 1answer 129 views ### Extension of Inverse Function Theorem from \mathbb{R} to \mathbb{R^n} Consider the Inverse Function Theorem in \mathbb{R}: Let O ∈\mathbb{R} be open for f:O → \mathbb{R}. If f is continuously differentiable, and for a x_0 ∈O, f'\left(x_0\right) \ne 0. Then ... 0answers 22 views ### Confusions on functions on open intervals Assume f \in C^1(\mathbb{R}^2) such that f(0,0)=0, f_x(0,0) \ne −1, and f_y(0,0) \ne 0. Assume g : \mathbb{R}^2 → \mathbb{R} such that$$g(x,y) = f(f(x,y),y).$Let$t$be a function defined ... 1answer 46 views ### Inverse Function Theorem to solve functions Prove that for$(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$close ... 2answers 65 views ### Cauchy-Riemann Equations + Locally Invertible implies non-zero derivative Suppose$C^1f:\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{\...
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 ...