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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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How to apply diffeomorphism theorem to obtain solvability of differential equation?

I've been reading about local and global diffeomorphism theorems recently and I started to wonder how shall I apply these theorems in order to solve nonlinear differential equations? Assume that we ...
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to show the existence of an inverse function

enter image description here this is an assignment problem in calculus on manifolds class the part (a) is little bit straightforward than part (b) so i have completed the proof of (a) but (b) was ...
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A function $f\in C^1(\mathbb{R})$ with $f'(x)\not=0$ for all $x$ with no global inverse.

I don't think one exists, but the author of my text says it should. Here's why I don't such an $f$ exists. Suppose $f$ didn't have a global inverse, clearly $f$ is not injective for else we could ...
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How to find $\frac{\partial f}{\partial x}(a,b)$ and $\frac{\partial f}{\partial y}(a,b)$ of a implicit function?

I need help with this problem: For each of the following functions $F:\mathbb{R}^3\rightarrow\mathbb{R}$, show that the equation $F(x,y,z)=0$ defines implicitly a countinuously differentiable ...
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Showing that a set cannot be expressed as the local graph of a $C^1-$function over the $x-$axis or the $y-$axis near the origin.

Consider the function $h(x,y)=(x−y^2)(x−3y^2)$, $(x,y)\in \mathbb{R}^2$. Show that the set $\{(x,y) | h(x,y) = 0\}$ cannot be expressed as the local graph of a $C^1-$function over the $x-$axis or the ...
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How to prove that $F(x,y)=x+y^2+\sin(xy)$ defines implicitly a continuously differentiable function?

I need help with this problem: Let $F(x,y)=x+y^2+\sin(xy)$. Proce that in a sufficiently small neighbourhood of $(0,0)$ the equation $F(x,y)=0$ defines implicitly a continuously differentiable ...
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$f(3t^3+2,et^2)=(3,6)$ for all $t\in\mathbb{R}$. Prove: $D_f(2,1)$ not invertible

Let $f\in C^1[\mathbb{R}^2 , \mathbb{R}^2]$ satisfying: $f(3t^3+2,e^{t^2})=(3,6)$ for all $t\in\mathbb{R}$. Prove: $D_f(2,1)$ not invertible. My try: we define $g(t)=(3t^3+2,e^{t^2})$. Then, $g\...
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Some difficulties in understanding the proof of the inverse function theorem.

The theorem and a part of its proof is given below: but it is not clear for me how he proved (d), could anyone explain this for me please? Also in this last part of the proof: why ...
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Why is continuous differentiability necessary for Inverse Function Theorem?

Inverse Function Theorem. Let $f: \mathbb{R}^{n} \to \mathbb{R}^{n}$ be a $C^{1}$ function. If $\det Df_{a} \neq 0$, there is open sets $U, V$ such that $f: U \to V$ is a diffeomorphism $C^{1}$ ($a \...
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An application of implicit fuction theorem

Let $f:\mathbb{R}\to \mathbb{R}$ a continuous and positive function such that $\int_0^{1}f(t)dt=5$. Show that there is an interval $J=[0,a]$ such that for all $x\in J$ there is an unique $g(x)\in [0,1]...
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Can determinant of Jacobian essentially bound from below implies locally bi-Lipschitz?

Let $F:\mathbb R^n\to\mathbb R^n$ be a Lipschitz map, so $DF:\mathbb R^n\to\mathbb R^{n\times n}$ is $L^\infty$ map. Suppose there is a $c>0$ such that $|\det DF(x)|>c$ for almost every $x\in\...
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Inverse Function Theorem from Rn to Rn

Inverse function theorem statement: Let $f: U\subset \mathbb{R}^n \rightarrow \mathbb{R}^n$ be $C^1$, $\underline{x_0}\in U,\ \ [Df(\underline{x_0})]$ invertible. Then $\exists W\in U, \ W$ open ...
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Proof that the inverse of an analytic function is analytic which uses only real analysis.

I would like to prove the following result: Let $f:R\to R$ be such that $f(x)=\sum\limits_{k=0}^\infty a_k(x-c)^k$ for all $x$ in some open set $O\subset R$. Suppose that $f'(c)\ne 0$. Then there ...
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real analytic inverse function theorem

I have a two part question about real analyticity and whether I have a an error in reasoning. Suppose I have a multivariate holomorphic mapping $f \colon U \to \mathbb{C}$, where $U$ is an open set in ...
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Higher order Inverse Function Theorem

Consider a smooth function $F : \Bbb{R}^n \to \Bbb{R}^m$ where $n \geq m$ are positive integers. Consider a curve $\alpha : [0,1 ) \to F^{-1}(0)$ such that $\alpha(0) = x \in F^{-1}(0)$. Then $F(\...
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Implicit function theorem implies inverse function theorem proof

I believe this will be a very long answer if anyone tries to write the full proof or anything so I'll specify which specific parts I am having trouble with to save people's time. I want prove three ...
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Injective derivative in neighborhood of a function

Let $f,g:U\to\mathbb{R}^n$ differentiable in $U\subset\mathbb{R}^m$ an open set. We write $|f-g|_1\leq \delta$ in $X$ when $|f(x)-g(x)|\leq \delta$ and $|f'(x)-g'(x)|\leq \delta$, $\forall x \in X$. ...
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If $f$ is $C^1$ s. t. $\Vert f(x) - f(y) \Vert \geq k \Vert x - y \Vert$, then $f$ is a diffeomorphism of $\mathbb{R}^{n}$

Let $f:\mathbb{R}^{n} \to \mathbb{R}^{n}$ be a function of class $C^1$ and suppose that there is $k>0$ such that $$\Vert f(x) - f(y) \Vert \geq k \Vert x - y \Vert$$ for any $x,y \in \mathbb{R}^...
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Inverse function theorem to prove onto

Let $f: \mathbb R^2 \to \mathbb R^2$ be $C^1$, $D_f(x)$ is invertible everywhere, and $\lim_{|x| \to \infty}|f(x)| = \infty$ Show that $\min_{x \in \mathbb R^2}|f(x)-a|$ exists, and that $f$ is onto. ...
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A function with a non-zero derivative, with an inverse function that has no derivative.

While studying calculus, I encountered the following statement: "Given a function $f(x)$ with $f'(x_0)\neq 0$, such that $f$ has an inverse in some neighborhood of $x_0$, and such that $f$ is ...
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Differential of inverse function to a tubular neighborhood

Suppose $S$ is a (regular) compact differentiable surface embedded in $\mathbb{R}^3$ so that tubular neighborhoods exist. Consider the diffeomorphism to one of them: $$F:S\times(-\epsilon,\epsilon)\...
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Inverse Function Theorem (Domain, one-to-one, application) [closed]

$f(x) = e^x + \ln(x)$ Q1) Find the Domain Q2) Show that $f$ is one-to-one Q3) Find $(f^{-1})(e)$ Q4) Find the derivative of the inverse function at $e$. I am pretty much at a loss for how to ...
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Misuse of inverse function theorem

Let $U=\{\binom{u}{v}\in \mathbb{R}^2|0<v<u\}$. Let $f:U\to\mathbb{R}^2, f(u,v)=(u+v,uv)$. Show that $f$ has a global reverse function, find $g=f^{-1}$ and its domain. Not a valid solution: For ...
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1answer
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Given that derivative of a function is bounded. Prove surjectivity

Given a differentiable function $f:\mathbf{R} \to \mathbf{R},$ such that $|f'(x)| < c < 1$. Consider a function $g:\mathbf{R}^2 \to \mathbf{R}^2$, such that $g(x,y) = (x+f(y),y+f(x))$. Prove ...
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Error in Statement of Inverse Function Theorem?

$\boldsymbol{6.8.2.}$ Theorem (Inverse Function Theorem). Let $G$ be an open subset of $\mathbb{R}^p$ and assume $f:G\to\mathbb{R}^p$ is a continuously differentiable function. If $a\in G$ with $b=f(a)...
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Fulfilling conditions of Inverse function(?)

Let's assume that $$h(g(t))=t$$ What conditions ae needed to say that $$g(h(t))=t$$ Is satisfied too? (Provided that $h$ and $g$ are continuous and derivatable, but not knowing whether they have ...
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Reference request for worked-out examples (solutions) on multivariable calculus (in particular inverse and implicit function theorem)

Reference Request : Is there any book or notes available where I can find a lot of examples / worked-out solutions of problems on multivariable calculus (the topics must include Inverse and Implicit ...
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Given a function $f:\mathbb{R}^2 \to \mathbb{R}^2$, to find derivative of its inverse at a given point

Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a function given by $$f\left(x,y\right)=\left(x^2-y^2,2xy\right),\,\, \left(x,y\right) \in \mathbb{R}^2$$ Question : Compute $Df^{-1}\left(0,1\right)$. I have ...
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Given a function $f:\mathbb{R}^2 \to \mathbb{R}^2$, to find its inverse near a given point

Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a function given by $$f\left(x,y\right)=\left(x-y,xy\right),\,\, \left(x,y\right) \in \mathbb{R}^2$$ Question : What is the inverse of $f$ near the point $\...
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1answer
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Sufficient conditions for a continuos function $f$ to be $C^1$

Hellow, I need some ideas for this problem: Let $f:\mathbb{R}^m\to \mathbb{R}^n$ a continuous function and let $g_1,\dots, g_n:\mathbb{R}^n\to \mathbb{R}$ fuctions in $C^1(\mathbb{R}^n)$ such that: $...
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1answer
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How to make squaring method of finding an inverse function to be invertible?

I've been trying to find an inverse of this function $$f(x) = e^{-\left(\displaystyle \frac{x}{\sqrt{1-x^2}}\right) \displaystyle \pi }$$ These are the approaches First approach using squaring ...
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1answer
161 views

Inverse Function Theorem: Proving Global Invertibility.

My question states: Prove that the following coordinate transformation is invertible everywhere, at all values of $(x, y)$ . $$u = \arctan(x - y)$$ $$v = \sinh(3x) + 2\sinh(y)$$ That is x and y ...
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Inverse function theorem consequence?

This is theorem 9.24 in Rudin, known as the inverse function theorem: Suppose $f$ is a continuously differentiable map of an open set $E \subseteq \mathbb{R}^n$ into $\mathbb{R}^n$, $f'(a)$ is ...
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Is a map with invertible differential a diffeomorphism onto its image near a boundary point?

Let $M,N$ be smooth manifolds of the same dimension, and suppose $M$ has a non-empty boundary. Let $f:M \to N$ be a smooth map, and suppose that $df_p$ is invertible for some $p \in \partial M$. ...
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Piecewise smooth function , Jacobian and locally invertible?

Say $f$ is a piecewise smooth function that is $f(x,y;\chi)=f_{1}(x,y;\chi)$ if $y\leq g(x;\chi)$ and $f(x,y;\chi)=f_{2}(x,y;\chi)$ if $y \geq g(x;\chi)$ where $g$ is a $C^2$ function and $\chi$ a ...
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James R. Munkres “Analysis on Manifolds” p.60 Theorem 7.4.

James R. Munkres "Analysis on Manifolds" p.60 Theorem 7.4. Let $A$ be open in $\mathbb{R}^n$; let $f : A \rightarrow \mathbb{R}^n$; let $f(a) = b$. Suppose that $g$ maps a neighborhood of $b$...
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Determinant does not vanish on an open subset.

Let $f : \mathbb{R}^2 \to \mathbb{R}^2$ be a $C^1$ map such that $f^{-1}(y)$ is a finite set for all $y \in \mathbb{R}^2$. Show that the determinant det $df(x)$ of the Jacobi matrix of $f$ cannot ...
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Use the implicit function theorem to prove that $f=f^{-1}$.

I've had some problems to prove this proposition: Let $f:\mathbb{R}^{n}\to \mathbb{R}^{n}$ a function of class $C^1$ such that $(f\circ f)(x_0)=x_0$ for some $x_0\in \mathbb{R}^{n}$ and $Df(f(x))Df(x)...
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show $\alpha^{-1}$ is continuous if $\alpha \in C^r$ is bijective, and $D\alpha$ is nonsingular

In the book of Analysis on Manifolds by Munkres, at page 196, it is given that However, if $\alpha$ is of class $C^r$, one-to-one and onto, and $D\alpha$ is nonsingular for $x \in U$, isn' $\alpha^{-...
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inverse function theorem and mapping of surface

The discussion of regular mappings in Section 7 of Chapter 1 translates easily to the case of a mapping of surfaces $F: M \rightarrow N$. $F$ is regular provided all of its derivative maps $F_{*p}: ...
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Explicit construction of Hamilton's counterexample for Implicit Function Theorem in Frechet Space

In his famous paper on Nash-Moser Theory, Hamilton mentioned in Counterexample 5.5.2 that a rotation $f: \theta\mapsto\theta + 2\pi/k$ on a circle can be "pushed a little bit" so that the new ...
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1answer
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Inverse function theorem regarding solutions to a cubic equation

Let $c_1, c_2, c_3 \in \mathbb{R}$ be such that the equation $X^3-c_1X^2+c_2X-c_3=0$ has got three distinct solutions $s_1, s_2, s_3 \in \mathbb{R}$. Show that there exists a neighborhood $U \subset ...
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1answer
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Diffeomorphism preserves dimension?

I want to apply the inverse function theorem to a general mapping that is a diffeomorphism. The inverse function theorem, as stated in my textbook, requires that the dimension of the range and domain ...
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Reading the proof of the Inverse Function Theorem

I am reading a proof of the Inverse Function Theorem from here In section 5 on page 13, Newton's approximation is used to derive the equation: \begin{align} g_{k + 1}(\mathbf{y}) = g_{k}(\mathbf{y}) ...
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3answers
173 views

Show function $f(x,y)=(x^2-y^2,2xy)$ is $1$-$1$ by Inverse Function Theorem

I'm trying to prove the problem below - which comes from Munkres' "Analysis on Manifolds" book in the section on the Inverse function theorem. Since its in the chapter on the Inverse Function Theorem ...
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1answer
55 views

Inverse Function Theorem in Immersions.

Let $\varphi: U \to \mathbb{R}^{n}$ of class $C^{k}$ ($k\geq 1$) in the open $U \subset \mathbb{R}^{m}$. If $a \in U$ is such that $\varphi'(a): \mathbb{R}^{m} \to \mathbb{R}^{n}$ is injective, then ...
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1answer
118 views

Inverse Function Theorem and Hessians Matrices

Let $f: V \to \mathbb{R}$ of class $C^{2}$ and $b \in V$ a critical point of $f$. If $\varphi: U \to V$ is a diffeomorphism $C^{2}$ with $\varphi(a) = b$, then the hessian of $f$ in the point $b$ and ...
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1answer
23 views

Is $f$ globally invertible on $R^2$?

$f(x,y) = (x^2y + 2xy, xy^2 + xy)$ Q. Is $f$ globally invertible on $R^2$? I found that $f_1(-2,1) = f_1(0,0)$. But, $f_2(-2,1) \not = f_2(-2,1).$ Is the case for $f_1$ enough to show $f$ is not ...
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prove the existence of negative $n$ th root and compute the derivative.

Let $n\in N$ be even. Prove that every $x>0$ has a unique negative $n$th root. That is, there exists a negative number $y$ such that $y^n=x.$ Compute the derivative of the function $g(x)=y.$ ...