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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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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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 ...
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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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Edwards Corollary III 2.8 $df_{a}$ is 1-to-1 $\implies$ $f$ is 1-to-1 in a neighborhood of $a$: versus the Inverse Mapping Theorem

In C.H. Edwards's Advanced Calculus of Several Variables we find Corollary III 2.8 Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be $\mathcal{C}^{1}$ at $a$. If $df_{a}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ ...
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Showing a function in $\mathbb{R}^n$ is locally 1:1

Before applying the inverse function theorem, we must show that it is locally 1:1. I am confused by two things: 1) what exactly does it mean by "local"? How is it different from "global"? 2) What ...
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If $|f'(t)| \leq k < 1$ for all $t$ then $\varphi(x, y) = (x + f(y), y + f(x))$ is a diffeomorphism of $\mathbb{R}^2$ onto itself

Show that if $f$ is a $C^2$ map such that $|f'(t)| \leq k < 1$ for all $t \in \mathbb{R}$ then $\varphi(x, y) = (x + f(y), y + f(x))$ is a diffeomorphism of $\mathbb{R}^2$ onto itself. Note that ...
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How can I say a function $F:\mathbb{R}^n \longrightarrow \mathbb{R}^m$ $n<m$ is locally homeomorphism?? (Implicit function theorem) !!

My doubts are about an application of the Implicit Function Theorem: I do not understand how a function $F: \mathbb{R}^n \longrightarrow \mathbb{R}^m$, with $n<m$ (!!!) can locally be an ...
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Applications of Inverse Function Theorem

I am searching for applications of the Inverse Function Theorem for smooth maps: Inverse Function Theorem. Let $E\subset \mathbb{R}^n$ be an open subset and let $f\colon E \rightarrow \mathbb{R}^n$ ...
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Question applying Inverse trigonometry to show ${d\theta\over{dt}}={v\sin\theta\over L}$

There is a car moving along a straight road at the speed of $v$. There is a tree with the distance of $L(t)$ from the car. The angle the car is facing towards the tree is $\theta(t)$. Show that the ...
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deducing matrix square root from inverse function theorem

This is a problem from Ted Shifrin's book on Mutivariable Mathematics, I have asked a similar question earlier, but as I am revisiting this problem after 3 years, I realized my old question and answer ...
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Showing inverse of multivariate holomorphic map is holomorphic using real analysis techniques

I'm grading for a graduate real analysis course and looking through the text, trying to do some problems before the term starts so I can stay ahead of things. I found the following exercise in the ...
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A function $f$ with Jacobian nonzero and satisfying $|f(x)-x|<1/3$ has a root. [duplicate]

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a $C^1$ function such that $\det(Df(x)) \neq 0$ and $|f(x) - x| < \frac{1}{3}$ for any $x \in \overline{B}_1 (0)$. Prove that there is $p \in \overline{B}...
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Generalization of inverse function theorem to allow unequal dimensions?

My book is An Introduction to Manifolds by Loring W. Tu. The last sentence here says being an immersion or a submersion at p is equivalent to the maximality of rk $[\frac{\partial f^i}{\partial ...
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I’m missing a step in a proof of the Holomorphic Inverse Function Theorem.

I’m looking at the proof of Theorem $1.7$ (Inverse Function Theorem), chapter VI of Lang’s Complex Analysis (which can be found on page $182$ in the fourth edition of the book), and I’m stuck with one ...
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Inverse restriction for polar coordinates in $\mathbb{R^2}$

I'm having trouble understanding why they set the domain and range ($U$ and $V$) as such. Because normally, if it's polar coordinates in one-dimension, you would set $0<\theta<\pi/2$ such that ...
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Proving a set is open in context of proof of a step in local Inverse Function Theorem

The context for this question is my answer to this question on the inverse function theorem. I'll try to replicate as much of the necessary information as possible: Let $E$ and $F$ be real Banach ...
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set of points of local invertibility of function.

Let $f:\mathbb{R^2}\to\mathbb{R^2}$ $\text{be a function given by}$ $f(x,y)=(x^3+3xy^2-15x-12xy,x+y)$. let $S$={$(x,y)\in \mathbb{R^2}:f\; \text{is locally invertible at}(x,y)$}.Then 1).$S=\mathbb{R^...
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Inverse function theorem.

Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$, $C^{1}$, such that $det(f'(x))\neq 0,\ \forall\ x\in \mathbb{R}^{n}$ and $f^{-1}(K)$ is a compact set for all $K\subset \mathbb{R}^{n}$ compact. Prove ...
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“Counterexample” for the Inverse function theorem

In a lecture we stated the theorem as follows: Let $\Omega\subseteq\mathbb{R}^n$ be an open set and $f:\Omega\to\mathbb{R}^n$ a $\mathscr{C}^1(\Omega)$ function. If $|J_f(a)|\ne0$ for some $a\in\...
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Is a differentiable (but not $C^1$) function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with invertible derivative everywhere an open map?

Is a differentiable (but not $C^1$) function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with invertible derivative everywhere an open map? I know that if we assume the function is $C^1$, then this is ...
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Proof of the proposition (Morse function in $\mathbb{C}$)

I was trying to prove the Proposition asked by OP. I was thinking to apply some version of inverse function theorem as proved in the Morse Lemma. But I am not able to do it. Can you please give me ...
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Doubt from Spivak's proof of inverse function theorem.

why must $y^i-f^i(x)=0$ for all $i$?
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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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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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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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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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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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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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170 views

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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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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54 views

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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327 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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181 views

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 ...