# 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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### 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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### 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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### 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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