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Questions tagged [implicit-function-theorem]

The implicit function theorem gives sufficient conditions to solve a given equation for one or more of the variables as functions of the remaining variables. The basic form of the theorem is that of an existence theorem. However, the contraction mapping proof of the theorem provides an error estimate for a sequence of approximating maps. Sometimes it is also termed the implicit mapping theorem. See http://en.wikipedia.org/wiki/Implicit_function_theorem

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Theorem implicit function

I have the following excercise: $f,g: \mathbb R^3 \to \mathbb R$ with: $f(x,y,z)=z^2-2y-xz$ $g(x,y,z)=zy+x^2$ a) Show that in an open environment of $(1,1,-1)$ there exist functions $h_1(z)$ and $h_2(...
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What is the condition for a manifold defined by a smooth implicit function to be a closed manifold? [closed]

We have an implicit function which describes a manifold surface, and I want to know the conditions for the implicit function so that the manifold defined by the conditions is a closed manifold. My ...
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Using implicit function theorem to find functions

I am working on a practice problem for a real analysis homework and could use some hints/help on its parts based on my current attempts and progress. Thank you in advance. The problem states: Given a ...
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Claim: Related Rates does not actually (ever!) involve implicit differentiation. [closed]

Claim: "Related Rates" does not involve implicit differentiation. Picky though it may seem, we ought to clear up the language on this. My question is... do you agree with this: "...
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What does this ratio of derivatives represent, geometrically?

I have a smooth real-valued function $F(u,v,w)$. In the neighborhood of a particular solution, I am able to find the unique implicitly-defined function $\widehat{w}(u,v)$ such that $F(u,v,\widehat{w}(...
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implicit function theorem to determine minimum of a function

Consider a function $F(x,y)$. We are given the following assumptions, for $y_0 < y_1$: convexity: $$\partial^2 F(x,y) / \partial y^2 > 0$$ initially decreasing: $$\partial F(x,y)/\partial y \...
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Condition on derivatives

I am working with a "well-behaved" optimization problem of the form: \begin{equation*} \max_{x} f( g_{1}( x) ,g_{2}( x) ,g_{3}( x) ,\mathbf{y}) \end{equation*} where $\displaystyle f:\mathbb{...
Weierstraß Ramirez's user avatar
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Deriative of algebraic implicit function is algebraic

Let $P(x,y)$ be a real polynomial with $P(0,0)=0$, and assume that $\partial_y P(x,y)$ is nonzero over $[-1,1]^2$. Then the equation $P(x,y)=0$ defines an implicit function $y=f(x)$ near $0$. In this ...
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Is the implicit function unique?

Does the equation $$xy^2+xz^3+\ln z=0$$ define the unique implicit function $z=g(x,y)$ in the neighborhood of $(0,1)$? If yes, compute $\dfrac{dz}{dx}(0,1)$ and $\dfrac{dz}{dy}(0,1).$ My attempt: Let ...
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Generic Intersection of Submanifolds

This is the geometric version of this linear algebra question. Given submanifolds $N_1\cdots N_n$ of $M$ and $p\in M$, show that the map induced by inclusions $f:\bigoplus_{i}T_pN_i\to(T_pM)^n/{\Delta}...
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Confusion in applying the Implicit function theorem

Consider the following equations $$\begin{cases} 2(x^2+y^2)-z^2=0\\ x+y+z-2=0\end{cases}$$Prove that the above system of equations defines a unique function $\phi: z\mapsto (x(z),y(z))$, from a ...
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Apply the implicit function theorem of the ratio $c_{t+1}/c_t$ that itselfs depend on the level $c_t$

I'm trying to compute the derivative $\frac{d c_{t+1}/c_t}{d (1+r_{t+1})}$ of this function: $$\frac{c_{t+1}}{c_{t}} = \left( \beta (1+r_{t+1}) +\gamma \frac{(s-c_{t}+z) ^{-\Sigma }}{c_{t+1}^{-\sigma}}...
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Relation between local defining functions for the boundary.

Definition $:$ Let $G \subseteq \mathbb C^n$ be a domain. The boundary $\partial G$ of $G$ is said to be smooth at $z_0 \in \partial G$ if there is an open neighbourhood $U = U(z_0) \subseteq \mathbb ...
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Find the equation of the tangent to a system using implicit function theorem

Here is my system of equations: $C: \begin{cases}x^2 + y^2 +z^2 = 14\\ x^3+y^3+z^3=36 \end{cases}$ Firstly I managed to show that for all $a \in C$, the implicit function theorem applies to express $...
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Question about the inverse function and the implicit function theorems.

The theorems in question are and A consecuence of (9.24) is and a consecuence of (9.28) is the following highlighted text I'm having trouble at how he arrived a these conclusions
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Conditions for the implicit function theorem being satisfied giving rise to $n-1$ dimensional manifold

I am trying to understand an example from Thirring's Classical Mathematical Physics, 2nd ed., p. 14. I want to understand how the condition on $M$ satisfies the condition for the implicit function ...
Apoorv Potnis's user avatar
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Implicit function theorem for eigenvalues: why simple eigenvalue matters?

I was reading about how the implicit function theorem may be used to express eigenvalues of real symmetric matrices as functionals of the matrix on a neighborhood. I got stuck in equation (8), page ...
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2 votes
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Solve nonlinear system of equations and show it has infinite solutions

So we have this system of nonlinear equations \begin{align*} \sin(x+u) - e^y + 1 = 0\\ x^2 + y + e^u = 1 \end{align*} and we want to show that it has infinitely many solutions $(x,y,u)$. I tried ...
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Can we take gradient of a curve?

Consider the case of planar curves in $\mathbb{R}^2$. They can be described by a function $f(x,y) = 0$. For example, a circle can be described by $x^2+y^2=1$. We can take the gradient of this function ...
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Weaker version of Implicit Function Theorem

Let $f: U \rightarrow \mathbb{R}$ be continuous on an open set $U \subseteq \mathbb{R}^2$, differentiable at a point $(a,b) \in U$, where $\partial_{y} f(a,b) > 0$ and $f(a,b) = c$. Prove that ...
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Regarding Implicit Function Theorem

I'm currently trying to understand the following equation. For $h(x) = n^{-1}\sum_{i=1}^pa_i/(a_i+x), a_i > 0, x > 0,\ n,p \in \mathbb N$, consider a equation: $$1 = \lambda/x + h(x)$$ and let ...
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Another question about implicit function theorem

Someone can help or in this question about implicit function theorem? Let $\Omega$ A bounded domain wich satisfies a uniform interior sphere condiciona. The authors says that - we may suppose, ...
Cézar Bezerra's user avatar
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What does "they depend differentiably on" mean?

Let $P(x) = a_{n}x^n + a_{n - 1}x^{n - 1} + \cdots + a_{0}$ be a polynomial with real coefficients. I have to prove that the simple roots of $P(x)$ depend differentiably on the coefficients of $P(x)$ ...
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Implicit function theorem for non $C^1$ mappings

I know that the inverse function theorem can be proved for differentiable mappings (not $C^1$) by requiring that $Df(x)$ has everywhere maximum rank (here is the reference https://terrytao.wordpress....
Lorenzo Vanni's user avatar
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Multi-variate cross-partial derivative of a inverse function

I have an invertible mapping $y=f(x,\theta)$ where $x,y\in\mathbb{R}^K$ with a scalar parameter $\theta\in\mathbb R$. Consider its inverse $x=g(y,\theta)$. I'm interested in the matrix $\partial^2g/\...
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Comparative static when optimization problems yield implicit functions

A worker's optimization problem is: \begin{aligned} \max_{e, s} \quad & v=\frac{(\lambda + i)U_{emp}+(1-e)U_{unemp}}{i(1-e+\lambda+i)}\\ \textrm{s.t.} \quad & \lambda = \frac{n(1-e)\left(\frac{...
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For $F:R^{n} \to R^{m}, m>n$, with local inverse $G$ at $x$ such that $G(F(x)) = x$, is $DG$ a left inverse of $DF$? Why/Why not?

Assuming both functions are differentiable in some neighboorhood, under which conditions, if any, is the Jacobian of $G$ the left inverse of the Jacobian of $F$, ie $(JG)(JF) = Id^{nxn}$? Since we're ...
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Implicit functions equality

Suppose that $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ is such that $f(x,y,z) = 0$ for some $x,y,z$. Then, I am asked to show that, if each variable $x,y,z$ can be defined as an implicit function of ...
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Sufficient conditions for Implicit Function Theorem for a composite function

Let $A: \mathbb{R^2} \to \mathbb{R^2}$ be a linear map and consider $f:\mathbb{R^2} \to \mathbb{R}$ such that $\nabla f(x,y) \neq (0,0); \forall (x,y) \in \mathbb{R^2}$. Determine sufficient ...
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Implicit function Theorem in a neighborhood of $0 \in \mathbb{R}$

Show that there exists a neighbourhood $U$ of $0$ in $\mathbb{R}$ and an unique $C^{\infty}$ function $g:U \to \mathbb{R}$ such that $g(0)=e$; and $\forall x \in U$,$x^2e^{g(x)}+g(x)^2e^x=1$. ...
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Is this version of Implicit Function Theorem correct?

I have come across this variant of the implicit function theorem in the book "Topics in Nonlinear Functional Analysis" by Ralph A. Artino. Theorem 2.7.2 (Implicit Function Theorem) Let $X, ...
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Using an implicit function in taylor polynomial

Let $f$ be the function defined on $R^2$ by $f(x, y) = x^5+y^3-x^2+2x-3y.$ (a) Show that $f$ is of class $C^\infty$ on $R^2$ and compute its gradient at a point $(x, y)\in R^2.$ (b) Show that, in a ...
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Lipschitz continuity of implicit functions

I am struggling on the following problem and I couldn't find a reference to answer that question. Let $F$ be a function of two variables such that $F\in\mathscr{C}^{1, \infty}(\mathbb{R}_+\times\...
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Overdetermined nonlinear system of equations with measure zero set of solutions

Consider an overdetermined system of nonlinear equations of the following form: $\pi_m(x)+g_m(y_1,\dots,y_N)=\pi_m(\tilde x)+g_m(\tilde y_1,\dots,\tilde y_N)$, where $m=1,\dots,M$ and $x,\tilde x, y_n,...
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Implicit function theorem / Implicit selections when Jacobian not invertible

I saw the attached result in the book by Dontchev and Rockafellar. It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...
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Implicit Function Theorem and cross second-order derivative

I have that $x^{*}(w,z)$ and $y^{*}(w,z)$ is the implicit solution to a the system $F(x^{*}, y^{*},w,z) = 0$ and $H(x^{*}, y^{*},w,z) = 0$. Using the implicit function theorem, I can prove that $\frac{...
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About Implicit Function Theorem and Lagrange Multipliers

I am studying the meaning of the multiplier in Lagrange Multiplier Method and got the following question. I tried it myself, but I am not sure if it is correct. I would really appreciate it if someone ...
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Necessary and Sufficient Conditions for Existing an Envelope for a Parametric Family of Implicit Surfaces

Let $F(x, y, z, \lambda)=0$ be a parametric family of implicit surfaces. Sometimes the envelope of the family exists as another surface, but at other times it may degenerate to a curve or a point, or ...
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Weak version of the implicit function theorem

If we are given a $C^1$ map $F:\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{m}$ and a point $(x_{0},y_{0})\in\mathbb{R}^{n}\times\mathbb{R}^{m}$ such that $F(x_{0},y_{0})=0$, the implicit function theorem ...
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Implicit function theorem on dense subset

Let $U \subset \mathbb{R}^n$ be open and we are given a differentiable function $$\Psi: U \times \mathbb{R}^m \rightarrow \mathbb{R}^m, \quad (x, A) \mapsto \Psi(x,A)$$ with $m>n$. We further ...
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Proof of Simpson's Paradox

I am studying Implicit Function Theorem and its application in Simpson's Paradox. I got the following problem. I tried it myself, but not sure if my answer is correct. I would really appreciate it if ...
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Show that $F(t) = (\cos t, \sin t)$ is locally one-to-one but not globally one-to-one.

I am studying Implicit Function Theorem and Inverse Function Theorem. The problem I want to ask is: Show that $F(t) = (\cos t, \sin t)$ is locally one-to-one but not globally one-to-one. I have two ...
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Overlap in Implicit Function Theorem

Consider $f_1(t,x_1,\ldots,x_n),\ldots, f_n(t, x_1,\ldots, x_n)$ for complex analytic functions $f_i$ around $\vec{a}=(t', a_1, \ldots, a_n)$ and $\vec{b}=(t^*, b_1,\ldots, b_n)$, such that $f_i(\vec{...
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Can every smooth manifold be written as the intersection of smooth hypersurfaces? [duplicate]

If $\vec{f}\in C^{\infty}(\mathbb{R}^D, \mathbb{R}^K)$, $K=D-d$, such that the Jacobian, $D\vec{f}(\vec{x})$ of $\vec{f}$ has full-rank (i.e. $=K$) on $M=\vec{f}^{-1}(\vec{0})$, then $M$ is a smooth $...
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Implicit function theorem by only one partial derivative is continuous.

Question: Let $f \colon \mathbb{R}^2 \to \mathbb{R}$ be a continuous function so that $\frac{\partial f}{\partial y}$ exists and is continuous on $\mathbb{R}^2$. Let $(x_0, y_0) \in \mathbb{R}^2 $ be ...
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Series, sum uniform convergence

Show that the series $\sum_{n=1}^{\infty}(e^{\frac{x}{n}}-1-\frac{x}{n})$ converges uniformly in $\left[-A,A\right]$ for any $A>0$.Show also that the sum of the series is a function with ...
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Implicit Inverse function Theorems on $\mathbb R^2$ and injective functions

I have the following question on the applications of Implicit Inverse function Theorems on $\mathbb R^2$ and injective functions: Let $U\subset\mathbb{R}^{2}$ be an open set containing $(0,0)$ and ...
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Inverse implicit function theorem

Let $f:\mathbb{R}^{2}\to \mathbb{R}$ be a twice continuously differentiable function such that $f(0,0)=0$ and $f_{y}(0,0)\ne 0$. By the implicit function theorem, there exists an $\epsilon >0$ and ...
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Generalization of implicit function theorem for non-reduced fibres with smooth reduction?

Suppose some power series $f_1, \dotsc, f_k \in \mathcal O_0 = \mathbb C\{x_1, \dotsc, x_n\}$ define a holomorpic map $$f: \mathbb D^n_{\epsilon} \to \mathbb C^k,$$ where $\mathbb D^n_{\epsilon}$ ...
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4 votes
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(Frechet) Differentiability of Implicit function in Banach spaces

I'm looking at the classical implicit function theorem in Banach spaces. So $X,Y,Z$ are Banach spaces and $F: U_{x_0}\times V_{y_0} \to Z$ continuous and continuously differentiable with respect to y. ...
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