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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Domain of definition in the implicit function theorem

My question is whether the following variant of the implicit function theorem holds: Let $U \subseteq \mathbb{R}^k$, $V \subseteq \mathbb{R}^m$ and $W \subseteq \mathbb{R}^n$ be open sets and $F$ a $C^...
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Implicit function theorem and a system of equations - am I doing this right?

Show that the system of equations $$x^2 u^2+xzv+y^2=0$$ $$yzu+xyv^2-3x=0.$$ defines implicitely the functions $u=u(x, y, z)$ and $v=v(x, y, z)$ in a neighborhood of the point $(u, v, x, y, z)=(0, 1, 3,...
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Prove that $f(x;y;z)=(x+y+z;x-y-2xz)$ can be resolve for $(x,y)=\phi(z)$

Let be $f(x;y;z)=(x+y+z;x-y-2xz)$ a function, prove that it can be resolve for $(x,y)=\phi(z)$ close to $z=0$. Find explicitly $\phi(z)$ I'm very lost for this problem, first I thought it could be ...
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T-periodic soluctions and orbits in an ODE

Guys I need help in this exercise, I thought about using the function theorem implicit in item a) but I was not successful ... can someone give me a light on the solution? Be $F:\mathbb{R}\times \...
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Does the implicit function theorem works on non-open subsets?

I have been looking the implicit function theorem's statement in various websites to solve the equation $$F(a,b)=0,\quad F:\mathbb{R}^{n+1}_+\rightarrow\mathbb{R},\quad a\in\mathbb{R}^{n}_+,b\in\...
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Jacobian of optimization under constraints

I have the following equation: $$r = \arg\min_x f(x,y) \hspace{5mm}\text{ constrained on } g(x,z) = 0$$ I want to obtain the Jacobian of the dependence of $r$ on $y$ evaluated at one of the solutions. ...
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Question about the implicit function theorem being a sufficient but not necessary condition

Given a function which we would want to solve for some $U_\delta(x_0)$. If the IFT does not hold, for example $\frac{\partial F}{\partial x}$ is not invertible at our point of interest, how could we ...
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Showing that a function has a distinct solution (IFT)

I am given the the function $F(t, x, y) := \left( \begin{array}{} 3x^3 +2y^5-5t^7 \\ 2x^3-y^5-t^7\\ \end{array}\right) = \left( \begin{array}{} 0 \\ 0\\ \end{array}\right)$ where I should show that it ...
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Implicit Function Thm

How do I apply the implicit function theorm to $$f(x,y,z)= \begin{bmatrix}(x-1)^2+y^2+z^2\\ (x+1)^2+y^2+z^2\end{bmatrix}$$ at the point $(0,0,1)$??? I'm not sure if I'm doing it correctly. I've tried ...
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Equilibrium for Implicit Best Response Functions

I am dealing with a problem in economics, specifically game theory, where $n$ agents have a best response $x_i$ given by an implicit function, as described below. We have $ i \in \{ 1, 2, ..., n \}$, $...
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Proof of continuity of a map from product of two spaces to $\mathbb R^n.$

$\mathbf {The \ Problem \ is}:$ Given , $U$ is an open subset of $\mathbb R^{m+n}$ and $f : U \to \mathbb R^n$ be a $C^1$ map ,then show that the map $F : (p,v) (\in U ×\mathbb R^{m+n}) \mapsto Df_p (...
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Proving existence and uniqueness for a $3 \times 3$ nonlinear system of algebraic-trigonometric equations

I have a nonlinear $3 \times 3$ system of algebraic/trigonometric equations for which I'm trying to prove existence and uniqueness of a solution. Specifically, I'm looking at the system \begin{align} ...
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A stronger result about Implicit Function Theorem: Factorization $F(x,y)=G(x,y)(x-g(y))$

In page 213(63) of the paper Spectral properties of Schrödinger operators and scattering theory by Shmuel Agmon, he used a stronger result of the implicit function theorem, which I'll state below in ...
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Prove $\exists\delta>0\ni\forall f\in C([0,1])$ with sup $|f|<\delta\ \exists g\in C^1([0,1])$ such that $g'+g^{30}=f$ (Use Implicit function theorem)

Using Implicit function theorem show that that there exists $\delta>0$ such that For all $f\in C([0,1];\Bbb{R})$ with $\text{sup }|f|<\delta,\ \exists g\in C^1([0,1];\Bbb{R})$ such that $$g'(t)+...
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Showing the existence of $\phi$ in the proof of the Implicit function Theorem

The Implicit Function Theorem I am working with: Let $f \in C^1$ from an open set in $\mathbb{R}^m \times \mathbb{R}^n$ to $\mathbb{R}^m$. Let $(y_0, x_0)$ be a point in this open set such that $f(y_0,...
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implicit function theorem with domain of definition not an open set or half open set

The following is a question about the nature of the domain of definition when we wish to apply the Implicit Function Theorem. My question has to do with the nature of the domain of definition, say $ S ...
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Being $f(x,y)=e^{2x}+\sin(x+\ln(y+1))-1$, prove the existence of $g(x): ]-e,e[\to \mathbb R$ with $f(x,g(x))=0$

I have the function $f(x,y)=e^{2x}+\sin(x+\ln(y+1))-1$ and I need to prove the existence of a function $g(x): \; ]-e,e[ \; \to \mathbb{R}$ with $e>0$ so $f(x,g(x))=0$ for all $x \in\;]-e,e[$ and ...
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$D=\{(a,b,c,d,e)\in\mathbb{R}^5 \mid ax^4+bx^3+cx^2+dx+e=0\}$ Prove that ${(1,2,-4,3,-2)}\in \operatorname{int}(D)$

I'm having some troubles with it: $$D=\{(a,b,c,d,e)\in\mathbb{R}^5 \mid \exists x\in\mathbb{R} : ax^4+bx^3+cx^2+dx+e=0\}$$ Prove that ${(1,2,-4,3,-2)}\in \operatorname{int}(D)$ Might use some hints, ...
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Proving there exists a unique solution for every $x\in\Bbb{R}$

Say we have $f(x,y)=x^8+3x^4y^3+y^8x^{20}+y$. I have to show that for every $x\in\Bbb{R}$ there exists a unique solution to the equation $f(x,y)=0$. So if I understand the question correctly, I take a ...
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Implicit theorem in $\mathbb{R}^3$ there is $\xi:(t_0-\varepsilon,t_0+\varepsilon)\to\mathbb{R}$ such that $F(t,\xi(t),\xi'(t))=0$

Let $F:\mathcal{U\subset\mathbb{R}^3}\to \mathbb{R}$ of class $C'$, let $(t_0,x_0,y_0)\in \mathcal{U}$, such that $F(t_0,x_0,y_0)=0$ and $\partial_y F(t_0,x_0,y_0)\neq 0$. Prove that for all $\...
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Implicitly Differentiating a System of Matrix Equations

Setup Let $\mathbf a$ be an arbitrary $m\times 1$ vector, $\mathbf B$ be an arbitrary $n\times m$ matrix, with $m>n$, and $\mathbf C$ be a symmetric $m\times m$ matrix. The scalar $\lambda$ is also ...
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Initial value problems for $y'=\frac{x+y}{x-y}$

The solutions of the differential equation $y'=\frac{x+y}{x-y}$, are given implicitly by the relation $$\ln x = \arctan\frac{y}{x}-\frac{1}{2}\ln(1+\frac{y^2}{x^2})+c,\enspace c\in\Bbb{R}.$$ I'm ...
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Prove or disprove that equations $x^2+y^2+z^2=3$, $xy+tz=2$, $xz+ty+e^t=0$ can be solved of $t$ near $(x,y,z,t)=(-1,-2,1,0)$

Prove or disprove that equations $x^2+y^2+z^2=3$, $xy+tz=2$, $xz+ty+e^t=0$ can be solved of $t$ near $(x,y,z,t)=(-1,-2,1,0)$ I thought that I can apply the implicit function theorem, but I found that ...
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Implicit function theorem and derivative

The implicit function theorem (in the $2$ dimensional case): Let $F\colon D \subset \Bbb R^2 \to \Bbb R^2$ and let $(x_0, y_0)$ be an interior point of $D$ with $F(x_0, y_0) = 0$. Suppose both first ...
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Some confusion in implicit function theorem and chain rule

Let $g:\mathbb{R}^3 \to \mathbb{R}$ and $p_0=(x_0, y_0, z_0)$ a point which holds the equation $g(x-y, y-z, z-x)=0$. Find sufficient conditions for $z$ to be extractable as a function of $x,y$ in the ...
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Implicit function theorem - derivatives

Let us have two function $n(\alpha)$ and $s(\alpha)$ and a set of two implicit equations $F_1(\alpha, n(\alpha), s(\alpha))=0$ and $F_2(\alpha, n(\alpha), s(\alpha))=0$. In this paper I've been ...
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Application of the implicit mapping theorem

Show that the equation $x^4 + y^4 -2xy=0$ defines a function $\phi(x)=y$ in a neighborhood of $(1,1)$. Proof. Let $f(x,y)=x^4+y^4-2xy$. Clearly $f\in C^{\infty}$, and $f$ is continuously ...
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Proving a point is interior

Let $D = \left \{ (a,b,c,d,e) : ax^4+bx^3+cx^2+dx+e=0 \text{ has a real root} \right \}$ Show that (1,2,-4,3,-2) is an interior point in $D$. My only idea was trying to somehow use the implicit ...
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Continuity in the Mean Value Theorem and the Implicit Function Theorem

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $\mathcal{C}^2$ function of $x\in\mathbb{R}^n$. Then we can use the mean value theorem (MVT) to formulate the following equality: $$f(x_1)-f(x_2)=\frac{\...
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Branches of implicitly defined cubic curves

Consider the following cubic curve defined by: $$y^3 - 3y + x^3 = c$$ for $c$ a real parameter. The implicit function theorem guarantees three functions, $\phi_1$, $\phi_2$, and $\phi_3$ with domains $...
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bijective $C^{1}$-image

Let $F: \mathbb{R^{3}}\to \mathbb{ R}$ a $C^{1}$-image and assume that $(dF) (x, y, z)\neq 0$ as soon as $F (x, y, z) = 0$. Define $O= \{(x, y, z) ∈ \mathbb{R^{3}}| F (x, y, z) = 0\}$. Prove that ...
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Variation of the inverse Function theorem for multiple variables to find existence of a function $L$

I have the following equation in $L$ $$ f[L(a,b)-L(\tilde a,\tilde b)]+1/2 f''[L(a,b)-L(\tilde a,\tilde b)](b^2+\tilde b^2)=f(a-\tilde a) $$ Here $f$ is an at least odd smooth function, we can add ...
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Prove that there is an $n \times n$ matrix $B$ such that $A=B^{2}+B .$

Given a positive integer $n,$ prove that there is $\varepsilon>0$ such that for every $n \times n$ matrix $A$ with $|A|<\varepsilon$ (Hilbert-Schmidt norm), there is an $n \times n$ matrix $B$ ...
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Using the implicit function theorem to show a level surface is noncompact

I was curious about finding conditions for a level surface to be compact and came up with this idea. In particular, let $F(x,y,z): \mathbb{R}^3 \to \mathbb{R}$ be a continuously differentiable ...
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Application of Implicit Function Theorem to the triangle

Well, I observe the triangle with side-lengths $a, b, c$ and angle $\alpha$, the opposite angle to $a$. The implicit relationship between them should be The Law of Cosines : $a^2-b^2-c^2+2bc\cos \...
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Generalization of Implicit Function theorem

Let $f\colon \mathbb R^n \times \mathbb R^m \to \mathbb R^m$, $\mathbb R^n \times \mathbb R^m \ni (x, y) \mapsto f(x,y)$ be continuously differentiable. The implicit function theorem guarantees, under ...
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Apply change of variable $x=x(y)$ to integrate $f(x,y)dx$.

Let $f:\mathbb{C}\times\mathbb{C}\to\mathbb{C}$ integrable in a contour $C\subset\mathbb{C}$, $$\exists \ \int_C f(x,y)dx \ \in\mathbb{C},$$ such that $\forall x\in C, \ \forall y\in\mathbb{C}, \ \...
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How to write this implicit equation explicitly for parameter $p$

I have this equation which is implicit in the parameter $p$, is it possible to write it explicitly in $p$ for some boundary cases? This is the equation: $6e^\chi=e^{\frac{Q}{6}+\frac{1}{6}(4+\frac{1+p}...
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Exercise using the Implicit Function Theorem

I'm trying to do the following exercise: Let $f:[0,2]\to \mathbb{R}$ positive such that $\int_0^1 f(x)\,dx=\int_1^2 f(x)\,dx=1$. Prove that, for each $x\in[0,1]$, there is a unique number $g(x)\in[1,2]...
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Implicit equation for the image of immersion $(\cosh(t),\sinh(t))$

how can I discribe a implicit equation for the Image of $$ f(t)=(\cosh(t),\sinh(t)), $$ knowing that this is an injective immersion? This doesn't look that hard, but I can't proceed. Thanks in advance!...
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An alternate to implicit function theorem?

So, suppose we have an algebraic curve of form: $$ x^2 +y^2 =1$$ Now, this is actually a level set for a function $F(x,y)=x^2 +y^2$. Suppose we took its gradient, then it's gradient would be ...
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Implicit function question: Given that $u=f\big(x,y,g(x,y)\big)$, express $∂^2u/∂y^2$ and $∂^2u/∂x∂y$ using partial derivatives of $f$ and $g$.

this is my first time using StackExchange so I don't really know if anybody would help me out but here I go! The question I had while solving practice questions for the [Implicit Function Theorem] is ...
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Show that the equation $x^2 -ze^{x+y+z} = 0$ defines a surface in the neighborhood of the origin.

Show that the equation $x^2 -ze^{x+y+z} = 0$ defines a surface in the neighborhood of the origin. Letting $f(x,y,z) = x^2-ze^{x+y+z}$ and noting that $\nabla f(0,0,0) =(0,0,-1) \ne 0$. Particularly $\...
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Is my answer ok? (Exercise 6 on p.79 in “Analysis on Manifolds” by James R. Munkres.)

I am reading "Analysis on Manifolds" by James R. Munkres. There is the following exercise (exercise 6) on p.79 in this book. Let $f:\mathbb{R}^{k+n}\to\mathbb{R}^n$ be of class $C^1$; ...
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About the implicit function theorem. (Is there any possibility that the following statement holds for some $f$?)

The implicit function theorem is the following theorem: Let $A$ be open in $\mathbb{R}^{k+n}$; let $f:A\to\mathbb{R}^n$ be of class $C^r$. Write $f$ in the form $f(\mathbf{x},\mathbf{y})$ for $\...
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Maximum and minimum of implicit function with a parameter.

Consider an equation $\frac{1}{y} + a\log{y} = x$, where $a \in \mathbb{R}$ is a parameter. I want to find the value of $a$ such that maximum (minimum) of $y(x)$ is the largest (smallest). The idea is ...
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(b) If $F(x,y)=0$ for all $(x,y)$, find $D_1g$ and $D_2g$ in terms of the partials of $f$ “Analysis on Manifolds” by Munkres

I am reading "Analysis on Manifolds" by James R. Munkres. There is the following exercise 3 on p.63 in this book. Let $f:\mathbb{R}^3\to\mathbb{R}$ and $g:\mathbb{R}^2\to\mathbb{R}$ be ...
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Implicit function theorem and a tangent

Show that the equation $x^3 - 2x + 2y^3 + y^2 = 0$ defines a curve of the form ${(x, y) : x = \varphi(y)}$ in $B^2 ((0, 0), \delta)$ form some $\delta > 0$. Find the equation of the tangent at $(0,...
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Help Implicit funtions theorem question

Let \begin{cases} u_t(x,t) + u(x,t)u_x(x,t) = 0, & \text{for } x\in\mathbb{R},\; t > 0 \\ u(x,0) = h(x), & \text{for } x\in\mathbb{R}. \end{cases} Suppose that $h\in \mathcal C^2 (\mathbb{...
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There exists $G$ s.t. $f \circ G$ is constant(proof completing)

Problem: assume that $f\in C^1(\mathbb{R}^2,\mathbb{R})$, prove that there exists a continuous injection $G:\mathbb{R} \rightarrow \mathbb{R}^2$ s.t. $f \circ G$ is a constant function. I 've come up ...

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