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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How do I prove that there is a neighbourhood $U$ of the orign in $\mathbb{R}^2$ and, $|y_{2}-y_{1}|\geq \epsilon|x_{2}-x_{1}|$.
Let V be a neighborhood of the origin in $\mathbb{R}^2$, and $f: V \rightarrow \mathbb{R}$ be continuously differentiable. Assume that $f(0,0)=0$ and $f(x,y)\geq -3x+4y$ for $(x,y) \in V$. Prove that ...
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Implicit differentiation conditions
Let $f:\mathbb{R}^2 \to \mathbb{R}$ be $C^1$ such that $f(0,0)=0$. Find conditions over $\frac{\partial f }{\partial x}(0,0)$ and $\frac{\partial f }{\partial y}(0,0)$ so that equation $f(x+f(x,y), y -...
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How do I use implicit function theorem to get second derivative of a function?
Consider
$x^2 + y^2 = r^2$
acc. to implicit function theorem,
$ y' = \frac{-x}{y}$
Now how would I find y'' by using implicit theorem again?
and what would be my multivariable function? , would ...
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Conclusion of implicit function theorem in local lie group
My teacher defines local lie group as follows:
Definition:
A complex n-dimensional local Lie group $G$ in the neighborhood $V ⊂ \mathbb{C^n}$
is determined by a function $\phi:\mathbb{C^n}
×\mathbb{C^...
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1answer
67 views
Uniqueness of O.D.E
Prove that the o.d.e. $(\frac{3}{2}\sqrt{|y|}+1+x^2)\frac{dy}{dx}+2xy=0$ has unique local solutions with $y(x_0) = y_0$ for any $x_0$ and $y_0$.
Does the existence and uniqueness theorem for o.d.e's ...
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Using implicit function theorem on Banach spaces to solve equations
I have seen it mentioned in the literature that one can often deal with a (quasilinear) non-linear PDE or a system of non-linear PDEs by perturbing to a linearised system and then finding solutions ...
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$\vec{y}=\vec{a}\times \vec{x}$. is it possible to define $\vec{x}$ as function of $\vec{y}$
$let\quad\vec{a} \in \mathbb{R^3}$ is it possible to define $\vec{x}$ as function of $\vec{y}$? $$\vec{y}=\vec{a}\times \vec{x}$$
So according to the solution the answer is not and I would like to ...
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Implicit function theorem, differentiable path
I need to show that the equations: $$x^2y+xy^2+t^2=1$$ $$x^2+y^2 -2yt=0$$ is difinding a differentiable path $\vec{\gamma}=({x}_{(t)},{y}_{(t)})$ at the point $(x,y)=(-1,1)$.
after that I should find ...
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Continously differentiable function is not injective
I learnt about the implicit function theorem and had to prove the following:
Let $F \in C^1(\mathbb{R}^2, \mathbb{R}).$ Show with the implicit function theorem that $F$ is not injective.
Proof:
...
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Problem in understanding proof of Implicit Function Theorem from Calculus of Manifolds
From the proof of the Implicit Function Theorem given in Spivak's "Calculus of manifolds" page 42, for $f: R^n \times R^m \to R^m$ which is continuously differentiable we define a $F:R^n \times R^m \...
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2answers
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Multivariable implicit function theorem proof
I am trying to understand the proof of the implicit function theorem for multivariable functions. If I have a function $F(x,y,z) = 0$ with the assumption that $z = f(x,y)$ and we want to find $\frac{\...
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How to explain that the intersection of a sphere and a cylinder cannot be parametrized wrt. the coordinates x, y and z.
I have $L = \{ (x, y, z) \in \mathbb R^3 \, | \,x^2+y^2+z^2 = 4, (x-1)^2+y^2=1 \}$, and I want to show that the curve cannot be parametrized as a smooth curve in the form of a graph in a neighborhood ...
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Exercise on the implicit function theorem.
Prove that the equation $x^2+y^2+\sin y=0$ defines a unique function $y=f(x)$ in a neighbourhood of $(0,0)$. Prove also that in $x=0$ there's a maxima for $f$.
I tried to do this exercise in two ...
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Using the implicit function theorem to prove that a function attains a certain value
Here' how I tried to do it but failed:
I don't see what the implicit function theorem has to do with this (this exercise is after the section on the implicit function theorem), but anyway, this is ...
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Implicit function solution for $ y=f(x)$
What does it mean with find the implicit solution to $y=f(x)?$ I know I'm supposed to check conditions and then use IVT to say $y$ can be solved uniquely for $x$. But what do they mean with solution, ...
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Maybe Implicit Function Theorem?!
Let $(x,y)$ be a curve on $R^2$ defined by the equation $f(x,y)=0$. Suppose $f$ is differentiable and $f_y$ never vanishes. Meanwhile, assume further that all the second-order partial derivatives of $...
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1answer
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How to find $F(u,v)$?
Let $u=f(x,y)$, $v=g(x,y)$, $f,g \in R^2(\Omega)$ and $\operatorname{rank}\left(\dfrac{D(u,v)}{D(x,y)}\right)\equiv 1$. For all $\vec x\in \Omega$, does there always exist a function $F(u,v)$ that $...
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Implicit function theorem for two equations
I have a question regarding the implicit function theorem for two equations, which can be arranged for a single equation. For example,
$F_{1}(x,y,a,b)-c=0 $,
$F_{2}(x,y,a,b)-d=0$
According to my ...
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A function that is not invertible but has an implicit form locally
I'm trying to find out the difference between the implicit function theorem and the inverse function theorem.
One of the obstacles of my understanding, is that I can't find a function that it ...
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1answer
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Application of Implicit Function Theorem to a function $\psi:U\subset\Bbb{R}^{2}\rightarrow \Bbb{R}^{4}$
Let $U$ be an open subset of $\Bbb{R}^{2}$ and
\begin{align*}\psi:U&\rightarrow\Bbb{R}^{4}\\
x\mapsto &(\psi_1(x),\psi_2(x),\psi_3(x),\psi_4(x))
\end{align*}
a $\mathcal C^1$ function.
...
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1answer
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$\forall x,y \in \mathbb{R^n}: x,y \in U => \left\lVert f(x) - f(y) \right\rVert \geq c \left\lVert x - y \right\rVert$ globally invertible
Let $f:U \subset \mathbb{R^n} \to \mathbb{R}^n$ be totally differentiable and there exists a constant $c > 0$, so that
$$\forall x,y \in \mathbb{R^n}: x,y \in U => \left\lVert f(x) - f(y) \...
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1answer
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Implicit function Theorem - I have an example which I think is wrong
I am trying to understand the implicit function theorem. Please consider the following system of equations which I
believe does not have a solution.
\begin{align*}
e^{-x} - y &= 0 \\
e^{3x} + 10 -...
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1answer
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the proof of implicit function theorem (Terence Tao)
Implicit function theorem: Let $E$ be an open subset of $\mathbb{R}^n$, let $f : E \to \mathbb{R}$ be continuously differentiable, and let $y = ( y_1, ... , y_n)$ be a point in $E$ such that $f(y) = 0$...
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2answers
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Weird subspace of a banach space, is this also banach?
Trying to solve an exercise I was wondering if it would be possible to use the implicit function theorem on a tricky space in order to immediately obtain some extra properties on the solutions. ...
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1answer
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Hessian matrix of the function defined with Implicit function theorem
Let $x=(x_1,...,x_n) \in \mathbb{R}^n, y\in \mathbb{R}$ and let $F(x,y)=F(x_1,...,x_n,y) \in C^2(\mathbb{R}^{n+1})$.
Suppose we have all the hypothesis for the existence of the function $f(x)=y$ ...
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How my teacher is deriving this equation?
I'm struggling in the resolution of this exercise. When he derives $f$, I don't understand the notation. What is $f$x1 supposed to mean? I think it means $\frac{\partial f1}{\partial x} $, being $f$1 ...
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2answers
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Prove that $xz^3-yz=x$ is a function of $z$ at $(1,0,1)$
Prove that $xz^3-yz=x$ is a function of z at (1,0,1).
Find $\frac{\partial \varphi}{\partial x}(1,0)$
Let the function be $f(x,y,z)=xz^3-yz-x$, by the implicit function theorem I found that:
$...
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1answer
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Find the tangent and the normal to an implicit function
Find the implicit function for $xy^2+4x^2y-12=0$ at $(1,2)$. Then find
the tangent and the normal to $y=\varphi(x)$ at $x=1$.
Let $y=\varphi(x)$ (1)
Now: $x\varphi(x)^2+4x^2\varphi(x)-12=0$ $\...
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1answer
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Continuity of the feasible set of a linear problem with linear constraints
Let ${\Omega}(t) \in \mathbb{R}^{m}$ be the set:
$ {\Omega}(t)= \{ \boldsymbol{x} \in \mathbb{R}^{n} : \boldsymbol{A}(t) \boldsymbol{x} =\boldsymbol{b}, c_1<x_i<c_2 \} $
where $\boldsymbol{...
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1answer
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Using the implicit function theorem to a system of equations.
Prove that the following system:
\begin{align}F(x,y,z)&=2x^2+y^2+1-z^2=0\\
G(x,y,z)&=2x^2+2y^2-z^2=0\end{align}
can be solved for $z$ and $y$ as functions of $x$.
Furthermore, ...
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1answer
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Converse of Implicit Function Theorem
The Implicit Function Theorem tells us that if we have a smooth function $f: \mathbb{R}^{n+k} \to \mathbb{R}^{k}$ such that $f(a) = 0$ for some $a \in \mathbb{R}^{n+k}$, then we can find points close ...
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Unit Circle and Implicit Function Theorem
Let $S^{1}$ be the unit circle in $\mathbb{R}^{2}$, which we can see as $f^{-1}(0)$, where
$$
f(x,y) = x^{2} + y^{2} -1
$$
The differential of this function at any point is $$
Df(x,y) =
\begin{...
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2answers
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Doubts regarding singularity and regularity of algebraic curves.
We say that a curve $F(x,y)=0$ is regular at $(x_0,y_0)$ if in a nbd of that point,the curve can be written explicitly as a continuously differentiable function of the form $y=f(x)$ or $x=g(y)$.We ...
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Implicit function, change of coordinates
How can you apply the implicit function theorem if the non vanishing partial derivative is not the last one?
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A function can be described locally as a graph of a $C^1$ function (Implicit Function Theorem)
Let S be the surface in R3 defined by the equation $z^2y^3+x^2y=2$.
Use the Implicit Function Theorem to determine near which points S can be
described locally as the graph of a $C^1$ function $z = f(...
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Task from Application of Implicit Function Theorem
Problem is: Show there is unique differentiable function $f \colon \mathbb{R}^3 \setminus \{ (0, 0, 0)\} \to \mathbb{R}$ for which $$\frac{x + y + z}{x^2 + y^2 + z^2} + f(x, y, z)^3 = e^{-(\frac{x + y ...
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Results and applications of Implicit Function theorem
It is well know that implicit function theorem is one of the central theorems in calculus and analysis. Currently, I am reading Steven G. Krantz's book on Implicit function theorem. In the book, some ...
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1answer
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Intermediate value Theorem for Curve in $\mathbb{R}^2$
I am stuck on this problem from a previous exam.
Let $f:\mathbb{R}^2 \to \mathbb{R}$ a $C^1$ function.
Assume:
$C=\{f(x,y)=0\}\subset \mathbb{R}^2$ is a path-connected curve.
$\forall (x,y) \in C :...
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Is my proof correct? (a little proposition related to the implicit function theorem)
I am reading "A Course in Analysis vol. 4" by Kazuo Matsuzaka.
The author uses the following proposition in his proof of the implicit function theorem without a proof.
I tried to prove the ...
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A Version of The Implicit Function Theorem
I want to ask if this version of the "implicit function theorem" holds.
Let $f:S^1\times S^1\to\mathbb{R}$ be continuous, for every $x\in S^1$, there exists $y\in S^1$, not necessarily unique, such ...
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1answer
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Using an equation in intercepts, write an implicit equation of the plane
Using an equation in intercepts, write an implicit equation of the plane which intersects the Cartesian coordinate axes X, Y and Z at the three points with coordinates P1=(2, 0, 0), P2=(0, 4, 0) and ...
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1answer
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Using the implicit function theorem
I am struggling with the following exercise:
Do the two equations
$$x+y-\sin(z) = 0 \text{ and }e^x-x-y^3=1$$
define two functions $y(x), z(x)$ in a neighbourhood of $x=0$ such that $y(0)=z(0)=...
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1answer
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Construct a parameterization of the inverse image of a regular value.
Let $f: U \subset \Bbb{R}^n \to \Bbb{R}^m$ be a $C^k$ function. Suppose that $c \in \Bbb{R}^n$ is a regular value of $f$. Show that
(a) $M = f^{-1}(c)$ is locally the graph of a $C^k$ function.
...
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1answer
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System of equations, implicit function theorem
Given the nonlinear system of equation:
$$2x+y^3+u^3-v^2=1 \\ x^2+3y-u^2-v^3=0$$ for $z=(x,y,z,v)\in \mathbb{R}^4$ and with a solution $z_0=(1,0,0,1)\in \mathbb{R}^4$, one can conclude that by the ...
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1answer
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Question regarding global invertible functions
Let $U \subseteq \mathbb{R}^n$ be open and let $f: U \to \mathbb{R}^n$ be continuous differentiable and one-to-one. Let $Df(x)$ be invertible for all $x \in U$.
I have already shown that the set $V:=...
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Confirmation of differentiability of fixed point with respect to a parameter
$a_{1}=\int_{(r_{1},r_{2})\in A_{1}(a_{2},\beta)} f(r_{1},r_{2})dr_{1}dr_{2}$,
$a_{2}=\int_{(r_{1},r_{2})\in A_{2}(a_{1},\beta)} f(r_{1},r_{2})dr_{1}dr_{2}$,
where $a_{1},a_{2}\in [0,1]$, $A_{1}(a_{2},...
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1answer
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About Implicit Function Theorem in “Principles of Mathematical Analysis” by Walter Rudin. I think $U$ is not necessary.
I am reading "Principles of Mathematical Analysis" by Walter Rudin.
I have a question about the implicit function theorem in this book.
Is the open set $U$ in the following statement so ...
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2answers
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Implicit function theorem - Tangent planes to an Ellipsoid
I have an ellipsoid, whose equation is $ \frac{x^2}{3}+\frac{y^2}{9}+\frac{z^2}{4} = 1$ and a plane whose equation is $ x + y + z = 0$.
The goal of this question is to find the tangent planes to the ...
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3answers
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Using the implicit function theorem in a system of equations.
I want to see for which points the system of equations $$\begin{cases}x^2+y^2+z^2=16\\2x-y+z=4\end{cases}$$
define $(y,z)$ as an implicit function of $x$, and then calculate $(y'(x),z'(x))$.
I know I ...
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Not sure how to eliminate variable from expression
I have an expression I'm trying to simplify, following along in a certain book, and I can't see how to do it. Here's what I've done so far.
I have a smooth real-valued function $F(x,y,z)$, which by ...
