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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Proving the set of zeros of a $F: \mathbb{R}^{n} \to \mathbb{R}^{m}$, $C^{1}$ func is a $n-m$ dimensional manifold with the Implicit function theorem
This is a follow up question to this question I asked earlier today, in which I asked whether the function $g$ we get from the implicit function theorem is a homeomorphism. As pointed out in the ...
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How to find a (local) $C^1$ bijection between two curves?
The following is not an exercise but something I was just curious about.
Consider the unit circle with radius $r$, $x^2+y^2=r$ and consider the level set of $x^2+ y^2+\frac{x^{11}}{5}=r$
Question:
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Implicit function theorem: when a level set intersects itself
Below is a figure of a Bernoulli lemniscate, the set of all $(x,y)$ in the level set where
$$
f(x,y):= (x^2 + y^2)^2 - x^2 + y^2 = 0.
$$
At $(x,y) = (1,0)$ and $(x,y)=(-1,0)$ the curve has vertical ...
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About a step in my proof of the implicit function theorem
I'm proving the implicit function theorem, here the exact statement:
Theorem (Implicit function theorem). Let $V, W$ be complete normed spaces, $\Omega \subseteq V \times W$ open and $f : \Omega \to ...
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Show that $f:\mathbb{R}^2\rightarrow \mathbb{R}$ that is $C^{1}$ is not injective. [duplicate]
I want to use the implicit function theorem to solve.
Attempt: Let $z=f(0,0)$. By $C^{1}$, $D_{(0,0)}f=\big(\frac{\partial f}{\partial x}(0,0),\frac{\partial f}{\partial y}(0,0)\big)$ is continuous.
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implicit function extrema
Show that there exists a $C^{1}$ function $\phi: R^{2} \to R$ such that $\{(x,y,z)\in R^{3}:x+y+xy+2z+sin(x+y+z)=0 \}=\{(x,y,\phi(x,y)): (x,y) \in R^{2} \}$.Then find local extrema of function $\phi$....
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A possible counterexample to the statement on p.340 in "An Introduction to Manifolds Second Edition" by Loring W. Tu.
I am reading "An Introduction to Manifolds Second Edition" by Loring W. Tu.
On p.340, the author wrote as follows:
On a smooth curve $f(x,y)=0$ in $\mathbb{R}^2$,
$y$ can be expressed as a ...
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Using the implicit function theorem to determine sign of derivative globally
I have a two functions $f(s,b)$ and $g(s,b)$ with the following properties for $b \in (\underline{b}, \overline{b})$:
$f(s,b)$ and $g(s,b)$ are continuously differentiable in $b$ and $s$
$f(s,b)$ and ...
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Value of algebraic generating function at $z=1$
We are given $p(z,u)$, a nonzero polynomial with real coefficients. Suppose we know that: (a) there is a generating function $g(z)$ that solves $p(z,g(z))=0$; (b) $g(z)$ has nonnegative coefficients; (...
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The author just wrote "Suppose that $f(V)=W$ holds". ("Introduction to Analysis II" by Mitsuo Sugiura.)
I am reading "Introduction to Analysis II" by Mitsuo Sugiura.
Theorem 2.1 (the inverse function theorem II):
Let $ U $ be an open subset of $ \mathbb{R}^n $, and let $ f: U \to \mathbb{R}^n ...
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Implicit function theorem open vs closed codomain for the implicit function $g$
In Wikipedia's formulation for the implicit function theorem see here, then codomain of $g$, i.e, the implicit function that comes out of the implicit function theorem is taken to be open. In the ...
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Application of Implicit Function Theorem: Why is this fixed point unique?
Theorem on pages 102 and 103, Devaney An Introduction to Chaotic
Dynamical Systems, 3d ed:
Let $f_\lambda$ be a one-parameter family of functions and
suppose that $f_{\lambda_0}(x_0)=x_0$ and $f'_{\...
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Application of implicit function theorem to get a diffrentiable structure on a level set.
We only covered the tangent bundle untill now and got this theorem to prove:
Let $m, n \in \mathbb{N}, U \subseteq \mathbb{R}^{m+n}$ be open, $f: U \to \mathbb{R}^n$ be a smooth map and $0 \in f(U)$. ...
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A consequente of the Implicit Function Theorem in $\mathbb{R}^2$ [closed]
For each $n \in \mathbb{N}$, let $f_n : \mathbb{R}^2 \to \mathbb{R}$ such that $f_n(0,0) = 0$ and $\frac{\partial f_n}{\partial t}(0,0) \neq 0$. By Implicit Function Theorem there exists $\delta_n >...
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If a combination of generators of the ideal of a smooth submanifold vanish, then the coefficients can be taken to be antisymmetric
Execuse me the word salad title, but I have no idea how to say this concisely. Suppose that $M$ is a smooth $m$ dimensional manifold, and $\Sigma$ is a smooth submanifold of codimension $0<n<m$. ...
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Boundedness of an Implicit Function (Algebraic Riccati Equation)
Consider the algebraic Riccati Equation which is dependent on some parameter $$ a = [a_{min},a_{max}] \in \cal A$$ $$ P(a)A(a)+A^T(a)P(a)-P(a)BR^{-1}B^TP(a)+Q = 0. $$ It is assumed that $(A(a),B)$ is ...
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Can one expect an implicit solution of a system of two equations in two variables?
Given two regular functions $f(x,y)$ and $g(x,y)$, consider the following system:
\begin{align} &f(x,y)=0 \\
& g(x,y)=0 \end{align}
over $\bar{\mathbb{Q}_p} \times \bar{\mathbb{Q}_p}$, where $\...
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The Use of the Inverse Function Theorem in the Proofs of the Implicit Function Theorem and The Rank Theorem
I've been wondering if there is a corollary of the Inverse Function Theorem (InFT) used often, but implicitly, in proofs of other theorems. $\def\RR{\mathbb{R}} \def\f{\varphi}$ I include the outline ...
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A global implicit function theorem on a minimisation problem
Let $g(x,y)\in C^1(\mathbb{R}^n\times\mathbb{R}^n,\mathbb{R})$ such that the least eigenvalue of $\nabla_{yy} g(x,y)$ is always greater than $1$, $\nabla_{yy} g(x,y)$ is invertible and $\nabla_{yy} g(...
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What does this ratio of derivatives represent, geometrically?
Suppose I have an arbitrary smooth real-valued function $F(x,y,z)$ with the property that all the first-order partials of $F$ are strictly positive everywhere.
The criterion of St-Robert says that $F$ ...
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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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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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Rudin PMA 9.28. Why is $\phi(y)'k =(g(y)' k, k)$?
I am revising The implicit function theorem from Rudin's PMA as I forgot the entire proof of this theorem and I couldn't remember or understand why $\phi(y)'k =(g(y)' k, k)$ I didn't encounter 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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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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