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 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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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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Doubt about the formulation of the implicit function theorem (and its application to manifolds)
Notation
Let $O \subseteq \mathbb{R}^{n+m}$ be open and denote $\mathbf{x} = \left( x_1,\ldots,x_{n}\right) \in \mathbb{R}^{n} , \mathbf{y} = \left( y_1,\ldots,y_{n}\right) \in \mathbb{R}^{m}$. In ...
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Properties of a set given by an implicit equation
First of all, english is not my first language so my post may contain some grammatical errors.
I'm a math undergraduate and I'm trying to prove a theorem for which I need to study the zeroes of a ...
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Understanding Implicit Function Theorem's proof proved by Pugh (smoothness of $y=g(x)$ part)
The following statements are from Pugh's book - Real Mathematical Analysis :
Fix attention on
a point $(x_0, y_0)$ ∈ U and write $f(x_0, y_0) = z_0.$ Our goal is to solve the equation $f(x, y) = z_0$ ...
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Find scalar that minimises spectral radius of a matrix
I have a matrix $A$ that is a quadratic function of a real scalar $\beta$ and real constant matrices $B,C,D$:
$$
A = B + \beta C + \beta^2 D
$$
I want to find the value of $\beta$ that minimises the ...
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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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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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I suspect that there might be another error in the proof of Theorem 2-13 in "Calculus on Manifolds" by Michael Spivak. Is my suspicion correct?
I am reading "Calculus on Manifolds" by Michael Spivak.
The following proof of theorem 2-13 is known to contain an error.
And it is said that the error is a well-known one. (But ...
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Theorem 2-13 in "Calculus on Manifolds" by Michael Spivak. Why does $f \circ F (x^1, \dots, x^n) = f(x,f(x,y)) = (x^{n-p+1}, \dots, x^n)$ hold?
I am reading "Calculus on Manifolds" by Michael Spivak.
I cannot understand this question and its answer about the following theorem.
2-13 Theorem. Let $f: \mathbb{R}^n \to \mathbb{R}^p$ be ...
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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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we may take to be of the form $A\times B$, such that $F:A\times B\to W$ has a differentiable inverse $h:W\to A\times B$. Spivak Calculus on Manifolds.
I am reading "Calculus on Manifolds" by Michael Spivak.
Spivak wrote:
which we may take to be of the form $A\times B$, such that $F:A\times B\to W$ has a differentiable inverse $h:W\to A\...
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Am I confused or is there something somewhat circular? ("Multivariable Mathematics" by Theodore Shifrin.)
I am reading "Multivariable Mathematics" by Theodore Shifrin.
I'm not sure if I can clearly express the vague sense of discomfort I'm feeling, but I will try to explain it below.
On p.188
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The implicit function theorem in $R^m \times R^n$: proving $C^1$ regularity
Let $F: A \subseteq \mathbb{R}^m \times R^n \rightarrow \mathbb{R}^n$ be a $C^1$ function, where $A$ is an open set. Let $(x_0, y_0)$ (where $x$, $y$ denote $m$- and $n$-dimensional vectors, ...
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