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
823 questions
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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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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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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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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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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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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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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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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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Is there a useful notion of "quasilinear" or "semilinear" ordinary differential equations?
There is a very standard taxonomy of partial differential equations that classifies them as "linear", "semilinear", "quasilinear", or "totally non-linear" PDEs, ...
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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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Answers don't match when using the implicit function theorem
I'm struggling with a question regarding the implicit function theorem:
Show that $$ x^y+\sin{y}=1$$ defines $y$ as a function of $x$ in a neighbourhood of $(1,0)$, and find $y'(x)$.
The first part ...
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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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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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Solving equations using implicit differentiation and differential equations
I appreciate this title sounds wrong.
I have a concept MCQ quiz which I give to students. One question asks "What is a differential equation?" One of the choices says:
taking an implicitly ...
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Doubt regarding Implicit Function Theorem in Spivak
On page 42 of Calculus on Manifolds by Spivak, he presents the following proof of the Implicit function theorem:
Proof: Define $F:\mathbb{R}^{n+m}\to\mathbb{R}^{m+n}$ by $F(x,y) = \bigl(x,f(x,y)\bigr)$...
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ODE and Implicit Function Theorem
Is my approach right to this question?
Let $F: U \subset \mathbb{R}^3 \to \mathbb{R}$ be a $C^1$ function in a open set $U$, let $(t_0, x_0, y_0)$ be a point in $U$ such that $F(t_0, x_0, y_0) = 0$ ...
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application of Implicit function theorem to show existence of a solution of a differential equation
Can we deduce anything about existence of a solution of a differential equation using Implicit function theorem. I feel we can but I am unable to setup things to apply implicit function theorem.
Let's ...
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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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Reference for global implicit function theorem in any dimensions
In the book Advanced Calculus 2E by Patrick M. Fitzpatrick, I found the global implicit function theorem in 1 dimension:
Suppose that the function $f: \mathbb{R}^2 \to \mathbb{R}$ is continuously ...
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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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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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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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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{...
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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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