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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Question about the inverse function and the implicit function theorems.

The theorems in question are and A consecuence of (9.24) is and a consecuence of (9.28) is the following highlighted text I'm having trouble at how he arrived a these conclusions
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Conditions for the implicit function theorem being satisfied giving rise to $n-1$ dimensional manifold

I am trying to understand an example from Thirring's Classical Mathematical Physics, 2nd ed., p. 14. I want to understand how the condition on $M$ satisfies the condition for the implicit function ...
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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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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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Can we take gradient of a curve?

Consider the case of planar curves in $\mathbb{R}^2$. They can be described by a function $f(x,y) = 0$. For example, a circle can be described by $x^2+y^2=1$. We can take the gradient of this function ...
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Weaker version of Implicit Function Theorem

Let $f: U \rightarrow \mathbb{R}$ be continuous on an open set $U \subseteq \mathbb{R}^2$, differentiable at a point $(a,b) \in U$, where $\partial_{y} f(a,b) > 0$ and $f(a,b) = c$. Prove that ...
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Regarding Implicit Function Theorem

I'm currently trying to understand the following equation. For $h(x) = n^{-1}\sum_{i=1}^pa_i/(a_i+x), a_i > 0, x > 0,\ n,p \in \mathbb N$, consider a equation: $$1 = \lambda/x + h(x)$$ and let ...
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Another question about implicit function theorem

Someone can help or in this question about implicit function theorem? Let $\Omega$ A bounded domain wich satisfies a uniform interior sphere condiciona. The authors says that - we may suppose, ...
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What does "they depend differentiably on" mean?

Let $P(x) = a_{n}x^n + a_{n - 1}x^{n - 1} + \cdots + a_{0}$ be a polynomial with real coefficients. I have to prove that the simple roots of $P(x)$ depend differentiably on the coefficients of $P(x)$ ...
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Implicit function theorem for non $C^1$ mappings

I know that the inverse function theorem can be proved for differentiable mappings (not $C^1$) by requiring that $Df(x)$ has everywhere maximum rank (here is the reference https://terrytao.wordpress....
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Using an implicit function in taylor polynomial

Let $f$ be the function defined on $R^2$ by $f(x, y) = x^5+y^3-x^2+2x-3y.$ (a) Show that $f$ is of class $C^\infty$ on $R^2$ and compute its gradient at a point $(x, y)\in R^2.$ (b) Show that, in a ...
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Implicit function theorem / Implicit selections when Jacobian not invertible

I saw the attached result in the book by Dontchev and Rockafellar. It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...
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