# 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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### 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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### 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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### 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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### 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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### 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 ...