# 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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### Prove that $f(x;y;z)=(x+y+z;x-y-2xz)$ can be resolve for $(x,y)=\phi(z)$

Let be $f(x;y;z)=(x+y+z;x-y-2xz)$ a function, prove that it can be resolve for $(x,y)=\phi(z)$ close to $z=0$. Find explicitly $\phi(z)$ I'm very lost for this problem, first I thought it could be ...
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### Implicitly Differentiating a System of Matrix Equations

Setup Let $\mathbf a$ be an arbitrary $m\times 1$ vector, $\mathbf B$ be an arbitrary $n\times m$ matrix, with $m>n$, and $\mathbf C$ be a symmetric $m\times m$ matrix. The scalar $\lambda$ is also ...
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### Initial value problems for $y'=\frac{x+y}{x-y}$

The solutions of the differential equation $y'=\frac{x+y}{x-y}$, are given implicitly by the relation $$\ln x = \arctan\frac{y}{x}-\frac{1}{2}\ln(1+\frac{y^2}{x^2})+c,\enspace c\in\Bbb{R}.$$ I'm ...
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### Prove or disprove that equations $x^2+y^2+z^2=3$, $xy+tz=2$, $xz+ty+e^t=0$ can be solved of $t$ near $(x,y,z,t)=(-1,-2,1,0)$

Prove or disprove that equations $x^2+y^2+z^2=3$, $xy+tz=2$, $xz+ty+e^t=0$ can be solved of $t$ near $(x,y,z,t)=(-1,-2,1,0)$ I thought that I can apply the implicit function theorem, but I found that ...
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### Implicit function theorem and derivative

The implicit function theorem (in the $2$ dimensional case): Let $F\colon D \subset \Bbb R^2 \to \Bbb R^2$ and let $(x_0, y_0)$ be an interior point of $D$ with $F(x_0, y_0) = 0$. Suppose both first ...
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### Some confusion in implicit function theorem and chain rule

Let $g:\mathbb{R}^3 \to \mathbb{R}$ and $p_0=(x_0, y_0, z_0)$ a point which holds the equation $g(x-y, y-z, z-x)=0$. Find sufficient conditions for $z$ to be extractable as a function of $x,y$ in the ...
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### Implicit function theorem - derivatives

Let us have two function $n(\alpha)$ and $s(\alpha)$ and a set of two implicit equations $F_1(\alpha, n(\alpha), s(\alpha))=0$ and $F_2(\alpha, n(\alpha), s(\alpha))=0$. In this paper I've been ...
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### Application of the implicit mapping theorem

Show that the equation $x^4 + y^4 -2xy=0$ defines a function $\phi(x)=y$ in a neighborhood of $(1,1)$. Proof. Let $f(x,y)=x^4+y^4-2xy$. Clearly $f\in C^{\infty}$, and $f$ is continuously ...
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### Proving a point is interior

Let $D = \left \{ (a,b,c,d,e) : ax^4+bx^3+cx^2+dx+e=0 \text{ has a real root} \right \}$ Show that (1,2,-4,3,-2) is an interior point in $D$. My only idea was trying to somehow use the implicit ...
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### Implicit function question: Given that $u=f\big(x,y,g(x,y)\big)$, express $∂^2u/∂y^2$ and $∂^2u/∂x∂y$ using partial derivatives of $f$ and $g$.

this is my first time using StackExchange so I don't really know if anybody would help me out but here I go! The question I had while solving practice questions for the [Implicit Function Theorem] is ...
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### Maximum and minimum of implicit function with a parameter.

Consider an equation $\frac{1}{y} + a\log{y} = x$, where $a \in \mathbb{R}$ is a parameter. I want to find the value of $a$ such that maximum (minimum) of $y(x)$ is the largest (smallest). The idea is ...
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### (b) If $F(x,y)=0$ for all $(x,y)$, find $D_1g$ and $D_2g$ in terms of the partials of $f$ “Analysis on Manifolds” by Munkres

I am reading "Analysis on Manifolds" by James R. Munkres. There is the following exercise 3 on p.63 in this book. Let $f:\mathbb{R}^3\to\mathbb{R}$ and $g:\mathbb{R}^2\to\mathbb{R}$ be ...
### There exists $G$ s.t. $f \circ G$ is constant(proof completing)
Problem: assume that $f\in C^1(\mathbb{R}^2,\mathbb{R})$, prove that there exists a continuous injection $G:\mathbb{R} \rightarrow \mathbb{R}^2$ s.t. $f \circ G$ is a constant function. I 've come up ...