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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implicit function theorem in multivariable function

My question that in multivariable calculus the implicit function theorem states that: if $F(x,y)$ and $y=f(x)$, $$\frac{dy}{dx}=-\frac{\frac{\partial }{\partial x}\left(F\right)}{\frac{\partial }{\...
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Monotonicity of implicitly defined function

Let $f(x,y):\mathbb{R}^2\rightarrow\mathbb{R}$ and $g(y):\mathbb{R}^2\rightarrow\mathbb{R}$ be $C^2$-differentiable functions. Let $f(x,y)$ and $g(y)$ be strictly decreasing in $y$, and let $f(x,y)$ ...
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What are the conditions for a function to be a unique implicit function?

My question is pertaining to part b. What are the conditions for a function to be a unique implicit function? Do we only have to check if the partial derivative at (x0, y0) evaluate to a number other ...
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Books/online notes for analytic maps in Banach spaces

Can anyone suggest some good books/online notes on (real) analytic maps between Banach spaces? I am looking for the basic definitions and the implicit function theorem in this setting. Thanks!
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The existence interval of the implicit function theorem

When I learned the implicit function theorem. I met a question about the extension of the existence interval. Consider the function $F(x,y)=0,x\in\mathbb{R}^{+},y\in\mathbb{R}^{+}$ with $F(x_0,y_0)=0$....
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Differentiating through Optimization Paths

I'm reading the paper "Optimizing Millions of Hyperparameters by Implicit Differentiation". The key contribution of the paper is to show that you can replace optimizing through the ...
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If $\frac d{dx} f(x,y)>0$, can I claim that $f(x,y)$ is increasing with respect to $x$?

I have an implicit equation $f(x,y)=0$; computing the derivatives, I see that $\frac d{dx} f(x,y)>0$ while $\frac d{dy} f(x,y)$ maybe positive, or negative. Question. Is this data sufficient to ...
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Does $x \circ y = id_V$ with $x \in GL(V)$ and $y \in L(V)$ implies $y \in GL(V)$?

Let $V$ be a real Banach space. Denote by $L(V)$ the real Banach space of linear continuous applications from $V$ to $V$ and by $\mathrm{GL}(V)$ the set of all linear bijective bicontinue applications ...
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Implicit differentiation: under what conditions can implicit differentiation not be used? is there a way too tell before solving?

My calculus I book states "in the examples and exercises of this section it is always assumed that the given equation determines y implicitly as a differentiable function of x so that the method ...
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How to find the second derivative with the implicit function theorem?

$z(x,y)$ is a function defined implicitly by the equation $$F(x,y,z)=5x+2y+5z+5\cos(5z)+2=0$$ I'm trying to find $\frac{\partial^2z}{\partial x \partial y}$ at the point $(\frac{\pi}{5},\frac{3}{2},\...
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how do you prove z is a differentiable function of x and y and then a diff. function of x with another equation

so i have the image attached because i’m not sure how to use latex. but basically o can do the first part because that was a straight forward application of the implicit function theorem. my problem ...
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Parametric inverse mapping theorem

Let $M$ and $N$ be smooth manifolds. I was reading a proof of a result in which we have a map $F:[0,\infty[\times M\rightarrow N\times N$ with the property that $F(0)\in \triangle$, where $\triangle$ ...
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Stitching up implicit function theorem to obtain global result?

Given $\mathbf{F}: \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^m$ which is continuously differentiable, consider a system of equations such that $\mathbf{F}(\mathbf{x}, \mathbf{y}) = 0$. Suppose ...
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If $z=g(x, y)$ is defined by $f(x, y, z)=0$ near the point $(a, b, c)$, find $\frac{\partial g}{\partial x} (a, b)$.

Problem: Suppose $f=f(x, y, z): \mathbb{R}^3\to\mathbb{R}$ is continuously differentiable, $f(a, b, c)=0$ and $\frac{\partial f}{\partial z}(a, b, c)\neq0$. If $z=g(x, y)$ is defined by $f(x, y, z)=0$ ...
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Convexity of implicit functions [duplicate]

Are there any sufficient conditions which guarantee us that a curve defined implicitly by the equation $F(x,y) = 0$ will be convex or concave? Assuming the conditions of the implicit function theorem ...
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Mistake in the second derivative of implicit function

I have came upon this segment in a paper: I am a bit confused. Is the formula for $\frac{d^2x}{dt^2}$ even true here? Also, it looks like they mistakingly used the notation $\frac{\partial^2 x}{\...
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An exercise on the implicit function theorem

I am trying to learn the implicit function theorem and this is one exercise about it; I have solved it and would be grateful for any feedback on my solution, thanks. Let $f\begin{pmatrix}x\\ y\\ z\end{...
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Question on inverse of implicit function

Let's consider the continuously differentiable functions $f:\mathbb{R}^n\to\mathbb{R}^n$ and $F: \mathbb{R}^{2n}\to\mathbb{R}^n$ where \begin{align*} F(x_1,x_2,\cdots ,x_n,y_1,y_2,\cdots ,y_n):=\...
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Proof - implicit function theorem $\implies$ inverse function theorem

Let's assume that $f:\mathbb{R}^n\to\mathbb{R}^n$ with \begin{align*} f(y_1,y_2,\cdots ,y_n):=\begin{pmatrix}f_1(y_1,y_2,\cdots ,y_n)\\f_2(y_1,y_2,\cdots ,y_n)\\\vdots\\f_n(y_1,y_2,\cdots ,y_n)\end{...
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Limit of implicitely defined function?

I have this implicit equation $F(x, y) = x^2y+e^{x+y} = 0 $. Now this defines a function $y=f(x)$ everywhere except for $x=0$. I need to compute the limit for $x \rightarrow 0^+$. I know that $y = -\...
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Showing a set is globally a graph

I have a following problem: Let $B^2$ be an open unit ball in $\mathbb R^2$ and $$ S = \{(x, y, z) \in B^2 \times \mathbb R: \frac12(x^2 + y^2)z^3 + xyz^2 + z - 2 = 0\}$$ Show that $S$ is globally a ...
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Asymptotic behaviour of implicit functions

Suppose we have an implicit equation $F\left(x,y\right)=0$ which we know defines $y = y(x)$ as a function of $x$. Are there sufficient or necessary conditions under which we can obtain information ...
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Finding all the points $(x,y,u,v)\in\Bbb R^4$ satisfying $F(x,y,u,v)=(0,0)$ so that $(u,v)=f(x,y),f:\Bbb R^2\to\Bbb R^2$

Function $F:\Bbb R^4\to\Bbb R^2$ is given by the formula $F(x,y,u,v)=(x^2+y^2+u^2+v^2-2,x^2-y^2+u^2-v^2).$ Find all points $(x,y,u,v)\in\Bbb R^4$ in which the equation $F(x,y,u,v)=0$ defines a ...
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finding derivatives of variables in multivariable taylor polynomial

Given: $$F(x,y) = -6 -4(x-4) +6(y-6) +8(x-4)^2 +9(x-4)(y-6) -4(y-6)^2 + R_2$$ and that $F(4,6)=-6$ find y'(4), y''(4), make sure that the function is applicable for the implicit function theorem. my ...
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Solving a non-linear system using the Implicit Function Theorem

I'm trying to do this exercise: Show that the system $$\begin{equation} w^2+2x^2+y^2 - z^2 - 6 = 0,\\ wxy - xyz = 0 \end{equation}$$ can be solved in terms of $w=w(y,z)$ and $x=x(y,z)$ in an ...
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Why must the inverse function be of this form?

I am reading a proof about the implicit function theorem that uses the inverse function theorem. There they make a statement that I do not understand. The inverse function theorem statement is: ...
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Solutions for $A^2=B$

Show that there exists a neighborhood $U$ of $I$ in $\mbox{Mat}(2 \times 2)$ such that for all $B\in U$ there are at least two solutions in $\mbox{Mat}(2 \times 2)$ for the equation $A^2 = B$. I have ...
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Do superellipses provide examples of submanifolds of $\Bbb{R}^2$ that are not smooth?

Consider the curve (a kind of Lamé curve or superellipse (https://en.wikipedia.org/wiki/Superellipse)) in $\mathbb{R}^2$ defined by the equation \begin{equation} |x|^n + |y|^n = 1, \end{equation} ...
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Dilemma in a proof of the Implicit Function Theorem

I was studying the Implicit Function Theorem (source Diferencijalni račun funkcija više varijabli by I. Gogić, P. Pandžić and J. Tambača, pages 91-92) and I stumbled across something. First, I'm going ...
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Prove Implicit Function Theorem directly from Constant Rank Theorem

For reference: ($\textbf{Constant Rank Theorem}$) Suppose $U_0\subset\mathbb{R}^m$ is open and $F:U_0\rightarrow \mathbb{R}^n$ is a $C^r$ map with constant rank $k$ (that is, its Jacobian matrix has ...
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Application of Implicit Function Theorem to $f(x,t)=x^3\log(t^2+1)+e^x-t$

Consider a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ given by $$ \begin{equation} f(x,t)=x^3\log(t^2+1)+e^x-t. \end{equation} $$ Show that for each value of $t$ there is a function $g_t$ such ...
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Implicit function theorem and constrained maximization

For simplicity, consider the following constrained maximization problem: \begin{equation*} \begin{aligned} & \underset{x}{\text{maximize}} & & F(x,y) \\ & \text{subject to} & & ...
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prooving that certain point is interior using the implict function theorem

I have a question that I got for homework that I have a difficult time to solve(I need to use the implicit function theorem) $$let \ D=\{(a,b,c,d,e)\in R^5 | \ ax^4+bx^3+cx^2+dx+e=0 \ has \ a \ real \ ...
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Understanding a lemma for Implicit Function Theorem

The following lemma is a key for Implicit Function Theorem, given in Topology and Geometry by Bredon; but I am unable to see what the theorem is stating, what its conditions imply geometrically. I ...
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System of Equations with 6 variables using Implicit Function Theorem

Use the implicit function theorem to discuss the solvability of the system $$3x + 2y + z^2 + u + v^2 =0 \\ 4x+ 3y+ z + u^2 + v + w + 2 = 0\\ x + z + u^2 + w + 2 = 0\\$$ for $u , v, w$ in terms of $x, ...
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Solving System of Equations using Implicit Function Theorem

Define a new set of coordinates $u, v, w$ in terms of $x, y, z$: $$u = x + xyz,\\ v = y + x\\ w = 2x + z + 3z^2 $$ Can the system be solved for $x, y,$ and $z$ in terms of $u, v, $and $w$ near $\begin{...
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implicit function on connected curves

Suppose $h(x,y)$ is Lipschitz continuous with continuous partial derivatives. Let $$h(0,y) = 0 \quad \forall y$$ and $\gamma : [0,1) \to \mathbb{R}^2$ be a curve beginning from the $y$ axis with $\...
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Does statement of Implicit Function Theorem imply that level set is a curve?

According to the Implicit Function Theorem ($2$ dimensional case): if $F:U\subset \mathbb{R}^2\to \mathbb{R}$ is a $C^1$ function defined on the open set $U$ and $(x_0,y_0)\in U$ such that $F(x_0,...
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Apply implicit function theorem on $z^3 - 3xyz = 1$ and $z = z(x,y)$

To solve this I first rearrange $z^3 - 3xyz - 1 = 0$ $\displaystyle\frac{\partial z}{\partial x}$ = $3z^2z'-3xyz'-3yz$ $\displaystyle\frac{\partial z}{\partial y}$ = $3zz' -3xyz'-3z$ $\displaystyle\...
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If $F(x+y+z, x^2+y^2+z^2)=0$ then find $\frac{\partial^2 z}{\partial x \partial y }$

If $F(x+y+z, x^2+y^2+z^2)=0$ then find $\frac{\partial^2 z}{\partial x \partial y }$. Attempt I think that here we must apply the implicit differentiation Theorem, but I´dont know how I should do it, ...
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Help proving implicit function theorem

I'm stuck at a step while proving the implicit function theorem for $\mathbb{R}^2$. I have $F:\mathbb{R}^2\rightarrow \mathbb{R}$ continuously differentiable and $F(x_0,y_0)=0$ and also $\partial_yF(...
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Question related to a proof of Implicit Function Theorem in Banach spaces

This post is related to Help me understand this proof of Implicit Function Theorem on Banach spaces So the quick question to all, (so including @Calvin Khor and @Jo Mo :)) is : why are you able to ...
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Questions about Rudin's rank theorem

I am trying to understand the rank theorem in Rudin's Principles of Mathematical Analysis. The theorem states: Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ ...
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Question about showing the equation defines an implicit function and give the Taylor expansion

I have a question while doing the exercise, I want to be sure if my step is correct, the exercise is: Show that the equation $$x^3 - y^3 -3xy + 1 = 0$$ defines an implicit function $\phi:x \to y$ in a ...
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Tangent plane of an implicit system of equations

The next system of equations implicitly define to the equations $u(x,y)$ and $v(x,y)$ $$xe^{2u+3v}-2uv=1$$ $$ye^{u-v}-\frac{u+1}{v+1}=2x$$ Find the tangent plane equation to the graphic of the ...
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Struggling to apply the Implicit function theorem

I'm struggling with applying the implicit function theorem for the following expression. As an economist I'm not sure if I'm missing something extremely easy, if so, I apologise. For other derivatives ...
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Application of Implicity Function Theorem

Show there is $r>0$ and continuously differentiable $f: (-r,r) \to \mathbb R$ with $f(0)=0$, such that the equation $$f(x)^2x+2x^2e^{f(x)}=f(x)$$ is fulfilled. Calculate $f'(0)$. Right now I ...
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Tangent line for the curve $x^3+y^3-3xy=0$

I'm trying to find the tangent line with respect to an arbitrary point. Let $f(x,y) = x^3 +y^3 -3xy$. We can apply the implicit function theorem here, to solve $y(x)$, in otherwords the equation ...
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Why there two different slopes for one level curve

For a given smooth enough function $G(x,y)$ the equation $G(x,y)=c$ defines the smooth curve, the level curve. Suppose a point $\left(x^{*}, y^{*}\right)$ lies on this curve, i.e. is a solution of ...
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Partial derivative matrix construct in the implicit function theorem ( Spivak )

This is part of the implicit mapping theorem in Spivak's calculus on manifolds. We let $f:\Bbb{R}^n \times\Bbb{R}^m\to\Bbb{R}^m$, be continuously differentiable in an open set which contains a point $(...
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