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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Find all the values that define an implicit function

Let $h\colon\mathbb{R}^2\to\mathbb{R}$ defined by $h(x,y)=x^2+y^3+xy+x^3+ay$ (where $a\in\mathbb{R}$). a) Find the values of $a$ for which $h(x,y)=0$ defines $y$ as a $\mathscr{C}^1$ implicit function ...
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example on solve implicit function

I was trying to find some examples of explicitly solving implicit functions. However, most I found was about implicit differentiation. For example, if we have a function $u(x,t)$, the implicit form ...
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Limiting behavior of $x$ given that $ 1=\sum^N_{i=1}\frac{\alpha_i}{x+M\alpha_i} $ as $N,M\rightarrow\infty$?

Consider the following implicit equation for $x$. $$ 1=\sum^N_{i=1}\frac{\alpha_i}{x+M\alpha_i} $$ I am trying to see what a limiting behavior of $x$ is, as $N$ and $M$ explode to infinity. I require ...
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Proving implicit function theorem using Kroneker-Rouchè-Capelli

I'm a physics student facing the implicit function theorem. My professor gave me an unintuitive proof of the implicit function theorem based on Banach fixed-point theorem. I need help formalizing a ...
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How to find the implicit function of $\,\mathrm dx/\mathrm dt=y+y^2-x^3=y+P_2(x,y)\;?$

How to find the implicit function of $\,\dfrac{\mathrm dx}{\mathrm dt}=y+y^2-x^3=y+P_2(x,y)\;\;?$ I am facing difficulty to find the implicit function in the given example (the picture is attached). ...
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Assumptions for gradient to be perpendicular to level set

The fact that a differentiable function is perpendicular to its level sets has been asked before. See: Gradient is perpendicular to level set and implicit function theorem and Why is the gradient ...
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Implicit function theorem for higher derivatives

We can find the statement of the theorem for continuously differentiable $f$ here https://en.wikipedia.org/wiki/Implicit_function_theorem. Its also interestingly remarked that if $f$ is $k$ times ...
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Coordinate system $(\phi, U, x_1, x_2)$ around $p \in S^1$ such that $S^1 \cap U = \{(x_1, x_2): x_2 = 0\}$.

Consider $S^1$ as a subspace $\Phi^{-1}(1) \subseteq \mathbb{R}^2$, where $\Phi(x_1, x_2) = x_1^2 + x_2^2$. Using the Implicit Function Theorem, show that for all points $p \in S^1$, there is a ...
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Implicit function theorem for $f(x,y,z)$: Can I find a function $g$ such that $y=g(x,z)$?

If I want to use the Implicit function theorem for a function $f:\mathbb{R}^3\rightarrow \mathbb{R}$. Is the order of the variables important? What I mean with that is, if I can solve $f(x,y,z)$ for $...
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Using the implicit function theorem for a function $f(x,y,z)$ , what if $\frac{\partial f}{\partial (y,z)} $ is not a square matrix?

I want to use the implicit function theorem for a function $f:\mathbb{R}^3\rightarrow \mathbb{R}$. The set $M\subset \mathbb{R}^3$ is defined by the equation $f(x,y,z)=\text{function rule}=0$ for ...
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What to do when IFT doesn't work

I am trying to solve a question and I am stuck on something. Consider the equations:$$e^{x}+y^{2}z=2,\ \ xy+2y+\frac{1}{z^{2}}=3$$I should find (i) If there exists a neighborhood of (0,1,1) such that ...
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Find derivative using implicit function theorem

Let $F(x,y)$ be a function such that $$F(x,y)=4+4(x -3) +4(y-7)-5(x-3)^2-(x-3)(y-7)-7(y-7)^2 +R_2$$ is the Taylor series. Using implicit function theorem and find $y'(3),y''(3)$ of the equation $F(x,y)...
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A weird theorem of existence

Im struggling with a problem which seems to be an application of implicit theorem function. It's really hard for me and I would love some help. Let $F: \mathbb{R}^2 \to \mathbb{R}$ a $C^1$ function. ...
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Comparative statics argmax of a function

I am trying to figure out methods to do comparative statics of a maximization problem. Let $$x^\star=\arg\max_x f\left(x,\mu\right).$$ Numerically, I see that $x^\star$ is non-decreasing in $\mu$. I ...
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Most interesting exercises about the implicit and inverse function theorems

I am a TA in a multivariable calculus course this semester. Right now I am writing the exercise session which deals with the implicit function theorem, inverse function theorem and open function ...
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Implicit function theorem: Show that the system defines $u,v$ as functions of $(x,y)$ near the point $(1,1,1,1)$

This question is taken from Vector Calculus by Marsden and Tromba. It is question 39 of Page 231. It reads as follows: Consider the system of equations \begin{align*} 2xu^3v - yv&=1 \\ y^3v+x^5u^...
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Tangent plane at a point to surface $\mathbf{F}=\mathbf{0}$

Let $\mathbf{F}:\mathbb{R}^5\to\mathbb{R}^3, \mathbf{F}\begin{pmatrix}x_1\\x_2\\y_1\\y_2\\y_3\end{pmatrix}=\begin{bmatrix}2x_1+x_2+y_1+y_3-1\\x_1x_2^3+x_1y_1+x_2^2y_2^3-y_2y_3\\x_2y_1y_3+x_1y_1^2+...
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How Optimizers Vary when Changing Parameters in the Constraints

I am trying to solve the following comparative statics problem. Assume I have four choice variables, $x_{1}, x_{2}, y_{1}$ and $y_{2}$, and two parameters $t_{1}$ and $t_{2}$. I have the following ...
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Implicit function theorem in the complex numbers

$F(x,y,u,v)=(x^2-y^2-u^3+v^2+4,2xy+y^2-2u^2+3v^4+8), F(2,-1,2,1)=0$ Do there exist functions $f,g$ defined in an environment U of $(2,-1)$ so that $f(2,-1)=2,g(2,-1)=1$ so that $F(x,y,f(x,y),g(x,y))=0$...
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Need help completing the solution of an implicit function problem

Let $f:\mathbb{R^3\rightarrow R^2}$ be given by $f(x,y,z)=(\cos(x)+5y-z^2-2, \exp(x)+2y-3z+3)$. Investigate whether the there exists a solution for $f(x,y,z)=(0,0)$ in a neighbourhood of $(0,1,2)$ ...
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The converse of the Implicit Theorem

According to the implicit function theorem(on $\mathbb R^2$ for simplicity), if $\displaystyle\frac{\partial f}{\partial y}\ne 0$ at $(x_0, y_0)$, then on a neighborhood of $(x_0, y_0)$, there is a ...
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Implicit Function Theorem for Multivariable Equation

I currently have an equation of the form $$F(a(x),b(x),c(x),\lambda)=0$$ and am trying to determine whether I can implicitly express $\lambda\in\mathbb{C}$ as a function of $a(x),b(x),c(x):\mathbb{\...
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Show that the below equation has exactly two local solutions.

Show that the equation $y=x^2+o(x^2)$ as $x\to0$ for given $y>0$ and for $(x,y)$ near $(0,0)$ has exactly two solutions given by $x= \pm \sqrt{y} + o(\sqrt{y})$ as $y\to 0$.
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How to apply implicit function theorem here?

I was looking at the following post: Prove the theorem of level sets as manifolds. And I am confused about how the implicit function theorem was used here. I understand that $\left( \frac{\partial f_i}...
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Generalization of implicit function theorem on smooth manifolds

Corollary 6.33 on Lee's Introduction to smooth manifolds is the following: Suppose $M$ and $N$ are smooth manifolds, $S\subset M\times N$ is an immersed submanifold with $\dim S = \dim M$, and $(p,q)\...
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partial derivatives and a $(u,v)=(1,1)$ neighbourhoods

I'm currently working through the problem: In a neighbourhood of $(u,v)=(1,1)$; show that the system: $$ \begin{cases} x=u+\ln(v) \\ y = v-\ln(u) \\ z=2u+v \end{cases} $$ defines a function $z=z(x,y)$...
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Implicit function theorem for overdetermined system of nonlinear equations

Consider a sufficiently regular ($C^1$ ?) function $$F:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{n+k}$$ with $k>0$. And assume an implicit function $y(x)$ is locally well defined by the ...
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Problem whit the use of Implicit function theorem

Problem Hi guys! I want to define for the part $a)$ $$x(u,v); \ \ y(u,v)$$ but in the problem change the order in the components. I calculate the Jacobian matrix and obtain an equation that I not ...
2 votes
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Implicit function theorem on equation involving cotangent

Given is the implicit equation $$ c(\zeta) = \frac{\alpha}{1 - \zeta q}\left(2 + \frac{1}{2}c(\zeta)\left[1- \cot\left(\frac{1}{2}c(\zeta)\right)\right]\right), $$ where $\cot$ is the cotangent ...
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Can Peano's existence theorem be proved by the implicit function theorem?

In The implicit function theorem:history,theory and applications written by Krantz & Parks, it's said that the implicit function theorem can prove the following existence theorem of ODE: Theorem ...
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Existence of coordinates on a neighborhood of a point using the Inverse/Implicit Function Theorems

This is an old comprehensive exam question I've been working on Let $\gamma_i:\mathbb{R}\rightarrow \mathbb{R}^3$ be $C^1$ curves for $i=1,2$ such that $\gamma_1(0)=\gamma_2(0)=p$. Assume $\gamma_1'(...
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If $m(d)=Fd$ and $l(d)=\mathbf{1}^T d$ then what is $\nabla_m (l)$?

Given a vector $d\in\mathbb{R}^p$, matrix $F\in\mathbb{R}^{q\times p}$, and unit vector $\mathbf{1}\in\mathbb{R}^{p}$, suppose the vector- and scalar-valued functions $m$ and $l$ are defined as $m(d)=...
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How do I apply Implicit function theorem correctly?

Reference:- An Elementary Course in Partial Differential Equations, T. Amaranth. How do I apply implicit function theorem here? My attempt:- I have gone through the generalized example of implicit ...
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Elimination of a variable between two explicit one-variable functions for parametric surfaces

Statement from the Text:- Suppose the given curve is $$x=f_1(t),y=f_2(t),z=f_3(t).$$ On eliminating $t$ between $f_1$ and $f_2$, we obtain a relation $\phi(x,y)=0.$ Similarly, between $f_1$ and $f_3$,...
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Finding $y'(x)$ using the Jacobian Determinant and Implicit Function Theorem

Problem text: Show that $$ x^{y} + \sin(y) = 1 $$ Defines $y$ as a function of $x$ in the surrounding region of $(1, 0)$ and find $y'(x)$. Textbook answer: $$y'(x) = \frac{-yx^{y-1}}{x^{y} \ln(x) + ...
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gradient of a function given its tangent plane at a point

I am confused about a solution of the following problem: Let $f$ be a function from $\mathbb{R^2}\rightarrow \mathbb{R}$ such that $f$ $\in C^1$. Furthermore, $9x-14y-2z=20$ is the plane tangent to ...
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From an explicit function to an implicit one

I've seen a lot of people asking questions about how to go from an implicit expression and get an explicit function. I wonder if we can do the opposite: starting with an explicit function $d(x,y)$ and ...
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Implicit function theorem for polynomials with functions as coefficients

Let $n ∈ N$ and $α_k : \mathbb{R} → \mathbb{R}$, $k ∈ [0, n]_\mathbb{N}$, be continuously differentiable functions, and consider the polynomials $P_x : \mathbb{R} → \mathbb{R}$, $P_x(y) := \sum^{n}_{k=...
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Proof of implicit function theorem for a complex function not necessarily holomorphic

Let $U,V\subset\mathbb{C}$ be domains, and $F(z,w):U\times V\to\mathbb{C}$ be continuous, and holomorphic in $z$ for every fixed $w\in V$. Let $(z_0,w_0)\in U\times V$ be s.t $F(z_0,w_0)=0$ and $\...
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The implicit function theorem is not applicable to the system of equations $3x^3+2y^5-5t^7=0$, $2x^3-y^5-t^7=0$?

I am given a system of equations: $F_1(t,x,y)=3x^3+2y^5-5t^7$ $F_2(t,x,y)=2x^3-y^5-t^7$ The task is to show that the system is locally uniquely resolvable in the point $(t,x,y)=(0,0,0)$ in respect to ...
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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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