Questions tagged [implicit-function]

This tag is for questions relating to "implicit function", a function or relation in which the dependent variable is not isolated on one side of the equation.

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Reference Request: Implicit Difference Equations

I know that there are some studies on implicit differential equations such as $$ f(x, y, y') = 0. $$ I did some search but found very few results on the discrete version---implicit difference ...
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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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Asymptotic integration of a function

The function $$\int_0^{\frac{\pi }{2}} \exp \left\{-\frac{1}{2}\left [ \sigma ^2 \left(\cos ^2(\theta )+\frac{1}{\cos ^2(\theta )}-2\right)+\frac{x^2 \cos ^2(\theta )}{\sigma ^2}+2 x \left(1-\cos ^2(\...
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What are some of the implications of a graph of an implicitly defined function being able to enclose an area that includes the origin.

Context: I am a high schooler who after being taught the formula for a circle at school and was taught to think of its graph as the locus of all points $(x, y)$ being a real constant distance away ...
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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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How do I solve this equation for mapping a sphere to a superellipsoid?

I want to programmatically create a superellipsoid in a 3D program (Blender, using geometry nodes) and to do so, I need to solve this equation for f: $$((\lvert x\rvert * f)^r + (\lvert y\rvert * f)^r)...
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Limit of derivative of implicit function

Define $f(x)=x(1-x)^s$, where $x\leq 1$ and $s>0$. Note that this is an inverted-U-shaped function with peak at $x=1/(1+s)$. Given $x$ not equal to $1/(1+s)$, define $y(x)$ implicitly by (i) $f(y)=...
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Is an implicit representation of 3D non-planar curve possible?

In a book that I am reading, Polygon Mesh Processing (page 1, last paragraph), the authors say this: [...] implicit definition is only available for planar curves, i.e., $\mathcal{C} = \{x \in \...
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How can I show that certain implicit functions can partition $\mathbb R^2$?

Consider the implicit function defined by the equation $$ f(x,y) = 0 \tag{1} \label{eq1} $$ where $f : \mathbb R^2 \rightarrow \mathbb R$ is some continuous function. Suppose that the implicit ...
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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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  • 151
3 votes
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Topology of sets defined by real-valued functions

Suppose I have a topological space $S$ and a continuous real-valued function $f:S \to \mathbb R$. I can define sets like: \begin{align} A &= \{x \in S : f(x) = 0 \} \\ B &= \{x \in S : f(x) \...
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Can 'f' be called a function in the given problem?

I know that if a variable $z = f(x,y)$, then $z$ or $f$ is a function of $x$ and $y$. Consider $f = xy^2+y=5.$ Clearly, $xy^2+y=5$ is a curve on the x-y plane. $y$ and $x$ are implicitly related, and ...
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Prove derivative of an implicit function

The question: $\sqrt{1+x^2} + \sqrt{1+y^2} = a(x-y)$ prove that $\dfrac{dy}{dx} = \sqrt{\dfrac{1+y^2}{1+x^2}}$ I tried doing it the normal way and got $\dfrac{dy}{dx} = \dfrac{(a\sqrt{1+x^2}-x)(\sqrt{...
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How to find bounds of solution to fixed point problem

I have a two-equation system of implicit functions, \begin{align} 0 & = f(x,y) = a y^\alpha \left(\bar{x} - x\right)^\gamma - x \\ 0 & = g(x,y) = a x^\beta \left(\bar{y} - y\right)^\gamma - y \...
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3 votes
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What type of curve is described by $4(\cos{x}+\cos{y})-6(\cos{2x}+\cos{2y})+8\cos{x}\cos{y}=7$?

Does the curve by the function $$4(\cos{x}+\cos{y})-6(\cos{2x}+\cos{2y})+8\cos{x}\cos{y}=7\\x=[-2\pi/3,2\pi/3],\; y=[-2\pi/3,2\pi/3]$$ belong to any known curve family? A collection of curves is found ...
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2 votes
3 answers
111 views

What type of curve is described by $\cos{x}+\cos{x}\cos{y}+\cos{y}=0$?

Does the curve by the function $$\cos{x}+\cos{x}\cos{y}+\cos{y}=0\\x=[-2\pi/3,2\pi/3],\; y=[-2\pi/3,2\pi/3]$$ belong to any known curve family? Examples of curves can be found in Wikipedia (Link1, ...
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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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Express y in terms of x for equation $y = 2 x \sin (x) \cos (y)$

The below equation gives beautiful graph in desmos. I was trying to find a way to draw this graph using Javascript but for that I first need to express y in terms of x but I am not able to figure out ...
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1 vote
2 answers
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Changing to polar coordinates bring the differential equation to $\Phi(\phi,\rho,\rho'(\phi)) = 0$ form.

I have a differential equation $y'=\dfrac{x+y}{x-y}$ and the problem says to change to polar coordinate system by assuming $x = \rho\cos\phi$, $y = \rho\sin\phi$ and $\rho=\rho(\phi)$ and bring the ...
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Solve for $y$, $\sin(x)=y+\cos(y)$

I’m helping a friend of mine solve an equation: Solve for $y$, $\sin(x)=y+\cos(y)$ Substituting $y$ multiple times isn’t what he wants, he wants an exact explicit function. I tried doing $\sin(x)-y=\...
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simple q, partial derivative writing formally with function having both implicit and explicit dependence on variable

I have the function $L$ which depends on other variables as well as $n$ explicitly and implicitly as below. $$\frac{\partial L}{\partial n} = \frac{\partial L}{\partial e}\frac{\partial e}{\partial n} ...
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8 votes
2 answers
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Finding the asymptotic behavior of a function defined implicitly

I encountered this when trying to solve a number theory problem. I have two variables $x,y$ related by $$(\ln(x))^{y+1}=(\ln(xy))^y$$ and I want to know how big $y(x)$ is as $x\to\infty$. Ideally I ...
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3 answers
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Trigonometry implicit differentiation

If ${y}+\sin(y) = \cos(x)$, then $$\frac{dy}{dx} \,=\, -\frac{\sin(x)}{1+\cos(y)},$$ so $\frac{dy}{dx}$ is not defined when $\cos(y)=-1.$ But $\cos(x)$ is differentiable at every real number, so ${y}+\...
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Why is Wolfram's (presumably more precise) Maclaurin series a much worse approximation than my own (presumably less precise) series?

I recently answered this question, concerning the existence and nature of the implicit function defined by: $$F(x,y)=x^3-y^3-3xy+1$$ For $x$ close to $0$ - roughly the range $|x|\lt0.4$ - I showed how ...
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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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Is $f(x, y) = 0x + 0y = 0$ the only implicit homeomorphic to $\mathbb{R}^2$?

Most of the time, implicits give you curves, for example $x^2 + y^2 = R^2$ or points $x^2 + y^2 = 0$. The expression in the title is always satisfied so it spits out $\mathbb{R}^2$, is there any ...
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Measurability of implicit functions

Let $g : \mathbb{R}^{d+1} \to \mathbb{R}$ be a continuous function such that $\forall x_1, \ldots, x_d$, the function $x \mapsto f(x_1, \ldots, x_d, x)$ is an increasing bijection from $\mathbb{R}$ to ...
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Young theorem for dependent variables

I have an implicit function $f \circ g$, where $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$. Suppose that both $y=g(x)$ and $z=f(y)$ are inifite times ...
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1 vote
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Parametric equation of egg curves (transformed ellipses): $\frac{x^2}{a} + \frac{y^2}{b}t_i(x) = 1$

I recently discovered 'egg' curves, such as the following $$\frac{x^2}{a} + \frac{y^2}{b}t_i(x) = 1$$ where $t_i$ could manifest as $t_1(x) = 1 + cx$ or $t_2(x) = e^{cx}$, with $c \geq 0$. I was ...
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2 votes
2 answers
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Show that $\partial ^2 y / \partial x \partial y = 0$ for implicit function $f(x,y(x))$?

I find Implicit function theorem (limited to scalar function) both fascinating and a bit puzzling. Simply said, total derivative of some $f(x,y(x))$ according to $x$ is: $$ \frac{d f(x,y(x)) }{ dx } = ...
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1 answer
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Area enclosed by bean curve [closed]

I just found this interesting article on Wolfram Mathworld. https://mathworld.wolfram.com/BeanCurve.html I am interested in the following implicit equation: $(x^{2}+y^{2})^2=a(x^{3}+y^{3})$ (The curve ...
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Why the implicit function theorem implies existence of $g(a)=b$ instead of $g(b)=a$? What do we need for the existence of both?

The implicit function theorem says Intuitively, in the equation $f(\mathbf{a},\mathbf{b})=0$, the role of $\mathbf{a}$ and $\mathbf{b}$ are symmetric, and I expect the theorem to deliver the ...
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-1 votes
2 answers
109 views

Solving a system of implicit functions

I am trying to separate $u(x,y)$ and $v(x,y)$ in the given system: \begin{equation} \begin{array}{cc} xu+yv=x+y \\ yu-xv=x-y \end{array} \end{equation} However, when I ...
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1 answer
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Proving an implicit equation has no orbits

Suppose we have the function $F(x,y)=x+y-a\log(x)$ where a is a constant and define the orbit by the implicit equation $F(x,y)=c$ where c is a constant. How can I formally prove that no orbit reaches ...
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Eliminate method for getting implicit function of biquadratic bezier surface

The bezier surface is the function of parameter like x = f(u,v). I have the function of biquadratic bezier surface which contains 9 control points x1 to x9. $$((1-u)^2)((1-v)^2)x_1+2u(1-u)((1-v)^2)x_2+...
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1 vote
1 answer
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Sketching the solution of an implicit quadratic equation

$$G(u,\lambda) = u^2+\lambda^2−9 = 0$$ where $u\in\mathbb{R}$ is the variable of interest and $\lambda\in \mathbb{R}$ is treated as a parameter. How can I sketch the solution branch for the equation ...
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2 votes
1 answer
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Using taylor series for approximating implicit functions

Given this implicit equation, where $a$ and $b$ are constants, and assuming that $\varepsilon$ is very small, $$x =y+\varepsilon(ay^3+byz^2) \tag{1}$$ I'm tryng to approximate the expression of the ...
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1 vote
1 answer
78 views

Integration by parts or substitution

Let $ f:\mathbb{R}\rightarrow \mathbb{R}$ be a derivative function with continue derivative so that $f(a)=0$ and $f(b)=6$. Let $g(x)$ also be $g(x)=\int_{6}^{f(x)} \sqrt[4]{1+t^{2}} dt $ Then, the ...
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1 vote
1 answer
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General strategy to proof (or disproof) existence of a Maximum Likelihood Estimator

most of the questions and topics I found about the MLE on this site focus on concrete examples, where mostly the standard strategy of maximizing via differentiating was the way to go. My situation is ...
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Plotting $2^x+2^y=2^x$ on Desmos

When I plot $2^x+2^y=2^x$ on Desmos, I get this graph: But I think it should not plot anything at all, since $2^y=0$ does not have a solution for $y$. Can someone guess what happened on Desmos's side?...
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1 vote
1 answer
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Area bounded by implicit function

The equation, $x^y = y^x ~(x≠y)$ does not seem to have an elegant method to separate the variables. It is intuitive from the graph shown below (Desmos output) that it should have a convergent area ...
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0 votes
1 answer
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Finding the second derivative of the function from the implict function theorem

I have $(f_1, f_2) : \mathbb{R}^n \to \mathbb{R}^2$ and say at $0$ the Hessian say at $(\mathbf{a}, \mathbf{b}) = (a_1, .., a_{n-2}, b_1, b_2)$ we have $(f_1(\mathbf{a}, \mathbf{b}), f_2(\mathbf{a}, \...
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Check if point lies within 2D Implicit Function

For a part of a project I am working on, I want to fill a shape defined by a 2D implicit function with evenly spaced points. To do this, I've realised that I will need some way of knowing if the point ...
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9 votes
1 answer
135 views

How to Paramaterize $2\cos(x/2)\cos(y/2)=1$?

$2cos(x/2)cos(y/2)=1$" /> This curve of $2\cos(x/2)\cos(y/2)=1$ looks like a circle squished in from the sides and top and bottom. I know how to parameterize the curve by dividing it into four 90 ...
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1 vote
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Two functions can only partition neighbourhood into at most 4 path connected sets

I have two $C^1$ functions $f$ and $g$ which go from $\mathbb{R}^{N-1}\rightarrow\mathbb{R}$. For some $k_1,k_2\in\mathbb{R}^N$, I know that locally about a point $x_0\in\mathbb{R}^N$ the sets; $$F=\{\...
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2 votes
2 answers
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How plot $\sqrt{|x|}+\sqrt{|y|}=1$

I am doing this question as a continuation of this following Previous question I don't know well how draw the plot of $\sqrt{|x|}+\sqrt{|y|}=1$. My idea: $$\sqrt{|x|}+\sqrt{|y|}=1\implies \sqrt{|y|}=1-...
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Does every plane curve have an implicit form?

One of our professors said every connected plane curve has an implicit form. That is, it can be written in this form: $F(x,y)=0$ After some searching, I found this article: Look at page 23: Conversion ...
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1 vote
1 answer
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Domain of definition in the implicit function theorem

My question is whether the following variant of the implicit function theorem holds: Let $U \subseteq \mathbb{R}^k$, $V \subseteq \mathbb{R}^m$ and $W \subseteq \mathbb{R}^n$ be open sets and $F$ a $C^...
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6 votes
3 answers
771 views

What exactly is an implicitly defined function?.

So I've just started to get into some calculus and I recently came across the topic of implicit differentiation. I am extremely confused on what implicit functions are and there is very little ...
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