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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How to calculate the first order correction to the asymptotic solution to a second order differential equation

Consider the following differential equation: $$ \frac{d^{2}y}{dx^{2}}=\frac{1}{2}\begin{cases} 1-e^{-\frac{y}{\epsilon}},\space\space\space x<0\\ e^{-\frac{1-y}{\epsilon}}-1,\space\space\space x&...
Chris's user avatar
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Using Lagrange multiplier for finding shortest distance to implicit surface constructed by metaballs

I am writing a software in which I need to find the shortest distance from an arbitrary point in 3D space to an implicit surface defined by a set of metaballs. I wanted to achieve this by using the ...
fieldmops's user avatar
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Arc length for continuous implicit differentiable functions

We have a continuous differentiable function defined as $$F(x,y)=0$$And I am looking for a formula for its arc length between $x$ values $a$ and $b$. Doing a quick search, I could only find formulae ...
Kamal Saleh's user avatar
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Implicit Equation for a Parametric equation of the following form

Say I have an two parametric equations: $$x(t) = 3 - 2\sin(t) + 3 \cos(t); y(t) = 4 - 3\sin(t) + 2\cos(t)$$ I tried to combine the sin and cos using $$a \sin{(x)} + b\cos{(x)} = \sqrt{a^2+b^2} * \cos{(...
Sebastian Clavijo's user avatar
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43 views

The area of closed curve made by sine functions that are rotated on some angle $\theta$ [closed]

I constructed interesting figure and want to know its function on Cartesian plane and the area enclosed by this figure. Suppose you have the sine function (explicitly, $m \cdot \sin(kx)$) from 0 to $\...
linus's user avatar
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1 answer
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Weak version of the implicit function theorem

If we are given a $C^1$ map $F:\mathbb{R}^{n+m}\rightarrow\mathbb{R}^{m}$ and a point $(x_{0},y_{0})\in\mathbb{R}^{n}\times\mathbb{R}^{m}$ such that $F(x_{0},y_{0})=0$, the implicit function theorem ...
ibr_'s user avatar
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Mapping the derivative of an implicit function on a 2D plane

I'm in the middle of my first calculus class, a week ago we covered how to find the derivative of implicit functions, and I'm still thinking about it. I completely understand what I am supposed to ...
accoustician's user avatar
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Which are the advantages of parametric modeling over implicit modeling?

I hope that this is the correct site to ask this question, but since I was unable to find one specifically for modeling, I decided to ask here (and please feel free to correct me, if any of my ...
j00hi's user avatar
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3 votes
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How do I find the horizontal tangents to the curve $x^3 + y^3 = 6xy$?

How do I find the horizontal tangents to the curve $x^3 + y^3 = 6xy$? James Stewart in section $3.6$ of the 7th edition (on page $167$) shows in a straightforward way that there's a horizontal tangent ...
user1145880's user avatar
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$y = x+\sin(x)$ Solid of Revolution y-axis with implicit function.

Given the equation $y = x + \sin(x)$, as far as I am aware, an explicit equation $x = f(y)$ can not be found. Is there still a way to compute the volume of the solid of revolution of this function ...
unnamed's user avatar
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Another question about partial derivatives with implicit functions

I am trying to solve the following problem: If $u(x,y) = x^u + u^y$, find $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$ My first thought was $$\frac{\partial u}{\partial x} = \ln(...
MrMagoo's user avatar
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1 answer
271 views

Finding the area surrounded by a part of the implicit equation $\sin (y^x) = \cos (x^y)$

Finding the area surrounded by the part of the implicit equation $\sin (y^x) = \cos (x^y)$ such that $y\le 2n-x$ where $n$ is the solution to $n^n=\frac{\pi}{4}$ where $n<0.5$ bounded by the $x$ ...
Dylan Levine's user avatar
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2 answers
176 views

Implicit function theorem: from local to global

Suppose that we have $F(x,y)=0$ satisfying the usual hypotheses of the IFT at $(0,0)$, but such that not only $F_y(0,0)\neq 0$, but $F(x,y)=0$ and $F_y(x,y)\neq 0$ for all $x\in U\ni0$, where the open ...
GGG's user avatar
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Solving a set of implicit equations involving Polylogarithms

I have the following simultaneous equations: \begin{aligned} &H(\lambda) = a\, \text{Li}_{3/2}\left(b\frac{H(\lambda)}{F(\lambda)}\right), \; \\&H(\lambda) = c\, \text{Li}_{3/2}\left(d \, \...
Harshit Rajgadia's user avatar
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1 answer
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What function $y=f(x,a)$ approximates the relation $1-y=(1-xy)^a$?

What function $y=f(x,a)$ approximates the relation $1-y=(1-xy)^a$? The domains of the variables are as follows: $x,y\in [0,1], a\in[1,\infty)$. Clearly $y=0$ is one solution, but I would like the ...
Oscar Delaney's user avatar
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1 answer
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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 ...
79999's user avatar
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How to prove the continuity of an implicit surface?

Let $f$ be an arbitrary function from $\mathbb{R}^2$ to $\mathbb{R}$. Let $S$ be an implicit surface defined by $$ S = \lbrace x\in\mathbb{R}^2,\quad f(x) = 0 \rbrace $$ How can I prove that the ...
T.L's user avatar
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1 answer
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Formal definition of implicit solution of a first order ordinary differential equation

My book formally defines implicit solution of first order ordinary differential equation $M(t,y)dt + N(t,y)dy = 0$ in the following way: Assume that $\Phi(t,y)$ is defined on some $Q \subset \mathbb{R}...
H-a-y-K's user avatar
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6 votes
3 answers
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What is really the TRUE definition of an implicit function?

First of all I would like to say that I have already found similar questions on stack exchange but somehow my confusion regarding the definition of an implicit function still linger. The title says it ...
mathadic's user avatar
1 vote
0 answers
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Inequality for solutions a solution of system of quadratic equations.

Let $v_{i,j}\in [0,1], i,j = 0,\ldots,n$ solve the following system of equations for $r>0$: \begin{align} 0 &= \tfrac 1 2 e_{i,j}^2 + e_{j,i} v_{i,j+1} - (r+e_{j,i}) v_{i,j}, \tag{*} \\ e_{...
Pavel Kocourek's user avatar
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1 answer
41 views

Limit as $(x,y)\to (0, 0)$ of implicit function

Question: Evaluate the following limit: $$\lim _{(x, y) \rightarrow(0,0)} \left(\frac{y^2}{x^4+y^2}=0.6\right)$$ I don't really know how to handle the fact that this is an implicit function and not ...
Leonard Mohr's user avatar
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Random variable transformation with implicit relation

I am interested in calculating the distribution $f_Y$ of a random quantity $Y$. $Y$ is a function of $N$ i.i.d. random variables $X_i$ with positive support distributed according to a density function ...
catatafish's user avatar
1 vote
1 answer
32 views

Find explicit form of function from implicit form with absolute value

I'm trying to find the function $\phi(x)$ in a physics problem, during which I have obtained the following implicit expression for $\phi$: $$\bigg|\dfrac{\phi-a}{\phi+a}\bigg|=b(x)+c$$ where $c$ is a ...
Wild Feather's user avatar
1 vote
1 answer
51 views

Power series around $w = 0$ of the implicit function $z(w) = w(1+z(w))^t$

I have to determine the power series around $w = 0$ of the implicit function $z(w) = w(1+z(w))^t, \ t \in \mathbb N$ and calculate the radius of convergence. I tried to compute $\frac{dz}{dw}$ because ...
syphracos's user avatar
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u-v Slope of the contour along a saddle point

Apologies if there is standard notation for the following that I am not aware of, but I will try to define everything that I am talking about. Suppose we are looking at the level sets of a function $\...
Ron Shvartsman's user avatar
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1 answer
229 views

Implicit numerical ODE solvers: still unconditionally stable if Newton iterations limited?

Implicit ODE solvers like Backward Euler are often described as unconditionally stable, which I believe means that the solution never blows up regardless of how large the solver step size is chosen. ...
user39728_i_said_user_39728_i_'s user avatar
1 vote
1 answer
63 views

Direction of Steepest Decent of an Implicit Function.

If given an implicit surface like $x^6z+x^3y^2+y^2z^3=65$ how would I go about finding a 3-D vector that points in the steepest downhill direction at a point? I know that the gradient of this ...
Dylan Owens's user avatar
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2 answers
69 views

What does the area of the figure formed by the implicit function $x^{n}+y^{n}=c$ approach as n approaches infinity?

What is the area of the figure formed by the implicit function $x^{n} + y^{n} = c$ approach as $n$ approaches $\infty$? ($n$ is an even positive integer) Of course, the area will likely be a function ...
Electriz's user avatar
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1 answer
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Finding an algebraic equation from a parametric curve

Given a closed parametric curve $c(t)=(c_x(t),c_y(t))$, on some interval $t\in [a,b]$, is it possible to find a closed form function $f(x,y)$, such that the set of points on the curve $C$ is equal to ...
Iain's user avatar
  • 177
1 vote
1 answer
193 views

Converting a Triangle Mesh into an Implicit Surface

I am aware of many methods (like the marching cubes algorithm) which, given a surface in $\mathbb{R}^3$ described by an implicit function $f:\mathbb{R}^3 \rightarrow \{0\} \subset \mathbb{R}$, convert ...
FabrizzioMuzz's user avatar
1 vote
1 answer
54 views

Characterizing Implicitly Defined Function (asymptotically)?

I would like to characterize an implicitly defined function $m_n(X)$ that satisfies $$ e^{-b m_n(X)} \sum_{i=0}^{n-1}\frac{(b m_n(X))^i}{i!} = e^{-X}. $$ In particular, I want to know how $m_n(X)$ ...
hungary_nem's user avatar
2 votes
1 answer
142 views

Singularity on power series expansion of implicit function

I am looking for help to understand the singular point coming up on my power series expansion of an implicit function: $$x^2 z^4 (x^2-(z-k)^2)=(z-k)^4(z^2-1)$$ where $z$ is complex-valued function of $...
ACEA's user avatar
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4 votes
4 answers
286 views

Proof of Implicit Differentiation (showing a statement is true)

Prove that if $x^2+y^2-2y\sqrt{1+x^2} = 0$, then $dy/dx = x/\sqrt{1+x^2}$. Whilst I have implicitly differentiated in terms of x in order to derive that $$dy/dx = (-x+2xy/\sqrt{1+x^2})/(y\sqrt{1+x^2}-...
bio's user avatar
  • 630
5 votes
1 answer
149 views

Implicit function differentiation methods

Q: If $$x\sqrt{1+y}+y\sqrt{1+x}=0$$ where $x,y\in\mathbb{R}$ then prove that: $$\dfrac{dy}{dx}=-\dfrac{1}{(1+x)^2}$$ My approach: $$x\sqrt{1+y}+y\sqrt{1+x}=0$$ $$\implies x\sqrt{1+y}=-y\sqrt{1+x}$$ $$\...
Myst1cal's user avatar
  • 257
0 votes
1 answer
40 views

Inverse function in $\mathbb{R}^n$ and bijectivity

I have some questions on invertability of functions in $\mathbb{R}^n$. If I want to show that a function $f(x, y, z)$ is not injective, is it enough for me to show that it's Jacobian $|D_f| = 0$ at ...
ludicrous's user avatar
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0 votes
1 answer
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Given a solution point with variables x, y , and z, how can you find f(x,y) of an implicit function?

$$g(x,y,z) = x^2 + y^2 - z^2 - xyz = 12$$ Given a solution point $(x,y,z) = (3,4,1)$, write the first order Taylor Formula centered at $f(3,4)$ A unique continuous implicit function $z = f(x,y)$ ...
1styearinrealanalysis's user avatar
1 vote
0 answers
56 views

Extruding an implicit/SDF?

Assume we have a convex implicit surface represented as an SDF $f:\mathbb{R}^3\rightarrow\mathbb{R}$, e.g. a sphere or a cube. Given a segment defined by 2 points $p_1, p_2$ I am interested in ...
Makogan's user avatar
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0 answers
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Surface described by an implicit function

I'm testing a surface mesh generator that takes a function $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ and builds the surface described implicitly by $f(x,y,z)=0$. I already tested with a torus, a ball ...
bobinthebox's user avatar
1 vote
2 answers
89 views

Second derivative of implicit equation in "How Not to Land at Lake Tahoe"

I was trying to replicate the math "How not to land in Lake Tahoe" problem by Barshinger (1992) in the American Mathematical Monthly. Essentially, he models the landing path of a plane with ...
djtech's user avatar
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0 answers
27 views

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 ...
Alexander Zhang's user avatar
4 votes
2 answers
129 views

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(\...
umby's user avatar
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2 answers
309 views

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 $f(x, y, z)=0$ ...
Dick Grayson's user avatar
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0 answers
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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 ...
Algo's user avatar
  • 147
1 vote
0 answers
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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)=...
Asen Ivanov's user avatar
0 votes
1 answer
94 views

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 \...
Harsh's user avatar
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0 answers
31 views

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 ...
mhdadk's user avatar
  • 1,185
4 votes
1 answer
273 views

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 ...
Algo's user avatar
  • 147
3 votes
1 answer
59 views

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) \...
bubba's user avatar
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1 vote
1 answer
72 views

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 ...
Curiouser and curiouser's user avatar
5 votes
2 answers
296 views

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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