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.
237
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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:
$$
\frac{d^{2}y}{dx^{2}}=\frac{1}{2}\begin{cases}
1-e^{-\frac{y}{\epsilon}},\space\space\space x<0\\
e^{-\frac{1-y}{\epsilon}}-1,\space\space\space x&...
- 425
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20
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Using Lagrange multiplier for finding shortest distance to implicit surface constructed by metaballs
I am writing a software in which I need to find the shortest distance from an arbitrary point in 3D space to an implicit surface defined by a set of metaballs. I wanted to achieve this by using the ...
- 11
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53
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Arc length for continuous implicit differentiable functions
We have a continuous differentiable function defined as $$F(x,y)=0$$And I am looking for a formula for its arc length between $x$ values $a$ and $b$. Doing a quick search, I could only find formulae ...
- 4,480
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1
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33
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Implicit Equation for a Parametric equation of the following form
Say I have an two parametric equations:
$$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{(...
0
votes
1
answer
43
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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 $\...
- 1
1
vote
1
answer
74
views
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 ...
- 13
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34
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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 ...
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15
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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 ...
- 103
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4
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119
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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 ...
- 625
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59
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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 ...
- 1
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1
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37
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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(...
- 1
3
votes
1
answer
271
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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$ ...
- 411
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2
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176
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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 ...
- 852
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70
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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 \, \...
1
vote
1
answer
45
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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 ...
0
votes
1
answer
34
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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 ...
- 135
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39
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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
$$
How can I prove that the ...
- 157
1
vote
1
answer
42
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Formal definition of implicit solution of a first order ordinary differential equation
My book formally defines implicit solution of first order ordinary differential equation $M(t,y)dt + N(t,y)dy = 0$ in the following way:
Assume that $\Phi(t,y)$ is defined on some $Q \subset \mathbb{R}...
- 641
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3
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609
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What is really the TRUE definition of an implicit function?
First of all I would like to say that I have already found similar questions on stack exchange but somehow my confusion regarding the definition of an implicit function still linger.
The title says it ...
- 65
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50
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Inequality for solutions a solution of system of quadratic equations.
Let $v_{i,j}\in [0,1], i,j = 0,\ldots,n$ solve the following system of equations for $r>0$:
\begin{align}
0 &= \tfrac 1 2 e_{i,j}^2 + e_{j,i} v_{i,j+1} - (r+e_{j,i}) v_{i,j}, \tag{*} \\
e_{...
- 1,530
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1
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41
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Limit as $(x,y)\to (0, 0)$ of implicit function
Question: Evaluate the following limit: $$\lim _{(x, y) \rightarrow(0,0)} \left(\frac{y^2}{x^4+y^2}=0.6\right)$$
I don't really know how to handle the fact that this is an implicit function and not ...
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answers
15
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Random variable transformation with implicit relation
I am interested in calculating the distribution $f_Y$ of a random quantity $Y$. $Y$ is a function of $N$ i.i.d. random variables $X_i$ with positive support distributed according to a density function ...
1
vote
1
answer
32
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Find explicit form of function from implicit form with absolute value
I'm trying to find the function $\phi(x)$ in a physics problem, during which I have obtained the following implicit expression for $\phi$:
$$\bigg|\dfrac{\phi-a}{\phi+a}\bigg|=b(x)+c$$
where $c$ is a ...
- 568
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 ...
- 464
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30
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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 $\...
- 394
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1
answer
229
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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.
...
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 ...
- 11
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2
answers
69
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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 ...
- 13
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1
answer
43
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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 ...
- 177
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1
answer
193
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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 ...
- 160
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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}. $$
In particular, I want to know how $m_n(X)$ ...
- 13
2
votes
1
answer
142
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Singularity on power series expansion of implicit function
I am looking for help to understand the singular point coming up on my power series expansion of an implicit function:
$$x^2 z^4 (x^2-(z-k)^2)=(z-k)^4(z^2-1)$$
where $z$ is complex-valued function of $...
- 51
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votes
4
answers
286
views
Proof of Implicit Differentiation (showing a statement is true)
Prove that if $x^2+y^2-2y\sqrt{1+x^2} = 0$, then $dy/dx = x/\sqrt{1+x^2}$.
Whilst I have implicitly differentiated in terms of x in order to derive that
$$dy/dx = (-x+2xy/\sqrt{1+x^2})/(y\sqrt{1+x^2}-...
- 630
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votes
1
answer
149
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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: $$x\sqrt{1+y}+y\sqrt{1+x}=0$$
$$\implies x\sqrt{1+y}=-y\sqrt{1+x}$$ $$\...
- 257
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1
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40
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Inverse function in $\mathbb{R}^n$ and bijectivity
I have some questions on invertability of functions in $\mathbb{R}^n$.
If I want to show that a function $f(x, y, z)$ is not injective, is it enough for me to show that it's Jacobian $|D_f| = 0$ at ...
- 621
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46
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Given a solution point with variables x, y , and z, how can you find f(x,y) of an implicit function?
$$g(x,y,z) = x^2 + y^2 - z^2 - xyz = 12$$
Given a solution point $(x,y,z) = (3,4,1)$, write the first order Taylor Formula centered at $f(3,4)$ A unique continuous implicit function $z = f(x,y)$ ...
1
vote
0
answers
56
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Extruding an implicit/SDF?
Assume we have a convex implicit surface represented as an SDF $f:\mathbb{R}^3\rightarrow\mathbb{R}$, e.g. a sphere or a cube.
Given a segment defined by 2 points $p_1, p_2$ I am interested in ...
- 3,088
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0
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32
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Surface described by an implicit function
I'm testing a surface mesh generator that takes a function $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ and builds the surface described implicitly by $f(x,y,z)=0$. I already tested with a torus, a ball ...
- 185
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2
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89
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Second derivative of implicit equation in "How Not to Land at Lake Tahoe"
I was trying to replicate the math "How not to land in Lake Tahoe" problem by Barshinger (1992) in the American Mathematical Monthly.
Essentially, he models the landing path of a plane with ...
- 13
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votes
0
answers
27
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Reference Request: Implicit Difference Equations
I know that there are some studies on implicit differential equations such as
$$ f(x, y, y') = 0. $$
I did some search but found very few results on the discrete version---implicit difference ...
- 433
4
votes
2
answers
129
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Asymptotic integration of a function
The function $$\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(\...
- 151
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309
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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 $f(x, y, z)=0$ ...
- 1,179
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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 ...
- 147
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0
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58
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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)=...
- 73
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94
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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 \...
- 123
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0
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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 ...
- 1,185
4
votes
1
answer
273
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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 ...
- 147
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votes
1
answer
59
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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) \...
- 42.4k
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1
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72
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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 ...
5
votes
2
answers
296
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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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