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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47 views

Proving there exists a unique solution for every $x\in\Bbb{R}$

Say we have $f(x,y)=x^8+3x^4y^3+y^8x^{20}+y$. I have to show that for every $x\in\Bbb{R}$ there exists a unique solution to the equation $f(x,y)=0$. So if I understand the question correctly, I take a ...
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1answer
50 views

Unable to solve definite integral if an implicit function

We were asked to find the area under the given function, as a challenging exercise: \begin{gather*} 6x^{3} +11x^{2} y+6xy^{2} +y^{3} =x \end{gather*} The first thing I tried to do, was probably naive, ...
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1answer
18 views

implicit functions in higher Dimensions

For Instance: Given the function: $\varphi(x) = x\,\varphi(x)^2+2\,x^2\,e^{\varphi(x)}$. It has to be shown that it exist a function $\varphi: U \to\mathbb{R}$ that solves it. In this case $f(x,y) = x\...
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1answer
44 views

What is the single equation for a helix?

Is there a way to describe a helix not by its parametric form $$ x=R\cos(t) ,\ y=R\sin(t) , \ z=ht , $$ but by a single equation like you can for a sphere with $ r^2 = x^2+y^2+z^2 $? Also the same ...
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0answers
70 views

Prove that $y^2+x^2+c^2x^2(1-x^2)=1$ is closed as a path. [closed]

How can I prove that the curve defined implicitly by $y^2+x^2+c^2x^2(1-x^2)=1$ and passes through the point (0,1) is closed as path when $c\in(0,1)$. Where closed means that for a parametrization $f(t)...
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0answers
40 views

Series for $f(x)=a\ln(1+b\sin(cx+df(x)))$?

This question is inspired by the two questions here and here. The answers to these questions show several ways to obtain approximations respectively give explicit fourier expansions for the functions ...
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0answers
62 views

Solving implicit function numerically

Given variables $x \in \mathbb{R}^{d_1}$, $y \in \mathbb{R}^{d_2}$ with $d_1 + d_2 = d$ and smooth functions $f_1, f_2 : \mathbb{R}^d \to \mathbb{R}$, I want to solve the equation $$g_1(x,y) = -(\...
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2answers
47 views

Numerical boundary-finding algorithms

I'm very familiar with numerical root-finding algorithms like Brent's method (scipy.optimize.brentq) or Chandrupatla's method. What if I am looking for a boundary ...
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0answers
31 views

Explicit example of a function given by an implicit equation

Let $\Phi (x) := \int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-\frac{y^2}{2}} dy$ the distribution Function of the standard normal distribution and $T,a>0$ and $c \in \left( 2 \Phi (\frac{-a}{\sqrt{T}}...
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1answer
29 views

Implicit function theorem - derivatives

Let us have two function $n(\alpha)$ and $s(\alpha)$ and a set of two implicit equations $F_1(\alpha, n(\alpha), s(\alpha))=0$ and $F_2(\alpha, n(\alpha), s(\alpha))=0$. In this paper I've been ...
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10 views

Find isovalue such that a volume integral under the isosurface has a specific value

Lets say i have a continiously differentialbe function $f(x, y, z)$ that is nonnegative everywhere and has a finite volume integral over all space that yields 1, $$ \int^\infty_{-\infty }dx\int^\...
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8 views

First order conditions when choice variables are functions of each other

I have a question regarding first order conditions of a simple maximization problem. Consider the following simple problem $\max_{x_1,x_2} \quad f(x_1,x_2)$ In this case, we know the first order ...
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1answer
34 views

How to calculate the implicit derivative for this equation

Basically, I am trying to find the implicit derivative for this problem, but I am stuck after a few steps. $$x^2y^2=x+y$$ I have calculated the derivatives of each term and used the product rule to ...
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0answers
80 views

Polynomial implicitization with alternating signed powers

For integer $n\geq 2$, consider a parameterization of the coordinates $(x_1, x_2, ..., x_{n})$ in terms of the parameters $(s_{1},s_{2}, ..., s_{n-1})$ given by $$x_{j} = \displaystyle\sum_{i=1}^{n-1}(...
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9 views

Differentiation of convex implicit function problem

Let $g: \mathbb{R}^2 \mapsto \mathbb{R}$ be a differentiable, convex function. Define $h(x)= \sup_y g(x,y)$. How to compute the value of $h'(0)$? Is there an expression in terms of $g$?
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30 views

Generalization of Implicit Function theorem

Let $f\colon \mathbb R^n \times \mathbb R^m \to \mathbb R^m$, $\mathbb R^n \times \mathbb R^m \ni (x, y) \mapsto f(x,y)$ be continuously differentiable. The implicit function theorem guarantees, under ...
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19 views

How to write this implicit equation explicitly for parameter $p$

I have this equation which is implicit in the parameter $p$, is it possible to write it explicitly in $p$ for some boundary cases? This is the equation: $6e^\chi=e^{\frac{Q}{6}+\frac{1}{6}(4+\frac{1+p}...
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1answer
32 views

Prove the equation implicitly represents a function

Unfortunately I got stuck solving the following exercise: Prove that the equation $xe^z + ye^{-z} + z = 1$ implicitly defines z as a function of (x,y), in the set $A\times[0,\infty)$ where: $A = \{(x,...
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34 views

Parametric polynomial surfaces to implicit form

My question concerns a multivariate generalization of this. Specifically, for integer $n\geq 2$, I have parametric polynomial equations of the form $$x_{1} = p_{1}(t_1, t_2, ..., t_{n-1}),\\ x_{2} = ...
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1answer
44 views

Constrained extrema and geometric interpretation of constrained stationary points

I'm studying constrained extrema for functions of two variables and with a single constraint(of equality). So I have a function $f$ and I have to find its constrained extrema in the restriction $E_0=\{...
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1answer
29 views

Implicit function graphing

I came across the function, $$z=\frac{4xy(x^2-y^2)}{x^2+y^2}$$ and I had to draw the level curve for this equation. Clearly the equation is already explicitly expressed in the form $z=f(x,y)$ and ...
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1answer
41 views

Elliptic-type integral with parameters: Maximum

I am stuck with the integral $$ I(\epsilon,\lambda)=\frac1\epsilon\int\limits_0^{\frac\pi2} \frac{1+\left(\lambda\epsilon^4-1\right)\cos^2\phi}{\sqrt{1+\left(\epsilon^4-1\right)\cos^2\phi\,}} \,\text{...
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1answer
23 views

Filling a region in the complex plane in Matlab

So I have plotted the the function defined implicitly by fimplicit in Matlab. I want to fill the region inside. How to do that? ...
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0answers
39 views

Limit of an implicit function

Consider the following implicit definition of $a$, $$\int_{a}^{a+\sqrt{\frac{a-b}{c}}} x \; p(x) \frac{(a-x)\left(a+\sqrt{\frac{a-b}{c}}-x \right)}{(x-b)(x-z)}dx = 0,$$ where $b<a<0<z$, $c>...
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3answers
496 views

Explaining the graph of $\sin(x^2) + \sin(y^2) = 1$

I had to plot the graph of the implicitly defined function $\sin^2 x + \sin^2 y = 1$ in an exam. This is not particularly difficult, but it got me wondering what the graph would look like when the ...
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1answer
76 views

Define pyramid and cylinder by functions f(x,y,z) >= 0

My thought process: Define the shapes individually For cylinder with radius 0.5 I got the function $(0.5^2-x^2-z^2, 2-y, 0.5-x, 0.5-z, y-0)$ Cylinder $r=0.25$ $0.25^2-x^2-z^2, 2-y, 0.25-x, 0.25-z, y-...
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20 views

Implicit Differentiation using Partial Differentiation

I have a question about implicit differentiation obtained from partial differentiation's chain rule. Below is the explanation from a calculus book. Suppose that an equation of the form $F(x, y)=0$ ...
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23 views

finding values of ordered pair $(p,q)$ for which following function is implicit.

A function $z=z(x,y)$ given implicitely by the equation $$\sqrt{z+1}+px^2y-y^2z=4+2y-qx$$ Then ordered pair of $(p,q)$ are Options : $(a)\; (1,2)\;\;\; (b)\; (2,1)\;\;\; (c)\; (2,2)\;\;\;(d)\;(-1,2)...
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0answers
50 views

Transforming implicit 2D region into explicit integral bounds

If I have a 2D region defined as follows: \begin{align} &(1) \qquad u^2-v^2 \leq a \\ &(2) \qquad u^2-v^2 \geq -a \\ &(3) \qquad u \geq 0 \\ &(4) \qquad v \geq 0 \end{align} and I ...
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0answers
37 views

Can all parametric equations be made implicit?

My question is simple - can all parametric equations be made implicit with enough rearrangement? If so, is there an easy/general method to do so? If not, how would one go about proving that they ...
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3answers
67 views

How to check if two functions only touch in one point?

I have two functions $f(x,y)=x^4+y^4 -xy$ and $g(x,y) = x^2$. I know that these two functions "touch" at $(x,y,z) = (0,0,0)$ My question is, how do I know if this the only point where the ...
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4answers
93 views

plot of $\sin(x) + \sin(y)= \cos(x) + \cos(y)$

I was playing arround with implicit plots of the form $f(x,y) = g(x,y)$, and I noticed that if you plot in the plane the following equation: $\sin(x) + \sin(y)= \cos(x) + \cos(y)$ you get the ...
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1answer
43 views

Using implicit function theorem to solve a system of equation

I have the following question: Consider the set $\Gamma \subseteq \mathbb{R}^3$ of solutions of the system \begin{equation*} \begin{cases} x+\ln{y}+2z-2=0 \\ 2x+y^2+e^z-1-e=0 \end{...
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0answers
22 views

Successive implicit functions

I am struggling with a chained or successive implicit function problem. Assume I have the relationship: $M=G(M)F+R$ I am optimizing: $min_{F,R}$ D(M)+c(F)+c(R) My FOC, accounting for the implicit ...
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2answers
71 views

Finding $\frac{dr}{du}$ and $\frac{dr}{dv}$ from $(1-\frac{r}{2m})\,\exp(\frac{r}{2m})=uv$

I have the following relation $$\left( 1-\dfrac{r}{2m} \right)\, \exp\left( \dfrac{r}{2m} \right)= u v$$ that defines implicitly $r$ as a function of $u$ and $v$. I need to find $\dfrac{dr}{du}$ and $\...
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0answers
18 views

Fourier transform of linear combination of variable

Suppose I have a function $f(x,y)$ which satisfies a pde in $x,y$. Will following kind of Fourier transform will make any sense? $$\int f(x,y) e^{-i\omega(x-y)}d(x-y)$$ in other words if I take $u=x-y$...
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1answer
23 views

Implicit curve/surface definition of a polynomial function that's rotated and translated

Supposing I have an $n^{th}$-order polynomial curve $$y = \sum_{i=0}^n c_ix^i$$ and an $n^{th}$-order polynomial surface $$z = \sum_{i,j\in\mathbb{Z}^+\!,\ i+j=n} c_{ij}x^iy^j.$$ Now suppose that in ...
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0answers
31 views

Dimension of solution space to nonlinear system of equations

My lecture notes leading up to the implicit function theorem state the following: We investigate underdetermined systems of equations $f(x) = b$ where $f$ : $\mathbb{R}^{n+k} \to \mathbb{R}^k$ and $b ...
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3answers
43 views

Implicit function the right approach

Given $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ what is $y’_x$? I’ve gone back and forth on this and I thought I could perhaps use the implicit function theorem, but then again is there a need to? I have ...
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1answer
33 views

Writing implicit equation from a graph [closed]

This the graph for which we have to write the implicit equation How to approach this problem? I don't think the figure is a circle, so please help me figure it out
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0answers
14 views

Existence of the limit of an implicitly defined curve

I am interested in characterizing the limit of a certain implicitly defined curve as I send a parameter in the defining equations to zero. However, I am not sure how to show that this limit even ...
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1answer
35 views

Inverse of the $y=x^x$ in implicit form? [closed]

I want to find the inverse of the function $y=x^x$ in implicit form and not by using Lambert W function. Can you tell me how to find it? Thanks.
2
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1answer
53 views

Limit of a function defined implicitly

Let a function $y:\mathbb{R} \setminus \left\{ 0\right\} \to \mathbb{R}$ be defined implicitly by the equation $F(y(x),x)=0$ for some $F:\mathbb{R} \times ( \mathbb{R} \setminus \left\{ 0\right\} ) \...
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0answers
11 views

Complexity of an implicitly defined function?

I am writing a paper in which the function $m_0$ is defined to be $$1 - \epsilon = \left(2n\alpha\cdot\binom{2n+m_0-2}{n-1}\right)^{-\frac{1}{1+m_0}}$$ where $\epsilon$ is fixed and $\alpha = O(n^d)$ ...
2
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1answer
77 views

Is there a general method to compute the area/volume enclosed by an implicit curve/surface?

If I have an implicit function $f_2(x,y) = C$ of a closed curve or an implicit function $f_3(x,y,z) = C$ of a closed surface, is there a general manner in which I can compute the area or volume ...
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1answer
27 views

Can someone help me express this parametric curve as an implicit curve $f(x,y) = C$?

I have the following parametric equation in polar coordinates: $$ r(\theta) = r_0 + \Delta r\cos(2\theta),$$ where $\Delta r$ is some perturbation to the radius $r_0$. Can anyone help me express the ...
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1answer
33 views

How do I solve the implicit equation $G(z) = z \sum_{k=0}^\infty f_K(k) G(z)^k$ for $G$?

I have run into a probability problem that leads to an implicit equation for a function $G: \mathbb{C} \rightarrow \mathbb{C}$. Given a known probability mass function $f_K$ on the non-negative ...
2
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1answer
57 views

An inverse function for $-\log x/\log (1+x)$

In the context of another question I asked on here a while ago, I came across the problem of inverting the function $$f(x) = -\frac{\log x}{\log (1 + x)}$$ for positive real $x$. Let $f^{-1}(x)$ ...
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0answers
16 views

Implicit differentiation of “similar” functions

The graph of the equation $x^2 + y^2 = 1$ gives a circle of radius $1$, centered at the origin: After we find its derivative: $$ \frac{dy}{dx} (x^2 + y^2) = \frac{dy}{dx} 1 \\ 2x + 2y \frac{dy}{dx} = ...
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0answers
8 views

Partial fraction decomposition involving implicit coefficient

This arised when I was attempting to solve an IVP using laplace transforms ; I'm not sure how to proceed when decomposing partial fractions involving implicit coefficient on the right hand side such ...