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.
172
questions
1
vote
0answers
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 ...
0
votes
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, ...
0
votes
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\...
0
votes
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 ...
2
votes
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)...
1
vote
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 ...
1
vote
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
$$g_1(x,y) = -(\...
1
vote
2answers
47 views
Numerical boundary-finding algorithms
I'm very familiar with numerical root-finding algorithms like Brent's method (scipy.optimize.brentq) or Chandrupatla's method. What if I am looking for a boundary ...
1
vote
0answers
31 views
Explicit example of a function given by an implicit equation
Let $\Phi (x) := \int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-\frac{y^2}{2}} dy$ the distribution Function of the standard normal distribution and $T,a>0$ and $c \in \left( 2 \Phi (\frac{-a}{\sqrt{T}}...
0
votes
1answer
29 views
Implicit function theorem - derivatives
Let us have two function $n(\alpha)$ and $s(\alpha)$ and a set of two implicit equations $F_1(\alpha, n(\alpha), s(\alpha))=0$ and $F_2(\alpha, n(\alpha), s(\alpha))=0$. In this paper I've been ...
0
votes
0answers
10 views
Find isovalue such that a volume integral under the isosurface has a specific value
Lets say i have a continiously differentialbe function $f(x, y, z)$ that is nonnegative everywhere and has a finite volume integral over all space that yields 1,
$$
\int^\infty_{-\infty }dx\int^\...
0
votes
0answers
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 ...
1
vote
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 ...
0
votes
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
$$x_{j} = \displaystyle\sum_{i=1}^{n-1}(...
0
votes
0answers
9 views
Differentiation of convex implicit function problem
Let $g: \mathbb{R}^2 \mapsto \mathbb{R}$ be a differentiable, convex function. Define $h(x)= \sup_y g(x,y)$.
How to compute the value of $h'(0)$? Is there an expression in terms of $g$?
0
votes
0answers
30 views
Generalization of Implicit Function theorem
Let $f\colon \mathbb R^n \times \mathbb R^m \to \mathbb R^m$, $\mathbb R^n \times \mathbb R^m \ni (x, y) \mapsto f(x,y)$ be continuously differentiable. The implicit function theorem guarantees, under ...
0
votes
0answers
19 views
How to write this implicit equation explicitly for parameter $p$
I have this equation which is implicit in the parameter $p$, is it possible to write it explicitly in $p$ for some boundary cases? This is the equation:
$6e^\chi=e^{\frac{Q}{6}+\frac{1}{6}(4+\frac{1+p}...
1
vote
1answer
32 views
Prove the equation implicitly represents a function
Unfortunately I got stuck solving the following exercise:
Prove that the equation $xe^z + ye^{-z} + z = 1$ implicitly defines z as a function of (x,y), in the set $A\times[0,\infty)$ where: $A = \{(x,...
0
votes
0answers
34 views
Parametric polynomial surfaces to implicit form
My question concerns a multivariate generalization of this. Specifically, for integer $n\geq 2$, I have parametric polynomial equations of the form
$$x_{1} = p_{1}(t_1, t_2, ..., t_{n-1}),\\
x_{2} = ...
1
vote
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=\{...
1
vote
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 ...
0
votes
1answer
41 views
Elliptic-type integral with parameters: Maximum
I am stuck with the integral
$$ I(\epsilon,\lambda)=\frac1\epsilon\int\limits_0^{\frac\pi2} \frac{1+\left(\lambda\epsilon^4-1\right)\cos^2\phi}{\sqrt{1+\left(\epsilon^4-1\right)\cos^2\phi\,}} \,\text{...
-1
votes
1answer
23 views
Filling a region in the complex plane in Matlab
So I have plotted the the function defined implicitly by fimplicit in Matlab. I want to fill the region inside. How to do that?
...
1
vote
0answers
39 views
Limit of an implicit function
Consider the following implicit definition of $a$,
$$\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,$$
where $b<a<0<z$, $c>...
21
votes
3answers
496 views
Explaining the graph of $\sin(x^2) + \sin(y^2) = 1$
I had to plot the graph of the implicitly defined function $\sin^2 x + \sin^2 y = 1$ in an exam. This is not particularly difficult, but it got me wondering what the graph would look like when the ...
0
votes
1answer
76 views
Define pyramid and cylinder by functions f(x,y,z) >= 0
My thought process:
Define the shapes individually
For cylinder with radius 0.5 I got the function $(0.5^2-x^2-z^2, 2-y, 0.5-x, 0.5-z, y-0)$
Cylinder $r=0.25$
$0.25^2-x^2-z^2, 2-y, 0.25-x, 0.25-z, y-...
0
votes
0answers
20 views
Implicit Differentiation using Partial Differentiation
I have a question about implicit differentiation obtained from partial differentiation's chain rule.
Below is the explanation from a calculus book.
Suppose that an equation of the form $F(x, y)=0$ ...
0
votes
0answers
23 views
finding values of ordered pair $(p,q)$ for which following function is implicit.
A function $z=z(x,y)$ given implicitely by the equation $$\sqrt{z+1}+px^2y-y^2z=4+2y-qx$$
Then ordered pair of $(p,q)$ are
Options :
$(a)\; (1,2)\;\;\; (b)\; (2,1)\;\;\; (c)\; (2,2)\;\;\;(d)\;(-1,2)...
0
votes
0answers
50 views
Transforming implicit 2D region into explicit integral bounds
If I have a 2D region defined as follows:
\begin{align}
&(1) \qquad u^2-v^2 \leq a \\
&(2) \qquad u^2-v^2 \geq -a \\
&(3) \qquad u \geq 0 \\
&(4) \qquad v \geq 0
\end{align}
and I ...
0
votes
0answers
37 views
Can all parametric equations be made implicit?
My question is simple - can all parametric equations be made implicit with enough rearrangement? If so, is there an easy/general method to do so? If not, how would one go about proving that they ...
0
votes
3answers
67 views
How to check if two functions only touch in one point?
I have two functions $f(x,y)=x^4+y^4 -xy$ and $g(x,y) = x^2$. I know that these two functions "touch" at $(x,y,z) = (0,0,0)$
My question is, how do I know if this the only point where the ...
4
votes
4answers
93 views
plot of $\sin(x) + \sin(y)= \cos(x) + \cos(y)$
I was playing arround with implicit plots of the form $f(x,y) = g(x,y)$, and I noticed that if you plot in the plane the following equation: $\sin(x) + \sin(y)= \cos(x) + \cos(y)$ you get the ...
0
votes
1answer
43 views
Using implicit function theorem to solve a system of equation
I have the following question:
Consider the set $\Gamma \subseteq \mathbb{R}^3$ of solutions of the system
\begin{equation*}
\begin{cases}
x+\ln{y}+2z-2=0 \\
2x+y^2+e^z-1-e=0
\end{...
0
votes
0answers
22 views
Successive implicit functions
I am struggling with a chained or successive implicit function problem. Assume I have the relationship:
$M=G(M)F+R$
I am optimizing:
$min_{F,R}$ D(M)+c(F)+c(R)
My FOC, accounting for the implicit ...
1
vote
2answers
71 views
Finding $\frac{dr}{du}$ and $\frac{dr}{dv}$ from $(1-\frac{r}{2m})\,\exp(\frac{r}{2m})=uv$
I have the following relation
$$\left( 1-\dfrac{r}{2m} \right)\, \exp\left( \dfrac{r}{2m} \right)= u v$$
that defines implicitly $r$ as a function of $u$ and $v$. I need to find $\dfrac{dr}{du}$ and $\...
0
votes
0answers
18 views
Fourier transform of linear combination of variable
Suppose I have a function $f(x,y)$ which satisfies a pde in $x,y$. Will following kind of Fourier transform will make any sense?
$$\int f(x,y) e^{-i\omega(x-y)}d(x-y)$$ in other words if I take $u=x-y$...
0
votes
1answer
23 views
Implicit curve/surface definition of a polynomial function that's rotated and translated
Supposing I have an $n^{th}$-order polynomial curve $$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.$$ Now suppose that in ...
0
votes
0answers
31 views
Dimension of solution space to nonlinear system of equations
My lecture notes leading up to the implicit function theorem state the following:
We investigate underdetermined systems of equations $f(x) = b$ where
$f$ : $\mathbb{R}^{n+k} \to \mathbb{R}^k$ and $b ...
0
votes
3answers
43 views
Implicit function the right approach
Given $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
what is $y’_x$?
I’ve gone back and forth on this and I thought I could perhaps use the implicit function theorem, but then again is there a need to? I have ...
0
votes
1answer
33 views
Writing implicit equation from a graph [closed]
This the graph for which we have to write the implicit equation
How to approach this problem? I don't think the figure is a circle, so please help me figure it out
0
votes
0answers
14 views
Existence of the limit of an implicitly defined curve
I am interested in characterizing the limit of a certain implicitly defined curve as I send a parameter in the defining equations to zero. However, I am not sure how to show that this limit even ...
-2
votes
1answer
35 views
Inverse of the $y=x^x$ in implicit form? [closed]
I want to find the inverse of the function $y=x^x$ in implicit form and not by using Lambert W function. Can you tell me how to find it?
Thanks.
2
votes
1answer
53 views
Limit of a function defined implicitly
Let a function $y:\mathbb{R} \setminus \left\{ 0\right\} \to \mathbb{R}$ be defined implicitly by the equation $F(y(x),x)=0$ for some $F:\mathbb{R} \times ( \mathbb{R} \setminus \left\{ 0\right\} ) \...
0
votes
0answers
11 views
Complexity of an implicitly defined function?
I am writing a paper in which the function $m_0$ is defined to be
$$1 - \epsilon = \left(2n\alpha\cdot\binom{2n+m_0-2}{n-1}\right)^{-\frac{1}{1+m_0}}$$
where $\epsilon$ is fixed and $\alpha = O(n^d)$ ...
2
votes
1answer
77 views
Is there a general method to compute the area/volume enclosed by an implicit curve/surface?
If I have an implicit function $f_2(x,y) = C$ of a closed curve or an implicit function $f_3(x,y,z) = C$ of a closed surface, is there a general manner in which I can compute the area or volume ...
0
votes
1answer
27 views
Can someone help me express this parametric curve as an implicit curve $f(x,y) = C$?
I have the following parametric equation in polar coordinates:
$$ r(\theta) = r_0 + \Delta r\cos(2\theta),$$
where $\Delta r$ is some perturbation to the radius $r_0$. Can anyone help me express the ...
0
votes
1answer
33 views
How do I solve the implicit equation $G(z) = z \sum_{k=0}^\infty f_K(k) G(z)^k$ for $G$?
I have run into a probability problem that leads to an implicit equation for a function $G: \mathbb{C} \rightarrow \mathbb{C}$. Given a known probability mass function $f_K$ on the non-negative ...
2
votes
1answer
57 views
An inverse function for $-\log x/\log (1+x)$
In the context of another question I asked on here a while ago, I came across the problem of inverting the function
$$f(x) = -\frac{\log x}{\log (1 + x)}$$
for positive real $x$. Let $f^{-1}(x)$ ...
0
votes
0answers
16 views
Implicit differentiation of “similar” functions
The graph of the equation $x^2 + y^2 = 1$ gives a circle of radius $1$, centered at the origin:
After we find its derivative:
$$ \frac{dy}{dx} (x^2 + y^2) = \frac{dy}{dx} 1 \\
2x + 2y \frac{dy}{dx} = ...
0
votes
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 ...
