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.
274 questions
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Is there a way to simplify an implicit function so that $x$ and $y$ will be on separate sides of the equation?
I have an implicit equation $(x^2+y^2)^2=a^2(x^2-by^2)$, where $a$ and $b$ are variables. I have tried to take this implicit function and separate the $x$ and $y$ values, but there is an unfactorable ...
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2
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Solving this equation for k, don't know where to begin [duplicate]
If $k$ and $N$ are both integers then is there an expression for the minimum of $k$ if $N$ is given and
$10^k\mod{N}=1$?
What I mean is, is there a function $f$ such that $f(N)=k$?
I haven't seen ...
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How do I convert this parametric curve into an implicit curve? [closed]
I have a continuous closed parametric curve
$$
\begin{align}
x &= \arccos\left(-\frac{Q × \sqrt{\frac{1}{3}} \left(\tan\left(\frac{π}{4} u\right)^2 - 1\right) - \sqrt{\frac{2}{3}} \left(\tan\left(\...
- 2,153
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Asymptotic development of implicit function
In his book Infinitesimal Calculus, Dieudonné presents the following theorem regarding the asymptotic development of an implicitly defined function (Chapter III, pp. 73-74):
(8.2) Let $g$ be a [real-...
- 3,519
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A possible counterexample to the statement on p.340 in "An Introduction to Manifolds Second Edition" by Loring W. Tu.
I am reading "An Introduction to Manifolds Second Edition" by Loring W. Tu.
On p.340, the author wrote as follows:
On a smooth curve $f(x,y)=0$ in $\mathbb{R}^2$,
$y$ can be expressed as a ...
- 10.4k
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82
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Proper way to write a solution of a differential equation
The solution of the separable DE
$ y^\prime = -\dfrac{2x}{3y} $
is: $$y^2 = \dfrac{2}{3}\left( C-x^2 \right),$$
which I know is an implicit solution.
My question is, if we were asked to solve for $y$, ...
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3
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116
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Isolating $y$ in $2x(x-y)+1=\exp(x^2-y^2)$ [closed]
I was studying implicit functions and, while playing around, stumbled across the curve $$2x(x-y)+1=\exp(x^2-y^2).$$ I'm trying to isolate $y$ in the equation but have been unsuccessful so far. This ...
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1
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graphing a surface of revolution
I'm looking at creating a 3D map of the equipotentials of a magnetic dipole (the level surfaces). So I set up a graph of the equation $\frac{1}{\sqrt{x^2+y^2}} - \frac{1}{\sqrt{(2-x)^2+y^2}} = k$ ...
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Implicit differentiation - why can you substitute the expression?
One of the popular proofs of the derivative of ln(x) is by implicit differentiation.
$$
\begin{align*}
y &= \ln x \\
e^y &= x \tag{2} \\
e^y dy &= dx \\
\frac{dy}{dx} &= \frac{1}{e^y} \...
- 99
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2
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104
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What iterative solution method to employ when recursion diverges?
I have a function for which I am trying to find the solution iteratively.
$y = x \cdot tan(x)$
I know $y$, I'm searching for $x$. I am trying to solve the problem by coding it, so I wrote a recursion ...
- 123
1
vote
1
answer
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Proving the uniqueness of a solution on an implicit curve.
Let $F(x,y) = x^3+y^3 - 3xy$ and we study the implicit curve $C$, defined by $F(x,y)=0$.
I am asked to show that, for $x<0$ this has a unique solution.
However, this is taught without a textbook or ...
- 6,178
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approximate solution for implicit equation with sum of exponentials
I would like to obtain an approximate solution to the equation
$$
\exp(x) = \exp(a_1 + z_1 y(x)) + \exp(a_2 + z_2 y(x))
$$
which implicitly defines $y(x)$. EDIT Assume that $z_1, z_2 > 0$, $a_1, ...
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Converting implicit equations with multiple exponentials to explicit functions using Lambert W
I have some implicit equations that involve exponentials that I'm seeking to convert to explicit functions using the Lambert W function. All equations should produce a single, real solution.
Note: ...
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1
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102
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Maximize a function implicitly defined
Given the fucntion
$$y(b,x)\text=\frac{1}{b} e^{-b x} + \sqrt{\frac{2}{\pi }} \left(b-\frac{1}{b}\right) e^{-b^2 \frac{x^2}{2}}$$
with $b>1$ and $x>0$, I am looking for a formula (also an ...
- 325
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1
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87
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How do I turn the equation for a curve in 3d into a parametric function?
I am working on a manim project and wanted to graph the curve of intersection between $\frac{x}{2}-\frac{3y}{4}+\frac{23}{4} = z$ and $\sqrt{x^{3}+x+23-y^{2}} = z$. In order to do this I have to turn ...
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69
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Converting a simple parametric surface to a function of two variables
I'm looking for a function of two variables that satisfy very specific conditions.
Namely,
$f(x,y) = 1$ if $(x,y)$ is a point on the unit circle,
$f(x,y) = 0$ if $(x,y)$ is a point on an ellipse with ...
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43
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Can a signed distance bound function surface be singular over a set of measure that is non-zero?
Motivation
This is a follow-up from my question here: Can an implicit surface be singular over a set of measure that is non-zero?.
I actually thought that the answer would be positive, but ...
- 2,408
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Can an implicit surface be singular over a set of measure that is non-zero?
Let $f:\mathbb{R}^3\to\mathbb{R}$ be a continuous function such that $S = f^{-1}(0) = \{p\in\mathbb{R}^3\,:\, f(p)=0\}$ is a surface. Let $f$ be continuously differentiable almost everywhere on an ...
- 2,408
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66
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Can "$\frac{dy}{dx}$" be defined uniquely under reasonable conditions for curves (relations) which aren't functions?
In solving differential equations, I have been taught that implicit solutions of the form $\{ (x,y) | R(x,y) = 0 \}$ are solutions only in the sense that any (single variable) differentiable function $...
- 1,029
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Finding critical points of a Morse function on an implicit manifold
In the special case that a manifold is given by some implicit equation
$$ F(x_0, x_1, \dots, x_N) = 0$$
Then a natural Morse function is to simply take $f(p) = X_i$, where $p = (X_0, X_1, \cdots X_i, \...
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2
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104
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Implicit differentiation choice
I was reading Calculus early transcendentals by Howard Anton, in which I encountered an example as follows,
Find the slope of tangents of a sphere $x^2+y^2+z^2=1$ in the direction of $y$ at points $(2/...
1
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1
answer
99
views
Second order partial implicit derivatives
I have some function $F(x, y, z) = 0$, and wish to find the second order cross derivative
$\frac{d^2z}{dxdy}$.
I've easily been able to get the second order derivatives $\frac{d^2z}{dx^2}$ and $\frac{...
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2
answers
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Implicit function equation $f(x) + \log(f(x)) = x$
Is there a function $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that
$$
f(x) + \log(f(x)) = x
$$
for all $x \in \mathbb{R}_{>0}$?
I have tried rewriting it as a differential equation ...
- 2,025
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321
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Find the area enclosed by the curve $(x^2+y^2-1)^3-x^2y^3=0$
The curve
$$(x^2+y^2-1)^3-x^2y^3=0$$
forms a heart shaped curve.
I want to find the area enclosed by it.
This curve is a sixth degree algebraic curve, so y cannot be found and x cannot be found. The ...
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118
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Condition on derivatives
I am working with a "well-behaved" optimization problem of the form:
\begin{equation*}
\max_{x} f( g_{1}( x) ,g_{2}( x) ,g_{3}( x) ,\mathbf{y})
\end{equation*}
where $\displaystyle f:\mathbb{...
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34
views
What is the scalar field the integral of which gradient norm is equal to the surface area of the surface that the scalar field represent?
Assume I have a scalar function $\phi(p): p \in \Omega \subset \mathcal{R}^3 \to \mathcal{R}$. I would like to use it to represent a 2D surface. For example, if $\phi(\cdot)$ is a signed distance ...
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Given $ye^{-{xe^{y-2x}}} = 2xe^{-x},$ where $x>\frac{1}{\sqrt{2}}, y<\sqrt{2}$. Show that $y=y(x)$ decreases.
Suppose $y$ is defined by the following implicit equation: $ye^{-{xe^{y-2x}}} = 2xe^{-x},$ where $x,y\geq 0.$
I want to show that $y$ decreases as $x$ increases, when $x>\frac{1}{\sqrt{2}}$ and $y&...
user1288879
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Finding the distance of the intersection point of two conics to $(-3,2)$
Consider the intersection of the curves $x^2+y^2+6x-24y+72=0, x^2-y^2+6x+16y-46=0$. Determine the sum of the distances of their intersection points and $(-3,2)$.
My first thought looking at this ...
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Solving explicitly $y(x)$ in an exact ODE
Given the following differential equation:
$$y^2+(2xy+1)\frac{dy}{dx}=0$$
I have found that the solution is given implictly by:
$$ (\ast) \quad xy^2(x)+y(x)=k \: ; k \in \mathbb{R}$$
If I was ...
- 1,152
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Implicit equation of all points that a circle that traces along a 2d parametric curve.
I want to find an implicit equation that contains points that fall within a circle that has an origin that follows a 2d parametric curve, which would look like you painted a circle along that curve. I ...
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3
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105
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Implicit equation of revolution $(x^2+y^2)^2=x$
I have a curve given as $(x^2+y^2)^2=x$ in the plane $z=0$. We then rotate this curve around the x axis and then must find a parametrization for the surface as well as find the tangent plane in the ...
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161
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Predictor Corrector Scheme for Implicit Runge Kutta
I want to solve an ODE system :
$$
\frac{dy}{dt} = f(y, t)
$$
Since my application requires method to be symplectic, I am using an implicit runge kutta method.
$$
y_{n+1} = y_n + h\sum_{i=1}^s{b_iK_i}
...
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Implicit equation of the limacon
I am studying from the textbook Elementary Differential Geometry by AN Pressley (2nd edition).
At the end of the first chapter the Author discusses the relationship between level curve and ...
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813
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is $x^2 + y^2 = 1$ a function?
I understand that implicit functions are functions where $y$ isn't isolated and isn't immediately expressed in terms of $x$. But I wonder if $x^2 + y^2 = 1$ can even be considered a function since it ...
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162
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Implicit Equation of a Line in $\mathbb{R}^3$
The equation $x^2+y^2=0$ in $\mathbb{R}^2$ gives a degenerate circle of radius $0$ which is simply the point at the origin. In $\mathbb{R}^3$, the equation $x^2+y^2=0$ would be the point at the origin ...
- 111
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80
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Implicit function theorem for non $C^1$ mappings
I know that the inverse function theorem can be proved for differentiable mappings (not $C^1$) by requiring that $Df(x)$ has everywhere maximum rank (here is the reference https://terrytao.wordpress....
- 395
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Comparative static when optimization problems yield implicit functions
A worker's optimization problem is:
\begin{aligned}
\max_{e, s} \quad & v=\frac{(\lambda + i)U_{emp}+(1-e)U_{unemp}}{i(1-e+\lambda+i)}\\
\textrm{s.t.} \quad & \lambda = \frac{n(1-e)\left(\frac{...
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1
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88
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Implicit functions and exponentials
I am working with real numbers and trying to express u as function of x, given the equation:
$x^2-u\,e^u = 4$.
I am a bit lost because of the exponential part. I read something about the Lambert ...
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82
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Continuity of implicit function defined by a continuous and strictly monotone function
Function $f:\mathbb R \mathsf x \mathbb R \rightarrow \mathbb R$ is given to be
Continuous everywhere.
Strictly monotone as per each free variable.
Such that the implicit function $g: \mathbb R \...
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2
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207
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Implicit integral equation involving square root
I have the following integral equation stemming from a physical problem (connected with BCS theory):
$$
1 = \frac{2}{\pi}u \int_0^1 dx \sqrt{\frac{1-x^2}{u^2\phi^2 + x^2}},
$$
where $u$ is a positive ...
- 203
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1
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An implicit function - Calculating second derivative
The equation:
$e^{x}+2xy^{2}-y^{5}=0$
defines a function $y$ of $x$ in an implicit way.
Calculate:
$y''(0)$
I have tried solving this, and got that the derivative of $y$ in respect to $x$ is:
$\dfrac{...
- 333
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0
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Finding implicit form of Nyquist curve without using a general algorithm for Groebner bases.
According to Conversion Methods Between Parametric and Implicit Curves and Surfaces, by Christoph N1. Hoffmann p.3
Every plane parametric curve can be expressed as an implicit curve.
In problems ...
2
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0
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94
views
Using Lagrange multiplier for finding shortest distance to implicit surface constructed by metaballs
I am writing a software in which I need to find the shortest distance from an arbitrary point in 3D space to an implicit surface defined by a set of metaballs. I wanted to achieve this by using the ...
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0
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84
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Arc length for continuous implicit differentiable functions
We have a continuous differentiable function defined as $$F(x,y)=0$$And I am looking for a formula for its arc length between $x$ values $a$ and $b$. Doing a quick search, I could only find formulae ...
- 7,051
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84
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Implicit Equation for a Parametric equation of the following form
Say I have an two parametric equations:
$$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
67
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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
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1
answer
151
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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 ...
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0
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105
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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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4
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593
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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 ...
- 705
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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 ...
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