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Questions tagged [implicit-function]

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Prove that $\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$ given a function [duplicate]

I am trying to solve the following exercise. I have the feeling that it is a very simple one, but I am lost. I do not know where to start, because you do not have an specific function. The exercise ...
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21 views

Implicit differentiation in multivariable calculus

What I don't understand is the disconnect between the reasoning for implcit derivation in single variable and multivaraible calc. in single variable calc they define say $y = f(x)$ for a small ...
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15 views

Implicit functions and partial derivatives

I am trying to solve the following exercise about implicit functions and their partial derivatives: If $u=x+y+z$, $v=x^2+y^2+z^2$ and $w=x^3+y^3+z^3$ prove that: $$\frac{\partial x}{\partial u}=\left(...
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Total Differential of an Equation

I want to find the total differential of an equation which has been defined as: $ Y = C((1-t)Y, M/P)$ where $t$ is a parameter and $M$,$P$, and $Y$ are variables. And Y is a function of C which in ...
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2answers
41 views

Implicit form for a logarithmic spiral

I was wondering if there is an implicit form for a logarithmic spiral. For example, if $$ x=e^{-t}\cos(t)\\y=e^{-t}\sin(t)$$ we can write $x^2+y^2=e^{-2t}$ and $y/x=\tan(t)$ which yields $$x^2+y^2=...
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2answers
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equation for an hexagon with rounded corners

I know a simple equation for a squircle, $x^4+y^4=a^4$ What would be the equation for an hexagon with rounded corners? In the equation, how can I take in account for the angle of rotation of the ...
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65 views

Convergence of an implicit series

I have an implicit solution for variable $y$ as a function of time $t$ and two other variables $y_0$ and $y_f$ which are initial and final values, respectively. $$ tC=\frac{1}{y_f^2}\ln\left(\frac{y}{...
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1answer
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How can I solve $x = e^{a+bx} + c?$

I need to solve this implicit equation for a physical system. I know that the similar equation $x = xe^x$ (solved with the Lambert W-function) doesn't have real roots, but since this equation has ...
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17 views

What is the class of implicit surfaces for which the distance to an external point is returned by its equation?

Let's define an implicit surface as the set of all points for which a function f : R3 -> R equals 0. A sphere around origin, for example, can be defined as ...
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3answers
38 views

How or why are implicit functions actually functions?

I'm a high schooler and today my teacher taught us the implicit differentiation, in which he gave us a very brief explanation of implicit function. I didn't quite get it at that time so I decided to ...
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1answer
24 views

array of $x$ and $y$ in an implicit plot in Matlab

If I use implicit plot of a relation (say $x\cdot y+\sin(x+y)=0$ ) and obtain a plot. How do I get the list/array of the values of $x$ and $y$ plotted?
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Finding random points and the $x$and $y$ intercepts of two implicit curves in Matlab

I plotted two implicit functions ($f_1$ and $f_2$) in Matlab using the function fimplicit(f1,[0 1 0 10]) fimplicit(f2,[0 1 0 10]) and I had the following figures ...
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0answers
42 views

Study of an implicit function

The problem consists of two questions : The first asks to prove that for any real number $x$, there exists a unique real numbers $t$ such that $$t_x^3x^2+t_x+x=0$$ I'm having problems with the ...
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30 views

Continuity of implicit function $y-\epsilon \sin{y} =x$

Suppose $0<\epsilon<1$ such that $$y-\epsilon \sin{y} =x$$ is defined as a continuous function for $x$, that is, $y=y(x)$ satisfies $$y(x)-\epsilon \sin{y(x)} = x$$ Prove that $y(x)$ is ...
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normal vector to a hypersurface reference

I'm three years university student and I saw the coarea formule and some others (green, superficial measure, green-gauss,...) so I need to compute normal vector to hypersurface. My semester will ...
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1answer
38 views

Given a cartesian equation get points in the plane

I hope this make sense. I'm trying to understand (I'm very newbie with curves) how could I get points from a cartesian equation. For example, given $(x^2+y^2)^2-2a^2\cdot(x^2-y^2)-a^4+c^4=0$ that is ...
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1answer
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How can we find some radius of circle which fully contains $x\arctan(x)-ax+y\arctan(y)-by=0$?

How can we find some radius of circle with center at origin which contains $x\arctan(x)-ax+y\arctan(y)-by=0$, where $\pi/2>a>0$ and $\pi/2>b>0$. I'm not sure how can we prove that ...
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2answers
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Find $\frac{\partial y}{\partial z}$ of the surface $g(s,t)=(s^2+2t,s+t,e^{st})$ near $g(1, 1) = (3, 2, e)$.

Consider the surface given by $g(s, t) = (s^2 + 2t, s + t, e^{st})$. Think of $y$ as a function of $x$ and $z$. Find $\dfrac{\partial y}{\partial z}(3,e)$ near $g(1, 1) = (3, 2, e)$. really ...
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1answer
67 views

How can we find minimum radius of circle which contains $\arctan^2(x)+\arctan^2(y)=a$?

How can we find minimum radius of circle which contains $\arctan^2(x)+\arctan^2(y)=a$, where I think $a<\pi/2$? For example I have some plots from WolframAlpha and I see it depends on $a$. But ...
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1answer
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How can we find some radius of circle so $-x\arctan(x)+0.2x-y\arctan(y)+0.9y=0$ will be fully inside this circle?

If he have this region $$ \begin{align} \ -x\arctan(x)+0.2x-y\arctan(y)+0.9y=0\\ \end{align} $$ How can we find some $R$ (maybe minimum) so this region will fully inside this circle $(x-a)^2+(y-b)^2=...
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How many points are required to fix a given curve from a locus

Say I have a locus on the $x,y$ plane defined by the parametric implicit function $$F(x, y, p_1, p_2, ..., p_n)=0$$ where $p_1, p_2, .., p_n$ are parameters. I now want to constrain $F$ to pass ...
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1answer
86 views

Implicit functions and related differential equations

I'm seeking guidance in derivation of implicit equation solutions to second degree differential equations.In the example below, differentiating twice just produced a tangle of terms which did not ...
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1answer
50 views

Lagrange inversion formula example unclear

The following example is from De Bruijn's Asymptotic methods in analysis (page 24). The considered equation is $x^t = e^{-x}$ The author wants to transform the equation into the form: $w=z/f(z)$, ...
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1answer
73 views

How to write a system of ODEs in proper state-space form?

I have the following system of differential equations: $\dot{x}_1=f_1\left(\dot{x}_1,x_1,x_2\right)\\ \dot{x}_2=f_2\left(\dot{x}_1,x_1,x_2\right)$ where $\dot{x}_1$ and $\dot{x}_2$ are time ...
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83 views

The solution of ODE is not the same as the implicit equation it was used for.

Lets say i have a variable named $p$. With this variable i can calculate the variables $\frac{dU}{d\varphi}$ and $\frac{dQ_{w}}{d\varphi}$ and with equation shown below also $\frac{dQ_{b}}{d\varphi}$. ...
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2answers
70 views

Is there a real function $f$ on $(0..1)$ such that $x·f·\log f = 1$?

I’m looking for a real-valued function $f$ on $(0..1)$ such that $x · f · \log f = 1$. Is there such function? Is it integrable on $(0..1)$? Why? This could yield an integrable function $f$ such that ...
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1answer
17 views

Function describing trajectory of a point on a rolling circle

Let us take a 2D unit circle $C$ with center in $M=(0,1)$ and fix a point on $C$, say $A=(0,0)$. Now we start to unroll the circle to the right and watch how $A$ moves; firstly, it turns around $M$ on ...
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0answers
40 views

Examples of smooth implicit curves and surfaces

I am currently constructing a method to approximate implicitly given plane curves and surfaces, which are smooth and single-sheeted. Now I have finished writing a Matlab function doing all the ...
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2answers
75 views

Integration of a function provided as an implicit function [closed]

Let a differentiable function $f$ satisfies the functional rule $$f(xy) = f(x) + f(y) + xy - x -y $$ for all values of $x,y > 0$ and $f'(1)=4$. Based on this statement there were three questions: ...
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2answers
154 views

Solving implicit equations in R [closed]

I have an equation which looks like this: $$\frac{1}{\sqrt{𝑥}}=−2\log⁡\left(𝐴+\frac{𝐵}{\sqrt{𝑥}}\right)$$ where $A$ and $B$ are constants How do I solve this in $\mathbb{R}$? Can this be solved ...
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0answers
98 views

Solving for an implicit function numerically?

I am trying to come up with a systematic way for solving an implicit "functional" equation. That is, given $$g(x) = f(x + a \cdot f(x))$$ how would one (numerically) recover the functional form of $...
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2answers
126 views

How to find the self-intersection point of $x^x=y^y(x,y>0)$? [duplicate]

As the figure below shows, the graph of the implicit function $$x^x=y^y,(x,y >0)$$ composes of a straight line and an arc, which of the two have an intersection point $P$. How to find the ...
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2answers
372 views

Integration of implicit functions [closed]

As we know an implicit function $f(x,y)$ is an expression containing $x$ and $y$ such that none of them can be seperated out. So my question is how can we integrate or simply find area under an ...
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2answers
47 views

Evaluating implicit functions numerically

A simple question. Are there any methods on how to evaluate an implicit function f(x,y) = 0 numerically? Does one fix one of the variables and then use newtons method or similar on f(x0,y) and then ...
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0answers
43 views

Tangent plane of implicit surface from normal derivative

Let $f : R^3\rightarrow R$ be the implicit equation of a surface embedded in $R^3$, where the surface is defined by the $0$-isosurface $\left\lbrace (x,y,z)\in R^3,\ f(x,y,z)=0 \right\rbrace$. I ...
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1answer
28 views

Bernoulli's Lemniscate: Extremes and Intersection with the axis

Given the equation: $$(x^2+y^2)^2 - 2p^2(x^2 + y^2) = k^4 - p^4$$ Determine the extremes and intersections with the axis. I know it requires the use of implicit functions but I'm not sure how to ...
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1answer
69 views

Implicit derivative, implicit integral?

An undergraduate student of mine asked me last year the following question. Let $f:\mathbb R^2\to \mathbb R$. The equation $f(x,y)=0$ defines implicitly a function $y:\mathbb R\to\mathbb R$ and we ...
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1answer
39 views

Solve for $x(t) : x(t)= b+at+ ∫ ( ∫\sin (wt-kx(t)) \, dt) \, dt$

$$ x(t)= b+at+\int \left (\int \sin (wt-kx(t)) \,dt \right ) dt$$ How do I extract $x(t)$ from the above equation. Any hint? Or is this equation has been already solved? Thanks in advance for any ...
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1answer
88 views

Transcendental and implicit functions

Def. An implicit function is a function that is defined implicitly by an implicit equation, by associating one of the variables (the value) with the others (the arguments). For example, $$F(x,y)=0$$$$...
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Analysis, implicit functions

Let $X=(x_0,y_0)=(1,1) \in \mathcal{R}^2$. Prove that there exists $\rho>0$ and continuously differentiable functions $u,v,w: B_{\rho}(X) \to \mathcal{R} $ such that $u(X) = 1, v(X) = 1, w(X) = -...
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1answer
64 views

Parametrizing $(2x+y)^2(x+y)=x$

I need to find $x=x(t)$ and $y=y(t)$ so that the implicitly defined curve on $\mathbb R^2$ $$(2x+y)^2(x+y)=x$$ is converted into an explicit function of the parameter $t$ that can be analysed using ...
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1answer
141 views

Is it possible to execute line integrals of non-conservative vector fields on curves defined by implicit relation such as $\sin(xy)=x+y$?

I want to execute the line integral (analytically) of a vector field over the curve defined by implicit function $\sin(xy)=x+y$ for some $x=a$ & $y=c$ to $x=b$ & $y=d$. The difficulties I ...
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0answers
188 views

Area of region in the Dog bone-shaped curve: $|x|^x=|y|^y$

EDITED: Q11 is answered. Please have a look at this origin post: Dog bone-shaped curve In case not to make the question too long and make it horrific, I decided to open a new post for $2$ new ...
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3answers
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Can a 'closed' curve be a function?

For a 'closed' curve (I don't know if there is such in mathematics called closed curve, but I mean a curve which is 'closed') e.g.1)It is well-known unit circle with equation$$x^2+y^2=1$$e.g.2)It is ...
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10answers
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Dog bone-shaped curve: $|x|^x=|y|^y$

EDITED: Some of the questions are ansered, some aren't. EDITED: In order not to make this post too long, I posted another post which consists of more questions. Let $f$ be (almost) the implicit ...
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0answers
113 views

Evaluating implicit functions numerically by transforming the problem into ODE

It just occured to me that if we need to evaluate an implicit function given by a nonlinear equation for a range of values, common root-finding methods, like Newton-Raphson, might not be the most ...
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0answers
50 views

Is there a simpler equation for this 2-d Analog to Coefficent of Variation?

I'm creating a test that measures how evenly distributed is closed curve $\mathcal{C}$. This requires the average ray length (from point $(u,v)$ to the boundary of the shape) inside $\mathcal{C}$ ...
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1answer
30 views

Get explicit, vector-valued function for a curve defined by an implicit expression.

$ \mathcal {B} = \{(x,y) \in \mathbb R^2 $ such that $ 0 = x^2 + y^2 + xy \exp (-x^2) \} $ $ \mathbf f(x,y) = f_1(x,y) \mathbf i + f_2 (x,y) \mathbf j = \mathcal {B} $ Please show explicit ...
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1answer
41 views

Finding the equation of aline in implicit form

If I am given two points, (-2,0) and (0,-1), how do I find the equation of the line in implicit form that passes through these two points? Additionally, how do I convert that equation to point-normal ...
0
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1answer
35 views

Implicit function and polynomials

Can someone give me an example what is the difference between an implicit function and curve with polynomials as coordinate system? I am learning basic math concepts and I do not understand those two ...