For questions about parametric equations, their application, equivalence to other equation types and definition.

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1answer
38 views

Show that $y^2 \frac{d^2y}{dx^2}+1=0$ for a set of parametric equations.

A function of $x$ is defined parametrically by $x=t-\sin(t)$ and $y=1-\cos(t).$ How do I answer this question, then? Show that $$y^2 \dfrac{d^2y}{dx^2}+1=0.$$
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1answer
58 views

Find all $a$ for which the equation $8x^6+(a-|x|)^3+2x^2-|x|+a=0$ has more than $3$ different roots.

Find all $a$ for which the equation $8x^6+(a-|x|)^3+2x^2-|x|+a=0$ has more than $3$ different roots. I found couple of important things: First a little rearrangement: $8|x|^6+(a-|x|)^3+2|x|^2-|x|+a=0$ ...
1
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1answer
70 views

What's the difference betwen parameterizations and variable substitution for solving integrals?

Asumming I have the following integral to solve in the complex plane: $$\int \frac{dz}{z+1} $$ while $|z|=5$ which means a contour of radius 5 around zero. Is it possible to solve this integral using: ...
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2answers
91 views

Describe a twisted parabolic trough

I want to describe a parabolic trough of the form $z=x^2$ and give it a twist, like a torsion in $y$ direction. Does anybody know how I can do that? Imagine this is the trough and the $z$ direction ...
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2answers
92 views

Show three ways that $f(z)=\frac{\overline{z}}{z-1}$ is not analytic

I need to show the complex function $$f(z)=\frac{\overline{z}}{z-1}$$ is not analytic in three ways; using Cauchy's equations, geometrically, and by integrating over the circle of radius 2. I used ...
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0answers
63 views

Polynomial parametrization of a quadric with two given points

Let $X^1, X^2 \in \mathbb{R}^3$ be two distinct points of the quadric surface defined by the implicit function $$ \phi(X)= X^T\cdot A\cdot X + b^T \cdot X+c=0, $$ where and A, b and c are unknowns. ...
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0answers
58 views

Find the length of the parametric curve (Difficult)

Find the length of the parametric curve $$x = t$$ $$y = f(t)$$ $$f(t) = \int_0^t {s \over (s^2-1)} \ \mathrm{d}s$$ $$0\leq t \leq 1/2$$ First I create the $x'$and $y'$ Then put it into the ...
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1answer
49 views

Question about continuity of a polynomial curve (Spline)

I'm getting a little bit confused trying to write my own algorithm for calculating a Spline. Let's start saying that for my application I need that the curve, interpolating between more points, must ...
4
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1answer
44 views

All $a$ that equation has at least one root. $a^2+7|x+1|+5 \sqrt{x^2+2x+5}=2a+3|x-4a+1|$

Find all $a$ such that the equation has at least one root. $$a^2+7|x+1|+5 \sqrt{x^2+2x+5}=2a+3|x-4a+1|$$ What have I done: substitution $t=x+1$ and some rearrangements ...
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1answer
27 views

Given a parametrization, find the function

Consider a parametrization of $x$ and $y$ by $t$ (all real variables), say $y=f(t)$, $x=g(t)$. Given a function $f$ and a function $h$, we would like to find the function $g$ such that $y=h(x)$. ...
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1answer
78 views

Looking for help for building a Spline's algorithm 10th order

I'm trying to code the following algorithm in C++ and need help to understand the build of Splines from a mathematical point of view (found on page 129 on this paper). $$ f(t) = \boldsymbol{t} \cdot ...
2
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0answers
38 views

Pursuit Curve, Parametric Equation

So its a classic problem: Object A starts at the origin (0,0) and moves straight up the y axis with a speed v. Object B starts at point (1,0), always moves towards object A and has a speed of 2v. ...
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2answers
24 views

Show that the surface $x^2+y^2=x$ using $\theta \space and \space z$ can be parametrised by $(\cos^2(\theta), \cos(\theta) \sin(\theta), z)$

I really have no idea how to do this: $x^2-x+y^2=0$ looks like it can be a circle given by: $(x-\frac{1}{2})^2+y^2=\frac{3}{4}$ mostly $x=r\cos(\theta) \space and \space y=r\sin(\theta)$ work as ...
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0answers
32 views

Parametrization of a 3D surface

While solving the following problem: The $x$ and $y$ coordinates of a point on the $Paraboloid$ $2z = x^2/a + y^2/b$ are expressed in the form $x = atanθ cosγ $, $y = btanθ sinγ $ where $θ$ is the ...
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2answers
60 views

Möbius band and Viviani Frill

I find a common rule that unites generation of Viviani Frill and the the Möbius Band. $$ \phi =\theta $$ where $ \phi,\theta $ are spherical coordinates. Please comment if this way looking at it ...
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0answers
35 views

Interpretation of parametrization

Let $f(t)=(x(t),y(t))'$ for $t\in[0,1]$, represents a parametric function. Let us consider a parametric equation (straightline) joining two points $a$ and $b$ in 2-dimension: $$f(t)=a(1-t)+bt.$$ ...
2
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2answers
19 views

Give the explicit form of the following parametrized surface

Let $\boldsymbol{X}:\boldsymbol{R}^2\to \boldsymbol{R}^3$ be the paramtrized surface given by$$\boldsymbol{X}(s,t)=(s^2-t^2,s+t,s^2+3t)$$ I'm trying to describe the parametrized surface by an equation ...
2
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2answers
48 views

Is parametric form of a given function unique? [closed]

Can we say that for any given function in single/multivariable, it is always possible to have a parametric form? (Elementary functions, complicated functions?) Given any function, is parametric form ...
2
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1answer
24 views

Is $\gamma(t) = (|t|,t)$ plot $y = x$?

I have this parametric curve : $\gamma(t) = (|t|,t)$ with $\gamma(t) : \mathbb{R} \to \mathbb{R}^2$ And I have to say if the plot is the line of equation $y = x$. Here's my answer: $x(t) = |t|$ ...
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1answer
54 views

What does $\mathbb{R}^2$ domain mean?

I have a parametric curve defined by : $\gamma\colon\left[ 0,+\infty \right] \to \mathbb{R}^2$ defined: $\gamma(t) = (\ln(t), 3\cdot\ln(6t))$ Now I have to say if the plot of this curve is a line of ...
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3answers
42 views

Determine if 3 parametric curves have same plot

I have to determine if those three parametric curves have the same plot: $\gamma_1(t) = (\cos(t), \sin (t))$ for $t \in \mathbb{R}$ $\gamma_2(t) = (\cos(t), \sin (t))$ for $t \in [0,2\pi]$ ...
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0answers
8 views

Is there a general way to find out if a curve defined by a set of parametric equations is transcendental?

Suppose we have a curve defined by a set of parametric equations $x_n = f_n(t)$ where $f_n$ may be transcendental. Is there a known general way to find out if the curve defined by those equations is ...
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2answers
39 views

derivative of closed parametric curve

Suppose, a parametric function $\beta:[0,1]\mapsto\mathbb{R}^2$ is a closed curve, that is $\beta(0)=\beta(1)$. For example $\beta(t)=(\sin 2\pi t,\cos 2\pi t)'$. Then my question: Is the derivative ...
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1answer
15 views

Question about second derivatives of parametric equations?

Consider the parametric curve given by $$x=4+t^2,\, y=−10t^2−10t^3$$ For $\dfrac{dy}{dx}$, I found $-5(2+3t)$ For $\dfrac{d^2y}{dx^2}$, I keep getting $\dfrac{d}{dt}\dfrac{-5(2+3t)}{2t} = ...
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0answers
27 views

Finding the Area of a Torus-like surface

I'm trying to find out the Area of the following surface: Let $C$ be the curve associated to a regular, simple path $\theta:[0,l]\rightarrow \Bbb R^2 $; also assume that ...
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0answers
5 views

Underlying connections between parametric surfaces and co-ordinate transforms

I've recently been learning about parametric surfaces and surface integrals involving various co-ordinates systems. I was wondering, is there a fundamental connection between parametrization and ...
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0answers
6 views

Partial differential version of $ G(y,m) = G( \frac{y}{m} + m^{1/4}, m^{1/2}) $

I figured there must be a relation between partial differential equations and parametric equations like the wave equation in physics. I was working on something and wondering if anyone could tell me ...
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0answers
21 views

Re-parametrization of triangle

Let $\beta(t)=(x(t),y(t))'$ for $t\in[0,1]$, represents a closed parametric curve. For example, let $\beta(t)$ is a circle. So one parametrisation may be $\beta(t)=(x(t),y(t))'$ where $x(t)=2\pi ...
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1answer
36 views

Problem involving distances on a parametrically defined curve

SO the problem is stated below: A curve is defined paramterically by: $$x(t)=a\cosh t$$ and $$y(t)=b\sinh t$$ where a and b are positive constants and $-\infty < t < \infty$ The expression ...
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1answer
33 views

Find an Equation of the Tangent Plane to the Given Parametric Surface at the Specified Point.

I am given $x=u+v$, $y=3u^2$, and $z=u-v$. I need to find the equation of the tangent plane at $(2,3,0)$. I understand that the equation of the tangent plane is ...
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0answers
27 views

Find all real numbers a such that equation 3^(x2+2ax+4a−3)−2=|(a−2)/(x+2)| Has exactly two different roots x1,x2 those belong to [−4;0]

Find all real numbers $a$ such that equation $${3^{(x^2+2ax+4a-3)}}-2=|{a-2 \over x+2}|$$ Has exactly two different roots $x_1,x_2 $ those belong to $[-4;0]$ Tried plenty different things to solve: ...
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2answers
29 views

How do you find a parametric representation for a specific surface?

I trying to find a parametric representation of the plane which goes through the origin and contains the vectors $i-j$ and $j-k$. I found the cross product for these vectors and found that the formula ...
4
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2answers
45 views

How do you parameterize a circle?

I need some help understanding how to parameterize a circle. Suppose the line integral problem requires you to parameterize the circle, $x^2+y^2=1$. My question is, if I parameterize it, would it ...
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3answers
31 views

Express angular position of the Earth as a function of time

Say I have for example the Earth orbiting the Sun (in circular orbit) and I want to express angular position (in radians) as a function of time. Would I be correct in thinking that $2\pi/t$ does the ...
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1answer
23 views

Where Am I Going Wrong on this Curvature Problem?

Massively deviating from the given answer on this one, have no idea what's going on. I'm sure that I'm using the correct formulas, but the answer provided is very different from what I come up with. ...
0
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1answer
49 views

Shannon entropy on the simplex

The state of a trit $$\{p_1,p_2,p_3=1-p_1-p_2\}$$ can be represented as a triangular simplex. The centre of the simplex is the maximally mixed state $$m=\{\frac{1}{3},\frac{1}{3},\frac{1}{3}\}$$. And ...
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0answers
20 views

Parameterized Curvature Problem

Massively deviating from the given answer on this one, have no idea what's going on. Find the curvature of the following function: x = 5cos(t) ; y = 4sin(t) ; t = $pi$/4 Formula for curvature is k ...
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0answers
20 views

Horizontal and Vertical Tangents of Limicons

I feel like I am over thinking this problem, and am probably just confusing myself... So I need to find the values of $t$ where the equation $r=a+b\cos(t)$ has horizontal and vertical tangents, for ...
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2answers
25 views

Can't Seem to Solve This Parametric Arclength Problem

I'm working on an arclength question right, now, and I'm consistently landing very close to the answer but not on it. I'm trying to find the arclength of the following: $x = 6e^tcos(t) ; y = ...
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0answers
34 views

volume under the surface with parametric equations

with parametric equations: What is the volume under the surface $z=x+y$ on the plane $xy$ and interior to the surface $$(x^2+y^2)^2=2xy\space, \space \space x>0, y>0 $$
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2answers
71 views

Can the volume of a partial sphere be re-written as $V= \frac{4\pi}{3}r^{2}h$?

I am doing a related rate problem where I have a spherical surface given by the spherical equation: $$V_{sphere} = \frac{4}{3}\pi r^3,$$ $$ρ = 4(1+cos(ϕ)),$$ bounded by the $xy$ plane I have a rate ...
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0answers
38 views

Is this the correct parametrization for the intersection between a sphere and a plane?

I'm studying Stokes' Theorem. I want to compute $\oint_C (-y^3,x^3,-z^3)$ where $C$ is the intersection between the sphere $x^2+y^2+z^2=4$ and the plane $x+y=0$. Applying the Theorem, I wanted to ...
2
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1answer
70 views

Verify Stokes' Theorem

I need to verify Stokes' Theorem for the vector field $F(x,y,z)=(x^2,xy,z^2)$, for the surface, $S$, given by the part of the plane $x+y+z=1$ that is inside the cylinder $x^2+y^2=x$. For this I ...
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1answer
39 views

Show in parametric form plane in $R^3$

I have this question in my book. Show in parametric form the plane of $R^3$ that determined by these points : $$(1,0,0)$$ $$(0,1,0)$$ $$(0,0,1)$$ Does $(0,0,0)$ found on this plane? My answer The ...
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0answers
39 views

Challenging Path Integral

Let $c$ be the curve of intersection of the plane defined by $x+y+z = a$ and the cylinder $x^{2} + y^{2} = a^{2}$ ($a > 0$). Evaluate the path integral: $\displaystyle\oint_{c} \sqrt{a^2 + xy} ...
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0answers
50 views

Rational dynamical system with nonnegative paramaters and with nonnegative initial conditions

let $A$ be a rational system of the form :\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...
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1answer
35 views

Is it possible to parametrise $x^{\frac{1}{x-1}}=y^{\frac{1}{y-1}}$?

I don't know if there is a process for parametrising $y^\frac{1}{y}=x^\frac{1}{x}$ and suspect it is not possible to do so. But if it is possible, is it also possible for the similar ...
2
votes
1answer
62 views

Surface integrals: Find the area of the portion of the cone $x^2+y^2=z^2$ above the $xy$ plane and inside the cylinder $x^2+y^2=ax$

I need to find the area of the portion of the cone $x^2+y^2=z^2$ above the $xy$ plane and inside the cylinder $x^2+y^2=ax$ . For this, I used cylindrical coordinates to parametrize the region: ...
1
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1answer
63 views

Let $S$ be the surface generated by the circles of radius $b$, find a parametric expression for $S$

Let $C$ be the curve associated to a regular, simple path $\theta:[0,l]\rightarrow \Bbb R^2 $; also assume that $((x'(s))^2+((y'(s))^2=b^2$ and let $S$ be the surface generated by the circles of ...
0
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1answer
15 views

How can I parameterize these angles

My angle starts at 90, goes down to 0/360, then down to 270. This is a 180 degree range of motion. How can I express these angles from 0 - 180 instead of 90 - 270, where 90 gives 0 and 270 gives 180. ...