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

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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: $$f(t)=a(1-t)+bt.$$ ...
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2answers
17 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 ...
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2answers
31 views

Is parametric form of a given function unique? [on hold]

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 ...
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1answer
22 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
42 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
38 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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6 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
20 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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0answers
24 views

Equation for intersection of two solids. [on hold]

Parametric equation $C$ for intersection of $r=2sin \theta $ and $4=x^2+y^2+z^2$
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1answer
9 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
12 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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3 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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4 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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17 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
31 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
13 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
21 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
21 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 ...
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2answers
37 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
24 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
14 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. ...
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0answers
14 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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16 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
11 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
21 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
20 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
42 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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21 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 ...
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1answer
62 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
28 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
25 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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39 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
29 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 ...
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1answer
34 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: ...
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1answer
61 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 ...
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1answer
28 views

Perpendicular line to plane

Consider the plane $\large2x+y-5z=7$ and the line with parametric equation $\large r=r_0+tu$ a) Give a value of u which makes the line perpendicular to the plane. So I'm confused on what to do. I ...
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8 views

Parametric Surfaces: rank of linear transformation is the plane spanned by $T_u$ and $T_v$

Let $\phi$ be a parametrized surface which is both simple and smooth in a point $(u_0,v_0)$. I want to prove that: i) The rank of the linear transformation $D\phi (u_0,v_0)$ is the plane spanned by ...
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1answer
12 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. ...
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1answer
42 views

Hysteresis Model with two real parameters

I would like to ask the following: I am trying to make a throughout analysis of a Hysteresis model in one dimension, with two real parameters: $\frac{dx}{dt}=f(x,ν,μ)=νx-x^3+μ$, where ...
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17 views

Existence of Lebesgue measure in parametric function space

I am thinking about this question but I can not solve. The question is: Can I define Lebesgue measure in the space of parametric functions and if the answer is yes what is that Lebesgue measure? Could ...
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0answers
19 views

Parametric derivative of $x^2+y^2+sin(4x)+sin(4y)=4$.

I am trying to parametrize $x^2+y^2+sin(4x)+sin(4y)=4$. I need to find a way of taking the intersections between $x^2+y^2+\sin(4x)+\sin(4y)=4$, and $\tan(nx)$, as n increases from $0\le{n}\le{2\pi}$. ...
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1answer
15 views

Equation of the Tangent Line and Area of Parametric Equation

I need to find the equation of the tangent line to the point (1,0) for the equation: $x=e^{-0.1t}cos(t) \\ y=e^{-0.1t}sin(t)$ I also need to calculate the area in the first quadrant bounded on the ...
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3answers
14 views

Find the slope of a parametric equation [closed]

Find the slope of a line whose parametric equations are $$x=2-t \\y=1+2t$$
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12 views

Parametric tangent lines?

$x=t^4+t$ and $y=t^5+2$ at the point (-1,1) $x'=4t^3+1$, $y'=5t^4$ $\frac{dy}{dx} = \frac{5t^4}{4t^3+1}$ Plug in the value of -1, and I get $y'(-1)= -5/3$ What do I do from here? I use ...
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Parametric curve arc length! Am I doing it right?

$x=3t^2+2$ and $y=2t^3-1$ on $[1,3]$ The formula for parametric arc length is $\int\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt$ $x'(t) = 6t,y'(t)=6t^2$ Under the radical, I ...
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1answer
14 views

Finding the points of a paramertized curve where a tangent line has slope 3?

I have a curve at $c(t) = (-5t^2-3t+4,t^3-9t+5)$ and given a slope for the tangent line of $3$. I would like to find the point $(x,y)$ where this occurs. What I did is took the derivatives of $x(t)$ ...
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13 views

parametric representations 3d object

I'm trying to model a 3 dimensional body that is sort of ellipsoidal and am looking for parametric representation of 3D objects similar to the quadratic surface representation of a sphere or ...
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2answers
41 views

How do I convert this parametric expression to an implicit one

I have: $$x=5+8 \cos \theta$$ $$y=4+8 \sin \theta$$ With $ -\frac {3\pi}4 \le \theta \le 0$ If I wanted to write that implicitly, how would I do it? I get that it's a circle, and I can easily write ...
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2answers
46 views

Ellipse with center in origin

The purpose is to fit data to a ellipse which center is the origin $(x_0=0,y_0=0)$. I found the general quadratic curve: $$ax^2+2bxy+cy^2+2dx+2fy+g=0$$ Reference: ...
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0answers
23 views

Non-linear integral equation

Show that the function $$x(t) = \frac{1}{{\sqrt {k \cdot m} }} \cdot \int_0^t {F(\tau ) \cdot \sin \left( {\sqrt {\frac{k}{m}} \cdot (t - \tau )} \right)\,d\tau } $$ satisfies the initial conditions ...