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

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0
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3answers
30 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 ...
1
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1answer
21 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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0answers
24 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
18 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
24 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
31 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
55 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
33 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
68 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
37 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
33 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
49 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}} \\ ...
1
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1answer
34 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
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1answer
51 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
13 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
53 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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0answers
21 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 ...
0
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0answers
22 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 $0\le{n}\le{2\pi}$. ...
0
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1answer
18 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 ...
0
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0answers
14 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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0answers
30 views

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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votes
1answer
16 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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2answers
43 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 ...
2
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2answers
56 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
25 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 ...
2
votes
1answer
24 views

Computing derivative of parametric equation

This is probably a silly question but I am just not sure if I understand what to do. So I have the parametric equations: $x=6\cos (t)-2\\ y=5\sin (t)+3$ I am asked to compute $\dfrac{dy}{dx}$ at ...
1
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1answer
27 views

When is $x=sin(at), y=sin(bt)$ symmetric to x and y axes?

Take the simple system of parametric equations, $$x=\sin(at)$$ $$y=\sin(bt)$$ where $a,b \in \Bbb{N}$. When is this curve symmetric with respect to both the $x$ and $y$ axes? In other words, what ...
2
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0answers
40 views

Famous parametric curves that are solutions to differential equations

I know that the cycloid satisfies the differential equation $ \left( \frac{dy}{dx} \right)^2 - \frac{2r}{y} + 1 = 0. $ Are there other famous plane curves that are also solutions to a differential ...
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0answers
20 views

Parameterization of a closed curve on a sphere

I'm looking for a parameterization of a closed curve C on a sphere. assume the projections of C on y-z, x-z, x-y plane are f(x), g(y), h(z), respectively, and ${\oint}f(x)dx={\oint}g(y)dy=0$, and ...
0
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1answer
80 views

Determine if a parametric equation's trajectory lies on a circle?

I have a question as follows: Determine whether the following trajectory lies on a circle. If so, find the radius of the circle and show that the position vector and velocity vector are everywhere ...
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0answers
19 views

Make equation on nonparametric form

I have the following Points $(-1,-2,-6)$ and $(-1,-2,-12)$ if I write the line on parametric form I get $$x = -1 + (0*t)\\ y = -2 + (0*t)\\ z = -6 + 6t $$ I know how to solve it if I have more ...
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0answers
50 views

Find a complete sufficient statistic

Here is my problem : Suppose theta is a nonrandom parameter satisfying theta > 1. Suppose further that, given theta, Y1 , Y2, ... , Yn are i.i.d. observations with each density f_\theta(y) = (\theta - ...
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1answer
59 views

Parameterizations of Lines.

Which of the following equations give alternate parameterizations of the line L parameterized by: r(t)=(1+2t)i +(2+2t)j -(1+4t)k? a. -(1+t)i-t*j+(3+2t)k b, (3-2t)i+(2-2t)j+(3-4t)k c. ...
7
votes
1answer
126 views

Length of parametric curve $\phi(t)=(f(t)\cos(t),f(t)\sin(t))$

Define the curve $\phi$ by $\phi(t):=(f(t)\cos(t),f(t)\sin(t))$, where $f$ be a strictly increasing infinitly many differentiable function . Find an explicit formula for the length of $\phi$ ...
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1answer
145 views

Parameterizing an ellipse

Given the ellipse $(x-1)^2 + \frac{y^2}{4}= 1$, parametrize the curve in polar coordinates. I've forgotten something very basic here. Can someone help get me started?
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2answers
34 views

Where does the line x = 2 − t, y = 3t, z = −1 + 2t intersect the plane 2y + 3z = 6?

form this 2y + 3z = 6 equation i take the x = 0. therefor 2 - t = 0 and ...
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0answers
32 views

An example of a space curve with given normal and osculating planes

I am student currently taking calculus 3 and I recently was given a quiz with a very difficult question. The question relates to the chapters in my book which talk about "Arc Length and Curvature" and ...
0
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1answer
25 views

Horizontal and vertical tangents to a parametric curve

I'm not sure what my procedure should be when solving this problem: find all points with a horizontal tangent find all points with a vertical tangent find all inflection points $$x(t) = ...
0
votes
2answers
21 views

Paramertized ellipses

I have $a^2 = (4x-8)^2 + 4y^2$ and $b^2=(4x+8)^2 + 4y^2$ which I switch between every $t=\frac{n\pi}2$ How do I draw this touching the origin, and moving outwards, noting that ...
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5answers
76 views

Evaluating a contour integral where C is a square

I've been working problems all day so maybe I'm just confusing myself but in oder to do this, I have to the take the integral along each contour $C_1-C_4$ My issue is how to convert to parametric ...
0
votes
1answer
33 views

How can I get a smooth distortion on a circle with a function g(x,y)

Let's say, $$f(x,y)=x^2+y^2=1$$ gives the unit circle. Now I would like to get a smooth distortion on the circle with a function $g(x,y)$. my guess is to consider the perimeter as one dimension, so ...
14
votes
1answer
245 views

Connected unbounded sets $S\subset \Bbb{R}^n$ such that $x\mapsto ||x||^t$ is uniformly continuous on $S$?

Spending the night perusing my old answers, and this question left me wondering about the following. Let's equip $\Bbb{R}^n$ with the usual Euclidean metric, and let us consider the map ...
0
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1answer
31 views

Equation for simple parametric curve

My math skills are rusty. I want to find the parametric equation for the 5 vertices curve below. It consists of an ellipse with a rotating axes. I get stuck after this: $$x = a \cos(t) \cos(\theta) - ...
0
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0answers
17 views

How to design own the parametric vector?

I try to design the parametric vector that looks like a roller coaster I know that my equation will like $r(t) = A\sin(t)i + Btj+ C\cos(t)k$, but i want ...
0
votes
1answer
25 views

curve with vanishing tangent vector assumption

I am just reviewing some assumptions in Parametric representations The book says we assume 3-d curve has non-vanishing tangent vector. Why do we need to assume this Simply if we take $R^3$ then ...
1
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0answers
30 views

Parameterize the equation

Find a way of parameterizing the following curve: $y^2=\sin x $. I have already tried $x(t) = (\sqrt t, \sin^{-1} t) $ but this only gives part of the curve because of the nature of the sqrt function ...
0
votes
2answers
24 views

Show that if $v_1$ and $v_2$ are any two vectors in this plane, then for any two scalars, $c_1v_1 + c_2v_2$ is also a vector in the plane

Let $a,\,b$ and $c$ be constants (not all zero) and consider the equation $ax + by + cz = 0$, which has a graph that is a plane that passes through the origin in $\mathbb{R}^3$. Show that if $v_1$ and ...
0
votes
1answer
24 views

Is this parametric equation correct?

Am I able to just put the planes point into the equation and leave it at that? or am I wrong here? My parametric equation: (x,y,z) = (3,-2,1)+ t(2,1,-3) + s(1,-2,4)