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

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Arc Length parametric curve

I have the following curve: $$x = \cos(t)$$ $$y = t - \sin(t)$$ $$0 \leq t \leq 2\pi$$ I have to draw the graph, point the direction and find its length. The solved the first two questions. The ...
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22 views

When is it possible to eliminate the parameter from a set of parametric equations?

I found this question on-line, I was unable to source it. Is it poorly written? For example, is there a case (C) where you can have a non-parametric form given, and also have a parametric form ...
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1answer
13 views

two equivalents equations with different representations in the plane

Consider the parametric curve $$C:\begin{cases} x = 4e^{t/4} \\ y = 3e^{t} \\ \end{cases} $$ A cartesian equation for this curve is $y=\frac{3x^4}{256}$. The problem is that ...
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1answer
32 views

Points of intersection of two parametric curves

I want to find all points of intersection of the parametric curves $$C_1:\begin{cases} x = t+1 \\ y = t^2 \\ \end{cases} $$ and $$C_2:\begin{cases} x = ...
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2answers
20 views

What does the Cartesian equation of a Parametric function tell us?

I've been taught that a Parametric function can be converted into a Cartesian one by eliminating the parameter $t$ but I've never been taught of how it specifically relates to the Parametric. Does it ...
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1answer
43 views

What is the correct math notation for the final solution while finding the rank of this matrix? [closed]

What are the respective different ranks of the matrix ? I tried with all parameters $a,b$ and $c$ being zero , and then $c$ being $0$. $$\left( \begin{array}{ccc} 1 & 4 & 3 \\ 5 & a & ...
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36 views

Verification of divergence theorem

At time t the velocity field of a fluid is given by $\bar V(x,y,z)=x^3 {\bf i}+y^3{\bf j}+z{\bf k}$ the outward flux integral $ Φ = \iint_S \bar{V}\cdot d\bar{S} $ where S is the surface of the ...
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22 views

Is it acceptable to call curves on parametric surfaces “isoparms”?

Let $\mathbf{r}(u,v):[a_0,b_0] \times [a_1,b_1] \to \mathbb{R}^3$ be a parametric surface. If $u$ and $v$ are fixed, is it allowed to call $\mathbf{r}(u,\cdot)$ and $\mathbf{r}(\cdot,v)$ "isoparms" or ...
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1answer
49 views

Why are there so many different vector and parametric equations for a line? [closed]

Please explain why there are many different vector and parametric equations for a line.
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1answer
22 views

Show that the line with parametric equations don't intersect

Show that the line with parametric equations $x = 6 + 8t$, $y = −5 + t$, $z = 2 + 3t$ does not intersect the plane with equation $2x − y − 5z − 2 = 0$. To answer this do i just plug in the $x$, $y$, ...
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1answer
20 views

Determining parametric equation given 3 points [closed]

Determine parametric equations for the plane through the points $$A(2, 1, 1), B(0, 1, 3), C(1, 3, −2)$$
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3answers
132 views

Three questions about the form $X^2 \pm 3Y^2 = Z^3$ and a related lemma

In Ribenboim’s Fermat’s Last Theorem for Amateurs, he gives the following lemma [Lemma 4.7, pp. 30–31]. Lemma. Let $E$ be the set of all triples $(u, v, s)$ such that $s$ is odd, $\gcd(u,v) = 1$ and ...
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2answers
22 views

Continuity and differentiability of a function defined parametrically

How do we check continuity and differentiability of a function defined parametrically e.g. $$x=2t-|t-1|$$ and $$y=2t^2+t|t|$$
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2answers
30 views

Area of described parametric region

The problem is as follows: A rope is tied to a cow and attached to the side of a circular silo with radius $r$. If the rope has length $\pi r$, what is the area of the land available for grazing ...
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1answer
25 views

Parabola that intersects two lines and matching the slope of the two lines?

If I have two lines with equations;$$x=0$$ $$y=0$$ $$z=t$$ and $$x=t$$ $$y=10$$ $$z=t$$ are there any parabolas that cross through the two lines and in which the parabola matches the slope of the ...
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2answers
55 views

How to parametrize $\left(4-\sqrt{x^2+y^2}\right)^2 +z^2=1$

How would I parametrize $$\left(4-\sqrt{x^2+y^2}\right)^2 +z^2=1$$ I am really struggling to parametrize this surface. Here is what I observed the surface is $$(4-r)^2+z^2=1$$ so perhaps we can try ...
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2answers
24 views

How to parametrise shapes such as petals and cardioids?

Okay for example I want to compute a line integral along the curve described in polar coordinates by $r=\sin(2\theta)$ so I will need to parametrise this curve. (In fact I only need to parametrise one ...
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1answer
47 views

Eliminating a parameter from 2 equations

The question given to me was actually of parametric differentiation, and the equations were: $$x = \dfrac{\sin^3 t}{\sqrt{\cos2t}}\ , \ \ \ \ y = \dfrac{\cos^3 t}{\sqrt{\cos2t}}$$ and we had to ...
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1answer
33 views

Need help with unit circle trig coordinates.

I'm in over my head and need some help with this question. Sorry if this is too simple for you but I'm really struggling. I can't for the life of me figure out how to write the angles A in terms ...
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1answer
31 views

Question about Flux and direction of normal

I am trying to do the following question; calculate the flux Suppose $$F(x,y,z)=(-x)i+(-y)j+(z^3)k$$ over the cone $z=\sqrt{x^2+y^2}$ between $z=1$ and $z=3$ with downward orientation My attempts: ...
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0answers
38 views

Can parametric equations graph all kinds of lines?

I saw this question which had a similar viewpoint, but was limited to straight lines and polynomials. Now we know that we can graph some pretty crazy stuff with parametric equations. For example: ...
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1answer
24 views

Parametric derivatives

Let $f(x) = \dfrac{2\sqrt{1+x^2}-5\sqrt{1-x^2}}{5\sqrt{1+x^2}+2\sqrt{1-x^2}}$. Hence, find $\frac{dy}{dz}$ when $y=\cot^{-1}(f(x))$ with respect to $z=\cos^{-1}{\sqrt{1-x^4}}$. To get this into a ...
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19 views

Non-Uniform grid

Let say I have $v_0\in [v_1,v_m]$ (say $v_0=0.04\in [0.004,0.24]$) I would like to find a $1$-to-$1$ map that map $[0,1]$ to $[v_1,v_m]$ and more cluster points around $v_0$ from two sides. It seem ...
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2answers
23 views

how to choose the suitable parametric form given a boundary?

Find the absolute minimum and maximum values of $g(x, y) = (x^2 + y^2)e^{(−x^2−4y^2)}$ on the set $A = \{(x, y) \in \mathbb R^2 \mid x^2 + 4y^ 2 ≤ 4\}$. here is the solution //see image so my ...
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2answers
40 views

Prove that if $x = \sqrt{a^{\sin^{-1} t}}$ and $y = \sqrt{a^{\cos^{-1}t}}$ then $\frac{dy}{dx}$ = $-\frac{y}x$

Prove: If $x = \sqrt{a^{\sin^{-1} t}}$ and $y = \sqrt{a^{\cos^{-1}t}}$ where $\sin^{-1}$ and $\cos^{-1}$ are inverse trig function, show that $\frac{dy}{dx}$ = $-\frac{y}x$ Unfortunately I ...
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2answers
65 views

Solving math problems involving extra variable (p)?

I have a very hard time solving problems in which you have to solve for the additional unknown variable. I would like to know whether there is some method I can learn or approach I can simulate in ...
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1answer
61 views

Solve $\tan(2t)=1$

My textbook is listing solutions to this equation as $2t=\pm \frac{\pi}{4}$ and $2t=\pm \frac{5\pi}{4}$ however this doesn't seem correct at all, I believe the only solutions should be ...
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17 views

Trouble finding the cartesian equation for a given parametric form

I'm given following parametric form $x = \cos(\sin(s))$ and $y = \sin(\sin(s))$ for $s \in \mathbb{R} \setminus 0$ I now need to determine the cartesian equation and draw the curve. I reasoned ...
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1answer
40 views

Why does Clockwise Rotation change the roles of Sine and Cosine?

Simple question: I've been asked find a parameterization of the circle of radius $2$ starting at $(2,0)$, moving in the counterclockwise direction. Simple enough I get $(2\cos(t),2\sin(t))$ because ...
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2answers
34 views

How to determine the period of the following Lissajous figure?

How do I determine the period of the following Lissajous figure? $$ x(t) =\cos(2t)-\sin(t)\\ y(t)=\cos(t-\frac{\pi}{3}) $$ Highly appreciated, Bowser
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1answer
67 views

Write $x^2+y^2=25$ as a vector valued function

How can I write $x^2+y^2=25$ as a vector valued function? At first, I tried letting $x=t$. Then, $y=\pm \sqrt{25-t^2}$. So, $r(t)=t \hat{i}+ \sqrt{25-t^2}\hat{j}$ Would this be correct? What ...
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0answers
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Parametric equations and specifications of a logarithmic triskelion (triple spiral)

There is a post in this forum that shows how to create an Archimedean triskelion: Parametric equations and specifications of a triskelion (triple spiral) ...
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1answer
33 views

Diffirent ways to write vector equations

Hi I am having some trouble with the following: When I am given some force F and it is in terms of components ie with respect to i, j, k then I have no issue using it to solve line integrals etc, my ...
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42 views

Find the exact area of the region enclosed by the curve of parametric equations.

Find the exact area of the region enclosed by the curve given by $$x=9-t^2$$ $$y=e^t$$ where $-3 \leq t \leq 3$ and the $y$-axis. I tried to take the integral of the $x$ function minus the $y$ ...
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2answers
40 views

Evaluating the integral $S=\int_0^12\pi t^4\sqrt{9t^2+4}dt$

I want to find the surface area $S$ by rotating the curve about the $x$-axis $$ \begin{cases} x = t^3 \\ y = t^2 \end{cases} ,t\in [0,1] $$ At some point I find ...
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1answer
24 views

Graph of parametric vector equation

The question is asking me to sketch the graph of the parametric vector function $$\vec r(t) = \vec at+\vec bt^2 $$Where $t$ is a real number, $\vec a$ and $\vec b$ are constant non-parallel non-zero ...
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1answer
51 views

Evaluate the integral $\int_C z^2 dz$ where $C$ is a graph of $y=\sin x$

$\int_C z^2 dz$ $C: y=\sin x$ $x=0$ to $x=3$. My attempt: I'm not sure if I have this correct - this is a line integral? If so, a line integral is the same concept as a regular integral in the ...
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0answers
22 views

Parametric equation for surface of revolution

Given a parametric curve that can be rotated about the $z$-axis: $$P(t) = \langle 0, a \sqrt{t} + b \sin(t), ct \rangle, \quad t \in [0,2 \pi)$$ Give a parametric equation and appropriate parameter ...
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1answer
14 views

Finding parameter for quadratic equation

Given $x^2 - 3ax + a^2 = 0$ and $$\frac{x_1^4-x_2^4}{\sqrt{5}x_1x_2} + x_1 + x_2 -20x_1x_2 - 4 = 0$$ Find $a$. The answer is $1$ ($a = 1$) I tried to present $x_1^4 - x_2^4$ as ...
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3answers
35 views

Converting a Parametric equation into a Cartesian one

I was working on converting an parametric equation into a Cartesian one and i cant seem to figure this one out. I was hoping you could help with that for this equation of a cycloid, Thanks $x = ...
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0answers
20 views

The different between Non-parametric statistics and Parametric statistics?

I have 1D data. I want to classify the data to $N$ cluster. The two common ways can use Using the mean/average value as a criterion to classify Assumption that the data follows a distribution with ...
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0answers
18 views

Parametric equations for mobius strip and klein bottle

How do you fund the parametric equations for a mobius strip and klein bottle? I know I can just look up the equations online but I want to know how you get those equations.
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1answer
37 views

Vector and Parametric equations of Planes

I really can't understand what is going on in this example. I understand the vector and parametric equations of a plane in R3, and I understand the examples below and above this one where they give a ...
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1answer
29 views

Problem using Parametric Equation of Semicubical Parabola

I've been working my way through an old A'Level maths book and am having a lot of difficulty with a problem given in the Chapter on Loci & Parametric Equations: "Find the equation of the tangent ...
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1answer
22 views

How to find right parameter and calculate the work done

Hi I am having trouble calculating the work done in moving a particle from $(-1,2,5)$ to $(1,0,1)$ where $F=yi+xj+zk$ on the curve C, where the curve C is the intersection of $z=x^2+y^2$ and the plane ...
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1answer
28 views

Find angle of line and time of impact for a line between two parametric circles.

I am trying to find the angle of a parametric line so that it will intersect a circular parametric curve when both of their parameters are equal. I also need to have the line's start position be ...
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2answers
37 views

How to predict symmerty of parametric curve

Suppose we are given a curve as $$x^{2/3} + y^{2/3} = 1 $$ In parametric form it can be written as $$x=\cos^{3}(\theta)$$ and $$y=\sin^{3}(\theta)$$ now how can we predict if curve will be symmetric ...
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0answers
31 views

Finding foci, asymptotes, and a vertices of a hyperbola given an equation

I'm given the equation $4x^2-y^2-24x-6y+23$ and asked to find the foci, vertices and asymptotes. The book showed me how to do it given an equation in the form of $(x^2/a)-(y^2/b)=1$, but didn't show ...
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1answer
26 views

Find conditions on $C$ and $C^{\prime}$ so that the spirals $r = Ce^{\varphi/a}$ and $r = C^{\prime}e^{\varphi/a}$ are the same

This question is related to one I asked here about the logarithmic spiral. In the linked problem, I had to find and sketch the image of the straight line $z=(1+ia)t+ib$, for $-\infty < t < ...
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2answers
45 views

Find and sketch the image of the straight line $z = (1+ia)t+ib$ under the map $w=e^{z}$

I need to find and sketch the image of the straight line $z = (1+ia)t +aib$, where $-\infty < t < + \infty$, $a,b\in \mathbb{R}$, and $a \neq 0$, under the map $w = e^{z}$. In order to ...