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

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Solving for the parametrization

I was wondering when evaluating line curves, and C is given by something such as $y = x^2$, how do you find the parametrization $<t, t^2, 0>$ ? ( I understand how z was found but not so much x ...
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1answer
21 views

equation of tangent line and parametric equation of tangent line (are they equivalent)

I am not sure about the steps to finding parametric equations of tangent lines and was wondering if these statements are equivalent Is there a difference if I am asked to find: equations of ...
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1answer
30 views

Is there an algorithm for parametrization of equations?

In this and this Math.SE questions askers wanted to parametrize their equations. It seems to me that one has to, without the algorithm, figure out a symbolic trick and then symbolically manipulate ...
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1answer
25 views

Parametric equation with image of a function.

Find all values of $a$ for which the image of the function $$y=\frac{\sqrt{a}-2\cos x+1}{\sin^2x+a+2\sqrt{a}+1}$$ contains $[2, 3]$. Now, I've already transformed it to ...
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1answer
11 views

Find the length of a curve specified by a series of polar co-ordinates.

I have a curve defined by a series of polar co-ordinates, $P_a(r_a,\theta_a)$ through $P_b(r_b,\theta_b)$. I would like to determine the length of this curve. Because the points are from ...
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14 views

Intersection of curves and constructing a plane

Can someone please help me with how to approach/solve this question? Show that the following pair of curves intersect, and construct a plane that is tangent to both curves at the point of ...
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2answers
28 views

Eliminating the parameter?

How would you eliminate the parameter where the x coordinate is in terms of t, but the t is squared. x= 3t - $t^2$ y= t + 1 I know to solve for y as a function of x, but I'm not sure how to do so ...
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3answers
40 views

How to convert this particular expression into some desired form?

The parametric equations of a curve are $$x=\cos(t) \cdot e^{-t} $$ $$y=\sin(t) \cdot e^{-t} $$ Show that $dy/dx =tan(t-\pi/4) $. how to solve this? I can get a $dy/dx$ but i cannot convert into the ...
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2answers
54 views

How to parametrize the curve $y^2 = x^2(x+3)$

I am emberassed to ask this, but I couldn't find a way. I want to write the curve $y^2 = x^2(x+3)$ as $$y=f(t) \quad \quad x=g(t) \quad \quad t \in \mathbb R$$ I guess I have to do something like ...
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1answer
23 views

Modelling the Möbius strip using implicit functions

While researching on Möbius strips I found its parametric representation on a lot of websites claiming it is easier. Can someone please explain what problems appear when modelling the Möbius strip ...
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1answer
27 views

How to parametrize circles on a sphere by the distortion of the equator?

I guess am having a very silly problem right now. Considering a unit sphere $S^2$ and, for example, a curve, in spherical coordinates, $c(t)=(1, \frac{\pi}{2},t)$ that goes around the equator how can ...
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2answers
166 views

What is parameterization?

I am struggling with the concept of parameterizing curves. I am not even sure if I know what it means so I tried to look some things up. On Wikipedia it says: Parametrization is... the process of ...
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1answer
83 views

Find the values of a and b such that the sytem has a unique solution and a two-parameter solution?

\begin{bmatrix} a & 0 & b & 2 \\ a & a & 4 & 4 \\ 0 & a & 2 & b \\ \end{bmatrix} Find the values of a and b such that the system ...
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1answer
32 views

Calculating the x, y coordinate a set distance between two points

I'm trying to calculate the x and y coordinates that are a set distance between the coordinates of two pixels in an image. For example, if I travel from my original location (x1=4, y1=3) to a new ...
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1answer
16 views

Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=\left(\frac{2xy-2xy^2}{\left(1+x^2\right)^2}+\frac{8}{13}\right)i+\left(\frac{2y-1}{1+x^2}+2y\right)j$$ Determine $$\int_C F\cdot dr$$ where $C$ is the path ...
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How to write explicity a curve on $S^n$?

I considered the $n$-sphere $S^n=\{x\in \mathbb{R}^{n+1}| \space ||x||=1 \}$ and $p\in S^n$. I want to write down explicity a curve $\sigma$ on $S^n$ passing through $p$ (for example one of the ...
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Is “imposing” one function onto another ever used in mathematics?

First of all, let me define what I mean by "imposing." Basically, I mean graphing some function with respect to some other function, rather than with respect to the x-axis. To be more specific, for ...
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16 views

Fundamental Theorem of Calculus for Line Integrals

Use the Fundamental Theorem of Calculus for Line Integrals to compute $\int_C F*dr$ where $$F(x,y,z)=(yz+2x)i+(xz+2y)j+(xy-2z)k$$ and C is the path from $(1,6,-1)$ to $(5,2,3)$ given by $x(t)=2t+1, ...
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25 views

What is the difference between a parametric equation and cartesian equation?

This may be a stupid question but I just can't seem to understand the difference between a parametric equation and a cartesian equation. I'm in first year multivariable calculus in university to give ...
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2answers
62 views

Parametrization by arclength

I could not re-parametrize the curve r[s_] := {-(5 + 2*Cos[2*s])*Sin[3*s], (5 + 2*Cos[2*s])*Cos[3*s], 2*Sin[2*s]} neither by hand nor with Mathematica. Is ...
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24 views

Parametrization of arbitrary objects to display on an x-y-scope

I am trying to find an approach for general parametrization of an arbitrary geometric object or closed curve. Though I am not sure if I am on the right path with that. Basically I have an geometric ...
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20 views

How to write an equation with a denominator parameter less and greater than 1

I am writing a paper, but I do not know what is the most compact and exact formula to express this equation: $$\left| A_y(v) \right|^2 = \hat{q} v^\gamma \quad \text{where} \quad \left\{ ...
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Working out the area of Australia through Calculus? [closed]

I was wondering if it would be possible, and if so how, to calculate the area of an abstract shape on a sphere using surface integrals and Parametric surfaces and such. I am looking in to this as ...
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1answer
35 views

Evaluate the countour integral $\int _\Gamma z dz$

Can someone please help me setup a) $\int _\Gamma z dz$ b) $\int _\Gamma \bar z dz$ and given the admissible parametrization of $\Gamma$ $\Gamma_1 : z_1: 2 + i(t - 1) ; 1 \leq t \leq 2$ and ...
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1answer
39 views

a question about how to parametrize a surface in $R^3$

Given a surface $$x^4/a^4+y^4/b^4+z^4/c^4=1$$,how can I parametrize the surface using X(u,v). I tried to use $x=a\sqrt{cos(\theta)sin(\phi)}$,$y=b\sqrt{cos(\theta)sin(\phi)}$,and ...
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1answer
25 views

Parametrization of this surface

Consider the following surface: $$A=\{(x,y,z,t)\in\mathbb{R^4}:0\leq x,y,z,t\leq1,\quad x+y+z+t=1\}$$ I need to parametrize it to be able to calculate its volume. Of course, I thought on seeing it ...
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1answer
10 views

Finding points on a tangent line of a parametric equation that are parallel to another parametric equation

So I got the tangent line of the first equation to be 12t/(3t^2+4) and I changed the second parametric equation to the cartesian form and got y= -(12/7x+5) with 12/7 as my slope. I equated ...
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1answer
18 views

Parametrics Arc Length

Can anyone explain the difference between the arc length and total distance? I'm using the textbook here and they seem to be the same formula. Please help me figure this out.
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Please check my calculations for finding a tangent plane to a parametric surface

Here's the question: Find the cartesian equation of the tangent plane to the surface $S : xy^2 + 3yz^2 − 2xyz = 1$ at the point $P(0, 3, 1/3)$. Here's what I did: The normal to the tangent plane at ...
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2answers
31 views

Parametric differentiation

The parametric equations of a curve are $$\begin{cases}x(t)=e^{-t}\cos t\\y(t)=e^{-t}\sin t\end{cases}$$ Show that $$\frac{dy}{dx}= \tan\left(t-\frac{\pi}{4}\right)$$ I did the differentiation ...
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1answer
50 views

Show that $(Y^2-X^3)|f$ if $f$ vanishes on the curve $C: (t^2,t^3)$, and determine what property of a field $k$ will ensure that the result holds.

Let $\phi: \mathbb{R^1}\rightarrow \mathbb{R^2}$ be the map given by $t \mapsto (t^2,t^3)$; prove directly that any polynomial $f\in \mathbb{R[X,Y]}$ vanishing on the image $C=\phi(\mathbb{R^1})$ is ...
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1answer
35 views

Find where a curve crosses itself?

I have the curve $x=t^2,y=t^3-4t$. I made a $t_2$ such that $t_2>t_1$ and from $x$ found that $0=t_1^2 -t_2^2$, from here I solved for values by basic guess and test and then subbed them into the y ...
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2answers
67 views

Find points on curve $r=2\sin\theta$ where the tangent line is parallel to the ray $\theta = \pi/4$

I was thinking to convert to cartesian coordinates and then find when the slope of the tangent line is $1$, but I get a messy equation $2\cos^2\theta -2\sin^2\theta=4\sin^2\theta\cos\theta$ I was ...
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2answers
20 views

Velocity of a curve given by parametric equations

In standard Cartesian equations, $\frac{dy}{dx}$ is the velocity function because it's the derivative of position. $$\frac{dx}{dt} = \sin^{-1}\left(\frac{t}{1 + t}\right) ...
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1answer
37 views

Value of x of which a slope is undefined for a parametric graph.

For what values of $x$ is the slope undefined for the graph $$x=8-t^3$$ $$y=t^2-6t$$ The slope should be undefined when $\frac {dx}{dt}=0$. $$\frac {dx}{dt}=-3 t^2$$ $$-3t^2=0$$ $$t=0$$ When ...
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1answer
42 views

Comet Problem Using Parametric Equations

A comet has an elliptical orbit that is $144$ billion miles across the $x$-axis and $48$ billion miles across on the $72$ years to complete one revolution. If the center of the coordinate system is ...
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1answer
15 views

Curve parametrization verification: Plane and paraboloid

I wanto to find a parametric equation of the intersection between the plane $x+y-z=2$ and the paraboloid $z=(1-y)^2$ I proposed $y=t$, wich implies $z=(1-t)^2.$ So: $$\begin{align} ...
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3answers
54 views

Parametric representation of a plane cut of a sphere at y=5

The sphere is given by $x^2+y^2+z^2=36$ Parametric Form: $$x=6\sin t\cos u$$ $$y=6\sin t\sin u$$ $$z=6\cos t$$ If the sphere is 'cut' at $z=5$ this problem is trivial. ...
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3answers
56 views

Calculate u in terms of time such that a particle maintains a constant speed following a parametric equation

I have a parametric equation given by: $x=\cos(6u)$ $y=\sin(4u)$ And I understand that the speed of a particle at any given t is: $\sqrt{\left(\dfrac{dx}{du}\right)^2 + \left(\dfrac {dy}{du} ...
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Parametrization of sphere including constant inclination $(\theta, i)$ geodesics

Find parametrization of sphere with respect to $\theta$ = constant meridians and i = constant inclination geodesic circles passing through N-S axis and E-W axis respectively. The Earth does not rotate ...
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1answer
33 views

Easy question on the limits of an integral

So I would like to ask how exactly do we determine what limits to take when integrating both Cartesian and parametric equations. So let's say we have a graph of $y=x^2$. If we wanted to take the area ...
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1answer
19 views

Finding parameters for a quadrature formula

To compute the integral $\int_0^1f(x) dx$ numerical I want to use the following quadrature formula: $$Q(f)=\omega_0f(x_0)+\omega_1f(1)$$ The question is how one should choose $\omega_0,\omega_1 ...
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22 views

parameterization of a horizontal line

say I need to parameterize the boundary of a set in order to optimize. The equation is f(x,y) = 3 + x − y + xy and the boundary is the set inclosed by the line ...
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1answer
27 views

Reverse Parametric Equations

I have just learned about parametric equations. I have gotten the concept of turning the parametric equations to regular/ordinary equations, but am having trouble doing the reverse in this problem: ...
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1answer
19 views

How can I show that the set of images of a parameterized curve is smooth?

I know that in order for a set S to be smooth, it needs to be connected and that for each point $\vec{a} \in S$, there needs to be a neighborhood $N$ such that $S \cap N$ is a class $C^1$ function. ...
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5answers
76 views

Find the equation of tangent line to the curve $x=\cos(t) + \cos(2t)$, $y= \sin(t) + \sin(2t)$ at the point $(-1,1)$.

I get the equation for the slope as $\frac{\cos(t) + 2\cos(2t)}{-\sin(t) -2\sin(2t)}$ but I'm unsure how to solve for the value of $t$. I know I need to sub in $-1$ and $1$ for $x$ and $y$ but the ...
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1answer
56 views

Gradient function of a circle

The parametric equations of a circle $C$ are: \begin{align*} x&=2+\dfrac{13}{5\sqrt{2}}\cos t\\ y&=1+\dfrac{13}{5\sqrt{2}}\sin t \end{align*} for $t\in[0,2\pi]$. I am stuck on this part: Find ...
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0answers
17 views

Smallest interval for graph

Parametric equation of a graph is x = cos(4t) , y = sin(6t) What is the length of the smallest interval $I$ such that the graph of these equations for all $t\in I$ produces the entire graph ...
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1answer
42 views

Why does the vector field $(\sin (\theta), - \cos(\theta), 0)$ indicate sideways motion?

If I study a physical system, such as a car, and let it drive forward a little bit, say a distance $m$, then I can draw out the right triangle and find the car's position at $(m\cos \theta, ...
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43 views

Line integral - new parametric equation

We know that $$\int_\gamma V \cdot dr = \int_a^b V(r(t)) \cdot r'(t) dt$$ with $V$being our vector field and $r$being the parametric equation for the curve $\gamma$. Let now $\hat{r} = r \circ \phi$ ...