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

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1answer
697 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 ...
1
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1answer
30 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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0answers
28 views

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 ...
27
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2answers
432 views

Is “imposing” one function onto another ever used in mathematics?

First of all, let me define what I mean by "imposing," and let me clarify that I've only studied this operation in 2D Euclidean space. Now then, to impose one function onto another, you need two ...
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2answers
88 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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0answers
31 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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0answers
41 views

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
39 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
43 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
33 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
53 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
30 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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2answers
46 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
67 views

Show that $Y^2-X^3\mid f$ 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
139 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
343 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
106 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
98 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
83 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
21 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
79 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
72 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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1answer
47 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
27 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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1answer
77 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: ...
0
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1answer
25 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
470 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
502 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
41 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
97 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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0answers
47 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 $\hat{r} = r \circ \phi$ ...
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2answers
80 views

Finding a local parameterization of a plane curve

I'm attempting to find a parameterization of $\frac{x_1^2}{a^2} + \frac{x_2^2}{b^2} = 1$. I find a tangent vector field: $X = \left( \frac{2x_2}{b^2}, -\frac{2x_1}{a^2} \right)$ (by taking the ...
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1answer
35 views

Understanding proof for Green's Theorem

In a proof for Green's theorem where we first assume that the area bounded by a closed $C^1$ curve $\gamma$ is of the following form: $$D = \{ (x,y) \ | \ x \in [a,b], \ \mu(x) \le y \le v(x) \}$$ ...
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1answer
85 views

finding parametric equations from a rectangular equation

Find the parametric equations for $x^2-4x+y^2-2y+5=2$, and graph. Hint: Complete some squares. I have completed squares and gotten $(x-2)^2+4=-(y+1)^2-2$ but I am confused with how to proceed. I know ...
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1answer
35 views

Parametrisize of vertecies (0,0) (a,0) (0,b)

Hi i have been given 3 vertecies. (0,0) (a,0) (0,b) The constants a and b are >=0. This forms a backwards triangle. The parametisation don't make sense to me, so basically what i am asking is for ...
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1answer
97 views

Parametric equation of a circle in 3 space - odd result

I have the following problem involving parameterized circular equations, but am getting strange answers and wanted to check if my approach made any sense. In 3D space, the parametric equation of a ...
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1answer
403 views

Finding tangents to a cycloid

here's the question I don't really get the second part of the question.. it uses parametric curve equation to solve. A curve $\mathcal C$, a cycloid, is defined by $x=r(\theta-\sin\theta), ...
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2answers
424 views

Torsion and curvature of the curve $X(t) = (at, bt^2, ct^3)$

Hey all I am looking for help on a problem. I will post it, and than I will add what I have tried and my ideas etc. The question has been up now for a few days, I'm sure someone out there can help! I ...
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1answer
14 views

Parametrize a surface using cylindrical coord.

Hi! I am having trouble parametrizing this tower. Specifically the radius which has to be a function of the height $Z$ $$0<z<H, 0 ≤ r ≤ R(2 − z/H), \quad 0<\theta<2\pi$$ I do not ...
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1answer
119 views

Finding out if vectors are Parallel or Orthogonal in Parametric Form.

I have two Parametric Form Vectors. Is it possible in that form to work out if the vectors are Parallel, Orthogonal or neither. Or do I have to have it in standard vector form $ (a_1, a_2, a_3)$ and ...
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0answers
20 views

Higher dimension leads to different optimization?

It appears z= f(x,y) has a global max/min at another particular (x,y). Using only one independent variable $x$ at fixed $y$ i.e., for z = f(x) I get another max/min point for $x$ optimum point ...
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1answer
95 views

arclength parametrization intuition

I have a question about parametric curves. I have learnt about arc length re parametrization and I understand how do the problems , for example finding the length of the vector and integrating with ...
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0answers
39 views

Lissajous Curve

$$ \gamma(t)= (x(t),y(t))=(sin(2t),sin(3(t)) $$ Justify that we can reduce the domain of study to [0, $\pi/2$], by specifying the necessary symmetries to obtain the whole curve. I'm not really too ...
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2answers
20 views

A curve is described in polar coordinates . Find parametric equations for $x$ and $y$ and plot the curve.

A curve is described in polar coordinates by the equations $$ r = t; \theta = 3 \cos t; 0 ≤ t ≤ 10 $$ Find parametric equations for $x$ and $y.$ I cannot convert it into parametric form
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1answer
384 views

Find Parametric Equations for a line passing through point and intersecting line at 90 degrees

Let $P$ be the point $(3,1,-2)$ and $L$ be the line given by $x=-4+2t$, $y=2+2t$, $z=1+t$. Find parametric equations for the line passing through $P$ and intersecting $L$ at a right angle.
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2answers
116 views

Parametrization of a circular arc in terms of the angle between the tangent line and the $x$-axis

I'm struggling with this problem: "Find a parametrization of the first quadrant part of the circular arc $x^2 + y^2 = a^2$ in terms of the angle between the tangent line and the positive x-axis, ...
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1answer
23 views

Complex Integration parametrisation

I'm trying to integrate $\int_\gamma (z^2-2)dz$ where $\gamma$ is a spiral that loops 3 times and ends at (3,0) on the Argand diagram. I have found the parametric equations for this contour to be ...
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0answers
55 views

Mean curvature of polar parametric surface

For the purposes of modelling a fluid mechanics experiment, I'm dealing with a convex surface parametrized by the azimuth $\theta$ and an arc length $s$ along the surface. The points on the surface ...
0
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1answer
44 views

Solving the cubic $2t^{3} - 3t_{1}t^{2} + t_{1}^{3} = 0$

The question is: "4. Find the equation of the tangent and the equation of the normal to the curve $x = 3t^{2}$, $y = t^{3}$ at the point whose parameter is $t_{1}$. Find the parameter of the point at ...
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2answers
205 views

Find the parametrization of the intersection of two surfaces.

I'm having trouble figuring this problem out since the $3x^2 + z^2$ is throwing me off. Especially because of the $z$. Find the parametrization for the curve of intersection between the cylinder ...