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

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1answer
19 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
14 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
19 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 ...
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0answers
27 views

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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0answers
13 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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0answers
23 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
59 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
22 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
18 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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0answers
18 views

Vector vs Parametric [closed]

When they were both introduced they appeared to be identical. Besides notation are parametric and vector treated the same? What are the differences, if any? $$<x(t),y(t)>$$ $$vs$$ ...
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0answers
39 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 ...
2
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1answer
34 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
38 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 ...
1
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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 ...
0
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1answer
9 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 ...
0
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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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0answers
18 views

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 ...
3
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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
34 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 ...
0
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2answers
62 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
12 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
35 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
40 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 ...
0
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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
51 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
53 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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0answers
19 views

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 ...
4
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1answer
31 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 ...
0
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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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0answers
18 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
26 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
18 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
68 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 ...
0
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1answer
51 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
40 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
42 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
31 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
24 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
25 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
22 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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0answers
14 views

parametrization: t values that lead to the same point

Given a parametrization $r(t)$, we can have several $t$ values that correspond to the same point on the curve... what's the method of finding all of these when dealing with trigonometric functions? I ...
0
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1answer
69 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
48 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
245 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 ...
0
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1answer
13 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
27 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
18 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 ...
0
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1answer
38 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 ...