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

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7
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104 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 $\phi$ ...
5
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0answers
66 views

Surface parametrization and calculating its area

I have to find the parametric equation of the surface of the sphere inside the cylinder and above the $z=0$ plane, as shown in this picture. $$ \text{Sphere: }x^2 + y^2 + z^2 = 1\\ \text{Cylinder: ...
5
votes
0answers
159 views

What is the parametric equations for the following closed curves?

First case Second case For the sake of simplicity, let the circle of radius $R$ be at the origin, the rectangle width and height be $w$ and $h$, respectively.
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votes
0answers
39 views

What's the next “recursion” here?

Plotting a single 3d helix is x = cos(t); y = sin(t); z = t; From this equation: x = [R + a cos(\omega t)] cos t y = [R + a cos(\omega t)] sin t z = h t + a sin(omega t) Comes the awesome ...
2
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0answers
22 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 ...
2
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0answers
37 views

Launch angle required to hit coordinate (x,y) with air resistance

Finding Angle of Elevation to hit X, Y and Wikipedia Angle required to hit coordinate work, but don't calculate air resistance. Is there a way to find the launch angle of a projectile required to hit ...
2
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0answers
29 views

Parameterizing an implicit curve

I have to parameterize this curve: $$F(x,y)=y-x^2+x-e^{-yx^2}=0$$ But I don´t know how to do it. thanks
2
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0answers
332 views

Relation between ellipse general and parametric equation

I am familiar with the fact that one can relate the eigenvectors and corresponding eigenvalues of an ellipse's quadratic equation matrix, to the pose of a circle in 3-space. When say quadratic ...
2
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0answers
243 views

Parametric Equation of a Hyperbolic Paraboloid

I need to make two trace plots of the hyperbolic paraboloid $z=x^2-y^2$. In the first plot, we set $z$ equal to a constant $k$, $z=k$. How do I find the parametric equation for this representation of ...
2
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0answers
756 views

Parametrization of square to calculate Dot-product in line-integrals and area-integrals, electric field from $\frac{dB}{dt}$?

I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly. I know $$\nabla\times E= ...
2
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0answers
1k views

Explain Triangle perimeter in polar coordinates

The question is to give a formula in $x$ and $y$ that gives all three sides of an equilateral triangle. The formula should not be true for points that are not part of the perimeter of the triangle. ...
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14 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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0answers
25 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 ...
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0answers
77 views

Helix around helix parametric equation?

I know the parametric equation for a 3d helix is: x = R cos t y = R sin t z = h t can somebody explain to me this parametric equation (image and equation from Wolfram) for a "Helix around helix" / ...
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0answers
35 views

Finding the mean value of y

I don't understand how to obtain the limits for the $t$-values considering that they gave us the $x$-values in the first part of the equation. I've considered substituting the $x$-values into the ...
1
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0answers
10 views

Comparing normal distributions using a two sample Kolmogorov-Smirnov test

I have used a two sample Kolmogorov-Smirnov test to compare the distributions of two sets of data. I know that the K-S test is a non parametric test, however the distributions of data I'm comparing ...
1
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0answers
28 views

Parametrization of a bounded solid.

So, I have a solid bounded by $z=\sqrt{x^2+y^2}, z=\sqrt{1-x^2-y^2}, z=2$ I had to parametrize it using spherical coordinates so I used $$\begin{cases} x(\rho, \theta, ...
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0answers
22 views

Removing parametrization from a system of equations

Consider the following system : $$ \begin{aligned} \frac{d^2t}{d\lambda^2} &= -f\left(t\right)\frac{d t}{d \lambda}\frac{d t}{d \lambda} -A\frac{d g\left(t,x\right)}{d \lambda}\frac{d t}{d ...
1
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0answers
39 views

Faster way of finding critical points?

So I am looking at parametric vector function. $$ \begin{vmatrix} \cos (t) & -\sin (t) & 0 \\ \cos f(t) \sin (t) & \cos f(t) \cos (t) & -\sin f(t) \\ ...
1
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0answers
44 views

Relation between $\sin(t)$($\cos(t)$) and $\sin(at)$ ($\cos(at)$) when both are rational

This question relates to Parametric equations where sin(t) and cos(t) must be rational. Suppose it is given that $\cos(t)$ and $\sin(t)$ are both rational and also $\cos(at)$ and $\sin(at)$, where ...
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0answers
132 views

Intersection between sphere and ellipsoid

I am failing since two days to compute and to plot the intersection of an ellipsoid in parametric notation ...
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0answers
29 views

Find all the values of real parameter “n”…

Let $S$ be the set of real solutions for the following equation:$$\log_2(1-x-x^2)=n\log_{1-x-x^2}2+2$$ Determine all the values of real parameter $n$ for which $S\cap(0;{1\over2})\neq\emptyset$.
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47 views

Mass and density function Calculus II Problem

A thin metal plate lies over the portion of the cylindrical surface $y^2 + z^2 = 4$ for $z ≥ 0$ between $1 ≤ x ≤ 4$. The density of the plate is given by $f(x,y,z) = z$. How do I calculate the mass ...
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0answers
181 views

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius “A” & “B”?

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius "A" & "B", which intersect at a distance of "H" from its Axis at an ...
1
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0answers
43 views

Line integral, Parametrization

I have this line $A=\{(x,y) \in R^2 : y^2+4x^4-4x^2=0\}$ , $(x>0)$ I parametrized it like that : $b(t) = (t, \sqrt{4t^2- 4t^4})$. And my $F$ is $F(x,y) = (x+y,-x)$. But when I calculate my ...
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0answers
69 views

Beth needs to make a crossing in her canoe

I have a math problem that has me stumped. I cannot seem to find a good starting point for this, and am flying blind with no check values. Maybe it's end of semester fog, but I'm struggling with ...
1
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0answers
21 views

Parameterisation Q

I'm looking to parameterise the expression $$f(r,\theta)=Usin(\theta-\alpha)(r-\frac{a^2}{r})-\frac{\Gamma}{2\pi}lnr-k=0$$ s.t. $z=re^{i\theta}$. I get a horrible expression if I parameterise for r. ...
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0answers
38 views

Can i formulate any curve with parametric equation?

Can i formulate any curve with parametric equation ? if not, so what kind of curves can be explained with parametric equations ?! Thanks in advance
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0answers
104 views

Splitting parametric curve equation into two ranges

I am examining the speed of motion on curves and in the textbook i am reading , the example was showing that using a parametric equation for an ellipse will result in a regular motion (as expected) , ...
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0answers
98 views

Vector Tangent to Curve of Intersection

I am having problems solving this. Find a vector tangent to the curve of intersection of $z = 4x^2 + y^2$ and $z=(27-x^2-y^2)^{1/2}$ at the point $(1,1,5)$. I'm able to do this kind of thing using ...
1
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0answers
18 views

Can one characterize which surfaces are capable of being described by a closed-form parameterization?

Speaking intuitively, I can visualize a lot of surfaces in my mind; but it seems that some of the ones I can imagine are not capable of being described by the 'usual suspects', i.e., elementary ...
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0answers
44 views

$\frac{dy}{dx}$ of a parametric curve

Given $x = sin^2(t)$, $y = cos^2(t)$, I need to find $\frac{dy}{dx}$ in every non-singular point of the curve. So $\frac{dy}{dt} = -2sin(t)cos(t)$ and $\frac{dx}{dt} = sin(2t)$. To find the ...
1
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0answers
115 views

Computing the surface area of a (piecewise) polynomial parametric surface

I'm wondering what kind of numerical integration (e.g. Gauss-Legendre quadrature) I should use to compute the surface area of a (piecewise) polynomial parametric surface. There are two cases. Case ...
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0answers
57 views

Maximum value for parameter

I am facing the following problem: A number of a adults, b children older than 12, and c children younger than twelve attend an event. The sum of all people a+b+c=100. The prices are \$6 per adult, ...
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0answers
29 views

MLE estimation of parameters, converting normalized observations to integers and back

I am fitting a model's parameters to grouped data by maximizing the likelihood equation: $L(\theta)=N!\prod_{i=1}^{G}\frac{p_i(\theta)^{n_i}}{n_i!}$ $\theta$ is the vector of parameters. $n_i$ is ...
0
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0answers
17 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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0answers
12 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
10 views

parametric representations 3d object

I'm trying to model a 3 dimensional body that is sort of ellipsoidal and am looking for parametric representation of 3D objects similar to the quadratic surface representation of a sphere or ...
0
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0answers
20 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 ...
0
votes
0answers
12 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
votes
0answers
18 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
12 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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0answers
8 views

Parametric equation ( Planes)

The three planes α, β and γ have the equations: α: x-2y+z=3 β:2x+y-3z=1 γ: x+y+az=1 where a is a real number 1) Given that α intersect the xy-plane (with equation z=0) in a straight line. Find a ...
0
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0answers
25 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
votes
0answers
12 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
0answers
15 views

Evaluate the line integral with Euler.

Need some help evaluating this line Intergral. $\int$$_c$ xy${e^y}$$^z$ dy Where C: x = 4t ; y = 3t$^2$ ; z = 3t$^3$ ; 0$\le$t$\le$1 Any help would be great. Thanks.
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67 views

Parametrization for intersection between sphere and elliptic cylinder

Given the sphere: $$x^2 + y^2 + z^2 = 12$$ and the ellyptical cylinder: $$(x-1)^2 / (7/3) + (y-2)^2 / 7 = 1$$ Give a parametrization for the intersection curve. I'm confused on how to do it. The ...
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0answers
61 views

Converting this 3D plane parametric equation to non-parametric

I had these 3 planes to put into parametric equation: (5, 4, −8),(1, 6, −3) and (7, −2, 5) so I put it into this parametric equation: (x,y,z) = (5,4,-8) + t(-4,2,5) + s(2,-6,13) But i am having ...
0
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0answers
35 views

Parametric Equation of a parabola from the derivative of the parametric equation of a circle

Find the velocity and trajectory to throw a ball from a Ferris Wheel to a friend standing below. The Ferris Wheel has a diameter of 16 meters and its highest point is 19 meters above the ground. It ...
0
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0answers
52 views

Find point on rotated curve

I have a curve $f(t)$ that has been rotated through an angle $\theta$, and also have defined a given offset $Y$ from the curve origin. Using the equation $Y=x*sin(\theta)+y*cos(\theta)$ which ...