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

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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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22 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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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
63 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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16 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
38 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
33 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
66 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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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
31 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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2answers
39 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
39 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
22 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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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
13 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
29 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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18 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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16 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
35 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
23 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
27 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
30 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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29 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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43 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 ...
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21 views

how to logically modify a variable so that it accurately fits a given curve

I have 2 sets of experimental data. Each set has 2 variables (A,B) and response data (C). A1 100 100 100 100 100 100 100 B1 11.3 10.1 8.9 8.1 7.7 6.5 5.3 A1/B1 8.8 9.9 11.2 12.3 13.0 ...
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1answer
44 views

Complex integral: how to parameterise a circle?

Evaluate $$\int_\gamma \bar{z}^2dz,$$ where $\gamma$ is the circle with centre $1$ and radius $1$ traced anticlockwise. One parameterises the circle $\gamma$ as $z=1+e^{it}$ for $t\in[0,2\pi]$ and ...
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1answer
20 views

Find a parametrisation of the arc of the circle with radius $r$ centred at $z_0$ between $\phi$ and $\theta$.

where $-\pi \leq \theta < \phi \leq \pi$. Im not sure how to start with this question.
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35 views

Finding the Euler parametrization of a curve

I have the following question as a homework problem for my differential geometry class: find the curvature and the explicit Euler parametrization of the ellipse $ \gamma(t) = (a \cos t, b \sin t) $ ...
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1answer
19 views

Parametrizaction of a Hyperboloid

I do not understand why when you revolve a hyperbola around a circle the respective parameters (cosh (v) and cos (u)) are multiplied by each other to get the parametric form of the hyperboloid. I ...
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74 views

For what $n$ is $\sum_{i=1}^\infty \frac{\cos (it)}{i^n}$ bounded and why doesn't a sine behave the same way?

I've been looking at a parametric curve $$\pmatrix{X \\ Y}=\pmatrix{\sum_{i=1}^N \frac{\cos (it)}{i^n} \\ \sum_{i=1}^N \frac{\sin (it)}{i^n}}$$ where, for the plots below, $N$ runs from $1 \rightarrow ...
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1answer
23 views

Is this a correct parametrization of a rectangle on the complex plane?

$z = 3 + i(2t - 1), t \in [0,1) \\ z = 3 - 6(t-1) + i, t \in [1,2) \\ z = -3 + i(1 - 2(t-2)), t \in [2,3) \\ z = 6(t-3) - 3 - i, t \in [3,4]$ I parameterized a rectangle with vertices at ...
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1answer
9 views

What is the parametrization of the set of points in $\mathbb{R}^2$ with $L^p$-(semi)norm $1$ for any $p$?

I'm looking for a curve $t_p: [0,L] \rightarrow \mathbb{R}^2$ that describes the set $T_p = \{ (x,y) \in \mathbb{R}^2 : |x|^p + |y|^p = 1\}.$
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2answers
76 views

How to compute $\int_0^2(1+4t^2+9t^4)^{1/2}\text{d}t$?

The original question was: find the length $\ell$ of the curve $\gamma$ given the parametric equations: $$x=t~~~~~ y=t^2~~~~~ z=t^3 $$ from $t=0$ to $t=2$
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1answer
49 views

Rectangular Hyperbola - Eliminating the Parameter

Question: The point P (2p,2/p) lies on the rectangular hyperbola C with equation xy = 4. (a) Find the equation of the normal to C at P. The normal at P meets C again at the point Q. The mid-point ...
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2answers
43 views

Speed of a parametric function?

I know speed = |velocity| Why is speed of parametric defined as $$speed = \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}$$ How is this derived? What is the principle here? Is ...
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2answers
43 views

Why is $\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$ [closed]

Why is this true? $$\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$$
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1answer
36 views

parametric equations, finding the range of t

When parametrizing a curve how doe we obtain the range of $t$? For example lets say we have the parametrization: $x(t) = 1+3t$ and $y(t) = 2+5t$. How do we find the range of t? $t\to[?,?]$
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25 views

Parameterization which is closed under addition

Suppose $\beta_1(t)$ and $\beta_2(t)$ are two parametric curves defined on $[0,1]$. Let $\beta_1^*(t)$ and $\beta_2^*(t)$ are two re-parametrized of the above curves. Now, I looking for a ...
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0answers
26 views

Parametrisation of the curve after a short time

I am trying to wrap my head around this differential geometry problem. I am given velocity V with components in the principle normal and binormal directions. Then I am given an approximation of the ...
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25 views

Calculating the Constant of Integration in Parametric, Vector-based Equations

I'm having trouble finding the constant of integration in parametric, vector-based equations. Given an equation: $$ a(t)\ =\langle \cos(t),\ \sin(t)\rangle $$ and $$ \int\ a(t)\ dt\ =\langle 0,\ ...
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1answer
31 views

Sketching a parametrised cone and a geodesic lying on it.

I just started a new module at University and I am having some trouble with parametrisation. I am given a parametrisation of a geodesic lying on a cone in notation $r(t)=x(t){\bf i}+y(t){\bf ...
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1answer
21 views

Show two parametrizations to be equal

Given the two curves \begin{align*}&\mathcal{C}\left\{\begin{matrix}u = t\\v = t\end{matrix}\right., & t\in [0,1]\\ \\ &\mathcal{C'}\left\{\begin{matrix}u = t^3\\v = ...
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1answer
72 views

Angle between position and velocity vectors is constant?

Is there a name for such a curve or can this even happen? I know when the velocity vector, $\mathbf{x'}$, and position vector, $\mathbf{x}$ are always orthogonal $\mathbf{x}(t)$ parametrizes a circle ...
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1answer
198 views

Find a parametrization of a hyperplane in $\mathbb{R}^4$ given by the equation $x+y+z+at=b$

Find a parametrization of the hyperplane in $\mathbb{R}^4$ given by the equation $x+y+z+at=b$ where $a,b$ are real numbers. I'm not sure about my answer: $$y \begin{pmatrix} -1\\ 1\\ 0\\ 0 ...
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1answer
45 views

Find the parametrization of the curve resulting from intersection of two surfaces

The question reads as follows: Find a parametrization of the curve resulting from the intersection of the surfaces: $z = x^2 - y^2$ and $z= x^2 +xy - 1$ My attempt: (Use y = t as a parameter, so ...
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0answers
35 views

What is the requirement for separable parameters in an LSQ fit?

I am trying to determine the amplitude of an amplitude modulated sinus as accurate as possible. My sampling frequency is sufficently high. The entire model looks as follows: $$ ...