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

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1answer
28 views

Evaluate $\iint_s\text{curl}\textbf F\cdot \textbf {n}dS$

Let $\textbf F=<xy,yz,zx>$ and $S$ be the upper half of the ellipsoid $\displaystyle \frac {x^2}{4}+\frac {y^2}{9}+z^2=1$. Evaluate $\iint_s\text {curl}\textbf F\cdot \textbf {n}dS$ I know the ...
2
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1answer
75 views

How to draw stick trefoil knot

http://commons.wikimedia.org/wiki/File:Stick_number_trefoil.png I am interested in plotting the stick trefoil knot. I don't know where to start. I am looking for equations or co-ordinates of ...
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1answer
40 views

Find all the intersection points of a vector parabola (in R3) and a sphere

Given that I have a vector in R3 (7t, 10t - 2t^2, 5t) | (These numbers are arbitrary for the sake of the process) A sphere centered at the point ( 15, 25, 10) with a radius of 20 There is a ...
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1answer
40 views

What is being represented by this 2 images?

image 1 image 2 It's possible that image 1 is showing some kind of methods for building polygons out of trigonometric functions ? It's also possible that image 2 is a quadratic bezier curve ?
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1answer
21 views

How can I reduce the interval of this parametric equation:$ x=t/(1+t^4) ; y=t^3/(1+t^4)$ to the simplest possible domain.

How can I reduce the interval of this parametric equation: $x=t/(1+t^4) ; y=t^3/(1+t^4)$ to the simplest possible domain? Can you explain this solution?
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1answer
28 views

Values of a limit with parameter

Evaluate the values of the limit as the changes of $\lambda \in \mathbb{R}$ $$\lim\limits_{x \to +∞} e^{x-x^3+\lambda\ln(x)}f^{(n)}x$$ with $$f^{(n)}(x)=p_n(x)e^{x^3-x}$$ where $p_n(x)$ is a generic ...
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2answers
21 views

Graphing an inverse parametrically

My calculus book has the following question: Graph the one - to - one function $ f(x) = x^2$ , where x is greater than or equal to zero, with its inverse. Now my answer is that the inverse is the ...
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2answers
27 views

Parametric Function for Coloring

I'm trying to write a parametric straightline function that changes its values between 0.529 and 0.933. Usually what I would do is: $r = 0.529+(0.933-0.529)*v$ where parameter $v= [0,1]$ This ...
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1answer
24 views

Parametrization Semicircle On Sphere

I need to find a parametrisation in terms of $t$ for a half circle on a sphere with radius $R$. The circle goes from $(R,0,0)$ to $(-R,0,0)$ and is going through the point $(0, ...
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1answer
25 views

Parametric equation for a curve $C_1$ in $\mathbb{R}^3$ such that the angle of its tangent $T$ and the $Y axis$ equals the angle of $T$ and a vector

I know that to find the angle between two curves in $\mathbb{R}^3$ at their intersection I differentiate both curves and evaluate at the intersection to find the slope of each tangent line and then do ...
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1answer
81 views

Find the area of the surface obtained by rotating the curve of parametric equations

Rotate about the $x$ axis $x = 2t-2/3t^3$ $y = 2t^2$ $0 \leq t \leq 1$ I did the integral of $\sqrt{(2-2t^2)^2+(4t)^2}$ and got $(2x(x^2+3))/3$ and then I did the integral of $2\pi 2t^2 ...
3
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2answers
50 views

Parametrizing a 3D surface

Find a parametrization of the surface $x^3 + 3xy + z^2 = 2$, $z > 0$, and use it to find the tangent plane at $x = 1$, $y = \dfrac{1}{3}$, $z = 0$. I know how to find the tangent plane once I have ...
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2answers
86 views

How to parametrize this region surface

$S$ is the portion of the plane $$x+2y-3z=3$$ in the octan bounded by the positive direction of the $x$ and $y$ axis and the negative direction of the $z$ axis. How can I parametrize this crazy ...
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2answers
43 views

Is it possible that $f_1,f_2,\dots,f_n$ are not all differentiable, but $\alpha:I\to \Bbb{R^n}$ is differentiable?

Consider the parametrized curve $\alpha:I\to \Bbb{R^n}$. These notes say that $f_1,f_2,\dots f_n$ being differentiable $\implies$ $\alpha$ is differentiable. I wonder why the converse is not true. Is ...
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2answers
86 views

parametrize surface region

S is the elliptic region of the plane $y+z=1$ inside the cylinder $4x^2+4(y-0.5)^2=1$. First parametrize $S$ using $(x,y,z)=G(u,v)$ and then calculate $\displaystyle \frac{dG}{du}\times ...
3
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2answers
131 views

Parametrization of the lemniscate

All over the net it is stated that the parametrization of the lemniscate with Cartesian equation: $(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)$ is: $$\varphi: t \mapsto ...
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1answer
51 views

Normal line to cycloid

A Cycloid is given by $$\left\{\begin{matrix} & x(t) = 3 \cdot (t-\sin t)\\ & y(t) = 3\cdot(1-\cos t) \end{matrix}\right.$$ I need to find the parametrized curve for the Normal line ...
0
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1answer
29 views

Find the parametric equations of the line of intersection…

Find the parametric equations of the line of intersection of the planes x - z = 1 and x + 2y + 3z = 1. I'm assuming it's something to do with cross product? Here's what I've set up: x y z 1 ...
3
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1answer
41 views

Evaluating a surface integral of a paraboloid

Calculate the average value of $(1+4z)^{3}$ on the surface of the paraboloid $z=x^{2}+y^{2}$,$x^{2}+y^{2} \leq 1$ I'm not sure on how to start this problem. I have already found the area of the ...
0
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3answers
34 views

eliminate parametric parameter to determine the Cartesian equation.

$$x = \sin^2(t), y = \cos(t)$$ I know that to eliminate parameter involving $\sin$ and $\cos$, we should reduce it to $x^2 + y^2 = r^2$. So, $x^2 = \sin^4(t)$, $y^2 = \cos^2(t)$ But I can't make ...
3
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1answer
54 views

Two curvature formulas when equal arc-length

all. So with a parametric curve $\vec{r}=\langle x(t),y(t)\rangle$, curvature is given by $$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$ When we have constant arc-length, an alternate ...
0
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1answer
24 views

Find the parametric curver from points (2,1) and (5,4)

I already found $x=2+3t$ and $y=1+3t$ but I don't know how to get the whole equation of $r(t)=$? and what the boundaries are $?<t<?$
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1answer
165 views

Find a parameterization for the circle of radius 2 in the xy-plane, centered at the origin, clockwise

Find a parameterization for the circle of radius $2$ in the $xy$-plane, centered at the origin, clockwise. I know to use $2\cos(t)$ and $-2\sin(t)$ but I'm not sure what to do after that
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1answer
76 views

System of parametric inequalities

I'm still struggling with the parametric inequalities.I'm doing some systems of parametric inequalities, but I can not understand how to proceed.This is the system: $x-2a<1+a$ $\frac x2 ...
0
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1answer
53 views

Equation of a polygon

I need a parametric equation for a filled polygon defined by 3 or more points. The closest I've got is by using 3 points in this equation - $polygon = p1 + u(p2-p1) + v(p3-p1)$. But by using points ...
0
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2answers
89 views

Find Vector and Parametric Equation

I'm having some trouble finding answers to these problems. When i try to find help online, all i find are (x,y,z) problems and I'm simply looking for a PreCalculus (x,y) problem solving technique: ...
0
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1answer
31 views

Parametric inequality problem

I would ask for help regarding a problem with the parametric inequalities. $\dfrac{(x+a)}{(a-1)}+\dfrac{(x-a)}{(a+1)}-\dfrac{x}{(a+1)}-\dfrac{2(x-1)}{(a-1)}\ge 0\ \text{for}\ a<-1$ Since ...
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2answers
73 views

Supremum Definition in context of a Rectifiable/Differentiable Curve

I am trying to prove that if I have 2 paramterizations of the same curve $\gamma$ and $\sigma$ (i.e. there is continuous bijective map $\phi$ such that $\sigma = \gamma \circ \phi$) then if the curve ...
0
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0answers
35 views

Inverse ease-in-out parametric function

I'm trying to create an inverse ease-in-out function that given values from 0 to 1, produces values from 0 to 1. Opposite of a typical ease-in-out function, though, I want it to start accelerated, ...
1
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3answers
72 views

Surface integral

Problem statement $$ \mbox{Calculate the surface integral}\quad \int_{Y}\ y\,\sqrt{z\,}\,\sqrt{4x^{2} + 4y^{2} + 1\,}\,\,{\rm d}S $$ where $Y$ is the surface $\left\{\left(x,y,z\right)\ \ni\ ...
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2answers
66 views

Paramtrizing a counterclockwise circle vs. a clockwise one

Does it make a different when you parametrize a counterclockwise full circle and a clockwise circle in the complex plane? For example, I am looking at computing an integral $\int_\gamma ...
0
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0answers
279 views

Wolfram and solids of revolution

I'm looking for the easiest method of having WolframAlpha calculate the volume of a solid of revolution. I've been working on a particular Project Euler problem for a long time. So far, I think I ...
0
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1answer
36 views

ratio of tangent to the ellipse

The tangent at point $P = ( a \cos \phi, b \sin \phi)$ on the ellipse $\frac{x^2} {a^2} + \frac{y^2}{b^2}=1$ meets the $x$ and $y$ axes at the points $X$ and $Y$, respectively. Find in terms of ...
0
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1answer
47 views

Parabolic flight. h(t) vs 2 parametric equations.

Often you have something like: $$h(t)=-16t^2+V_0t+C$$ I have little experience with parametric equations, but I have also seen parabolic functions represented this way: $$x=x_0 + V_{0_x}*t$$ ...
1
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1answer
93 views

How to construct a parametric cubic B spline?

If I am given n+1 control point Pi(xi,yi), Po .... Pn , how do I construct a parametric relationship to draw a curve ? From what I understand , a parametric relationship is that you can express x and ...
1
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1answer
75 views

Express parametric curve as graph of a function

I have a parametric curve in $\mathbb{R}^2$ given by $$ t\mapsto f(t)\left(\begin{array}{c}1\\1\end{array}\right)+\sqrt{-f'(t)}\left(\begin{array}{c}1\\-1\end{array}\right),\quad ...
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0answers
53 views

How are massive parametric functions that make a picture (example inside) created?

[Here's] (http://www.wolframalpha.com/input/?i=sergey+brin+curve) an example of what I'm talking about. The equation might take a second to load. How are these generated, computer algorithms? I ...
1
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1answer
130 views

Asymptote of a parametric equation (with Arctan)

I need to find the asymptotes of a parametric equation. My book says you have a vertical asymptote when $y\to \infty$. But the parametric equation is the following: $$x= \frac 13t^3-\pi,y= \frac ...
1
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1answer
30 views

Does this limit of arc length exist?

We have a parametrized curve $\gamma: \mathbb{R} \rightarrow \mathbb{R^2}$ given by $\gamma (t) = \langle e^t\cos (t), e^t\sin(t)\rangle$. I want to compute the arc-length of this curve on $[a,b]$ in ...
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2answers
52 views

Eliminate the parameter from the parametric equations

$$x=\frac{3t}{1+t^3} , y=\frac{3t^2}{1+t^3} , t \neq -1,$$ and hence find an ordinary equation in x and y for this curve, The parameter t can be interpreted as the slope of the line joining the ...
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3answers
57 views

Find rectangular equation from parametric equation???

Find rectangular equation from parametric $ x = t^{2} + t $ $ y = t^{2} - t $ I tried finding the equation but I am stuck here: $ x - t^{2} = t $ $ y = t^{2} - t $ $ y = t^{2} - ({x - t^{2}}) ...
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0answers
41 views

How to parametrise this surface integral

This is the question: $ S $ is the boundary of the region $ \{(x,y,z):0≤z≤h, a^2 ≤x^2+y^2 ≤b^2 \}$ where $ h,a,b$ are positive and $a<b$. ${\bf F(r) } = \exp(x^2+y^2){\bf r}$ where $ {\bf ...
0
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0answers
30 views

Parametric Derivatives

I recently wanted to derive a parametric equation which is as follows: $x(t) = t \cos(\frac{p\pi}{2t})\\ y(t) = t \sin(\frac{p\pi}{2t})$ I derive both equations in terms of $t$ but when I go to ...
2
votes
1answer
437 views

Finding a vector parametric equation given P and Q equations?

Find a vector parametric equation $r⃗(t)$ for the line through the points $P=(3,5,4)$ and $Q=(1,4,7)$ for each of the given conditions on the parameter $t$. If $r⃗ (0)=(3,5,4)$ and $r⃗ (7)=(1,4,7)$, ...
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2answers
100 views

parametrization of plane in $\mathbb R^3$

Parametrize the plane in $\mathbb R^3$ with direction vectors $\hat u$ and $\hat v$ and through the point $p$ as in representation as the range of a $C^1$ function $f:\mathbb R^2\to\mathbb R^3$. ...
0
votes
1answer
28 views

Can you give a general expression for the rate of rising water poured in an object?

Given an 3-d object defined by a set of parametric equations (x, y, z), can you write a formula expressing the rate that a liquid rises as it's poured into this object at a constant flow rate? Assume ...
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2answers
44 views

Removing parameter $t$ from $z$-axis

How do I remove the parameter $t$ from the $z$-function in the following: $$\begin{align}x&=a\cos{t}-a\\ y&=a\sin{t}\\ z&=nt\end{align}$$ (where $n,a$ are arbitrary coefficients) So far I ...
3
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2answers
163 views

Arc length paramatrizations satisfy original system of differential equations?

Say we have a system of differential equations $$ \begin{cases} x'''(t)+f(t)x'(t)=0\\ y'''(t)+f(t)y'(t)=0 \end{cases} $$ on an interval $[a,b]$, along with the restriction that $$ x'(t)^2+y'(t)^2=1 $$ ...
0
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1answer
69 views

How can I increase the resolution of a 3d plot in Sage to see all the details of the noodle? [closed]

I want to plot my noodle with a high resolution to see all the details. How can I achieve this with Sage? Here is my parametric, piecewise function: ...
2
votes
3answers
84 views

Parametrization of $y^2 - x^2=1$

I have found parametrizations for the level curve $y^2-x^2=1$, however, I have a question regarding one of them. From the Pythagorean trigonometric identity $\cos^2 x + \sin^2 x =1$ we obtain ...