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

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1answer
13 views

How to find parametric equation between two points in line integral?

[In this example how can we find parametric equations of x and y.] [1] [question]: http://i.stack.imgur.com/lTOnW.png [1] [Solution]: http://i.stack.imgur.com/l8ao7.jpg
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1answer
15 views

Convert line parametrization into two equations

Consider the following parametrization on $\mathbb{R}^3$ $$g(t) = (t^2,t\cos(t),t\sin(t))$$ This is a line, and as such can be characterized by two equations. I already found the first one to be ...
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0answers
11 views

Derive the parametric form of the locus of point where difference between distance to two points is constant

Given two points $P_1=(x_1,y_1)$ and $P_2 = (x_2,y_2)$, the locus of the point whose (signed) difference between the distance to the two points is a constant $\Delta$ is one branch of a hyperbola ...
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0answers
21 views

How do I parametrize this equation? [on hold]

$\left(x^2 + \dfrac{9y^2}{4} + z^2 - 1\right)^3 - x^2 z^3 - \dfrac{9y^2 z^3}{200} = 0$ This is the equation for a 3D heart, I believe, and I need to parametrize it in terms of u and v. Thanks for any ...
3
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2answers
33 views

Parametric version of a simple equation

I have a simple relation that I need to plot in a plane. I could do it, but I believe that I don't get the best way. A plane curve is defined implicitely by the following equation : ...
-1
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1answer
23 views

Parametric Equations: Having trouble with finding two tangents (Calculus) [duplicate]

curve C defined by these parametric equations $x = t^3 - 3t^2$ $y = t^3 - 3t$ I need to find the equations of two tangents at the point $(-4,2)$ I have attempted to solve this problem myself but I ...
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1answer
33 views

Finding the equations of the tangents to a curve defined by a parametric equation [on hold]

The curve $C$ defined by parametric equations $$C:\begin{cases} x = t^3 - 3t^2 \\ y = t^3 - 3t \end{cases}$$ I need to find two tangents at point $(-4,2)$ and find their equations for curve $C$. So ...
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1answer
14 views

Deriving parametric equations for a cubic equation

I've been looking at elementary cubic equations for curves and seem to understand them well enough. Going the other way and driving parametric equations has been mystifying. For example: given a ...
2
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0answers
26 views

Graphing/visualizing a complex parametric plot without using mathematica

I am trying to visualize the parametric plot in $\mathbb{C}$ of the curve $\gamma$ defined for $t\in[-\infty,\infty]$ as $$\gamma(t)=\exp\left(-t^{2}+\frac{t}{\sqrt{1+t^2}}i\right).$$ I think I find ...
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1answer
33 views

What's the name of this simple, closed, planar curve?

A simple, closed, planar curve can be given by the following parametric function: $$ \gamma(t)=\left(\cos t,\sin t+\frac{\sin^2t}{2}\right) $$ This function on $t=0$ to $t=2\pi$ gives the following ...
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0answers
20 views

How to determine the smallest period of a parametric curve?

Consider the polar function $r(\theta) = \sin(3\theta)$, and the parameterization of its graph given by $x = \sin(3\theta)\cos(\theta), \;y=\sin(3\theta)\sin(\theta)$. Upon inspection, one can observe ...
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4answers
328 views

Horizontal tangent line of a parametric curve

Suppose $x=t^2,y=t^3$ is a parametric curve. Here's a quote from my textbook: The origin, which corresponds to $t=0$, is a singular point of the parametric curve, because $dx/dt=2t,dy/dt=3t^2$ are ...
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1answer
27 views

Finding parametric equations of rectangular equation

Is there a general process to follow when finding the parametric equations of a normal rectangular equation ? I know that one rectangular equation might have many parametric equations, but are there ...
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2answers
65 views

Parametric Surface

A surface is given by $$r(u,v) = \langle u, v^2, uv\rangle$$ (a) Evaluate the unit normal vector, $\vec n$, to the surface at the point corresponding to $u=2$ and $v=1$. I've done this by ...
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1answer
23 views

finding cartesian equations of parametric equations

find the Cartesian equation for the parametric equation $$x=\frac{1}{\sqrt{1-t}} \text{ and } y=\frac{t}{1-t}$$ I tried cross multiplying but I cant seem to find the equation in terms of $t$ to ...
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0answers
29 views

Does anyone known the parametric equations for Cloud Gate?

I would like to use Mathematica to plot the famous Chicago "Bean." I couldn't find parametric equations anywhere and was wondering if anyone knew them. Thanks!
0
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1answer
36 views

Find arc length of $(x-1)^{2/3}+(y-2)^{2/3}=1$

I'm trying to find the arc length of the curve defined by $$(x-1)^{2/3}+(y-2)^{2/3}=1$$.My first approach was try to set 'y' in terms of 'x' and then apply the formula ...
2
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0answers
21 views

Calculate $\int\int_sf(x,y,z)dS$ for $x^2+y^2=9$, $f(x,y,z)=e^{-z}$

Calculate $\int\int_sf(x,y,z)dS$ for $x^2+y^2=9$, $0<z<6$; $f(x,y,z)=e^{-z}$ I am completely confused on this. I know I can parameterize $x^2+y^2=9$ into... $x(r,\theta)=rcos\theta$ and ...
0
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0answers
21 views

Minimizing arc length on unit sphere (geodesics)

I just completed a Calculus IV course and taught myself basic Calculus of Variations, and wanted to extend some of the basic principles of optimization from planes to surfaces. The arc length ...
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1answer
28 views

How can I draw a Bézier Curve through a set number of points?

For high school Mathematics Pre-Specialist, I have been given the task of writing a mathematical investigation based on the following three questions: Quadratic Bezier curve enables a smooth curve to ...
2
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0answers
69 views

Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...
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1answer
31 views

Question about calculating the area underneath a “3d” curve.

I'm trying to calculate the area underneath a curve after a $z$-component has been added. Suppose we have the equation: $$y = -x^4 - x^3 + 3x^2 -x + 4$$ on the interval $[-2.38, 1.76]$ (the roots ...
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0answers
18 views

How could I determine the form of a function that chases another function?

This is a problem that a teacher told me about that's been bothering me for a while. I'm positive that this has been explored before because it seems way too useful for physicists to not have come up ...
0
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1answer
24 views

Projecting a parametric curve to a plane

I have a parametric equation: x = t^3 and y = t + 2t. I would like to do a line integral of this curve up to the plane z = 5. Basically, I would like to find the area of the "walls" formed when ...
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1answer
38 views

Tangent line for a parametric curve

I am given that $x=ae^{-t}$ and that $y=be^{2t}.$ I'm asked to find the tangent line at $t=0.$ I have said that $$\frac{dx}{dt}=-ae^{-t}, \frac{dy}{dt}=2be^{2t}$$ Thus ...
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3answers
42 views

Function parameters and cartesian curves

Given $$x = \cos t + \cos 2t \,,\; y = \sin t + \sin 2t ,$$ find the tangent line for the parameter at point $(-1, 1),$ and draw a graph of the curve. To find the point you could simply do ...
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2answers
32 views

Finding parameterization of surface that maps parameter distance to arc length

I have a surface in $\mathbb{R}^3$ defined by four corner points $p_i$ and with known normals at each corner $n_i$. I've also constrained the contour of each edge to be a circular arc, which can be ...
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1answer
15 views

Need help understanding what the curve made by two or three intersecting surfaces looks like

I have trouble visualizing what curves are traced out by the intersection of multiple surfaces in $R^3$. for example take the parametric equations $ <cos(t),sin(t),sin(t)$ > Clearly this would ...
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2answers
37 views

Arc Length parametric curve

I have the following curve: $$x = \cos(t)$$ $$y = t - \sin(t)$$ $$0 \leq t \leq 2\pi$$ I have to draw the graph, point the direction and find its length. The solved the first two questions. The ...
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0answers
20 views

When is it possible to eliminate the parameter from a set of parametric equations?

I found this question on-line, I was unable to source it. Is it poorly written? For example, is there a case (C) where you can have a non-parametric form given, and also have a parametric form ...
0
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1answer
13 views

two equivalents equations with different representations in the plane

Consider the parametric curve $$C:\begin{cases} x = 4e^{t/4} \\ y = 3e^{t} \\ \end{cases} $$ A cartesian equation for this curve is $y=\frac{3x^4}{256}$. The problem is that ...
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1answer
31 views

Points of intersection of two parametric curves

I want to find all points of intersection of the parametric curves $$C_1:\begin{cases} x = t+1 \\ y = t^2 \\ \end{cases} $$ and $$C_2:\begin{cases} x = ...
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2answers
20 views

What does the Cartesian equation of a Parametric function tell us?

I've been taught that a Parametric function can be converted into a Cartesian one by eliminating the parameter $t$ but I've never been taught of how it specifically relates to the Parametric. Does it ...
0
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1answer
43 views

What is the correct math notation for the final solution while finding the rank of this matrix? [closed]

What are the respective different ranks of the matrix ? I tried with all parameters $a,b$ and $c$ being zero , and then $c$ being $0$. $$\left( \begin{array}{ccc} 1 & 4 & 3 \\ 5 & a & ...
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0answers
36 views

Verification of divergence theorem

At time t the velocity field of a fluid is given by $\bar V(x,y,z)=x^3 {\bf i}+y^3{\bf j}+z{\bf k}$ the outward flux integral $ Φ = \iint_S \bar{V}\cdot d\bar{S} $ where S is the surface of the ...
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0answers
20 views

Is it acceptable to call curves on parametric surfaces “isoparms”?

Let $\mathbf{r}(u,v):[a_0,b_0] \times [a_1,b_1] \to \mathbb{R}^3$ be a parametric surface. If $u$ and $v$ are fixed, is it allowed to call $\mathbf{r}(u,\cdot)$ and $\mathbf{r}(\cdot,v)$ "isoparms" or ...
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1answer
48 views

Why are there so many different vector and parametric equations for a line? [closed]

Please explain why there are many different vector and parametric equations for a line.
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1answer
22 views

Show that the line with parametric equations don't intersect

Show that the line with parametric equations $x = 6 + 8t$, $y = −5 + t$, $z = 2 + 3t$ does not intersect the plane with equation $2x − y − 5z − 2 = 0$. To answer this do i just plug in the $x$, $y$, ...
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1answer
20 views

Determining parametric equation given 3 points [closed]

Determine parametric equations for the plane through the points $$A(2, 1, 1), B(0, 1, 3), C(1, 3, −2)$$
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3answers
127 views

Three questions about the form $X^2 \pm 3Y^2 = Z^3$ and a related lemma

In Ribenboim’s Fermat’s Last Theorem for Amateurs, he gives the following lemma [Lemma 4.7, pp. 30–31]. Lemma. Let $E$ be the set of all triples $(u, v, s)$ such that $s$ is odd, $\gcd(u,v) = 1$ and ...
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2answers
19 views

Continuity and differentiability of a function defined parametrically

How do we check continuity and differentiability of a function defined parametrically e.g. $$x=2t-|t-1|$$ and $$y=2t^2+t|t|$$
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2answers
29 views

Area of described parametric region

The problem is as follows: A rope is tied to a cow and attached to the side of a circular silo with radius $r$. If the rope has length $\pi r$, what is the area of the land available for grazing ...
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1answer
24 views

Parabola that intersects two lines and matching the slope of the two lines?

If I have two lines with equations;$$x=0$$ $$y=0$$ $$z=t$$ and $$x=t$$ $$y=10$$ $$z=t$$ are there any parabolas that cross through the two lines and in which the parabola matches the slope of the ...
0
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2answers
47 views

How to parametrize $\left(4-\sqrt{x^2+y^2}\right)^2 +z^2=1$

How would I parametrize $$\left(4-\sqrt{x^2+y^2}\right)^2 +z^2=1$$ I am really struggling to parametrize this surface. Here is what I observed the surface is $$(4-r)^2+z^2=1$$ so perhaps we can try ...
0
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2answers
20 views

How to parametrise shapes such as petals and cardioids?

Okay for example I want to compute a line integral along the curve described in polar coordinates by $r=\sin(2\theta)$ so I will need to parametrise this curve. (In fact I only need to parametrise one ...
0
votes
1answer
41 views

Eliminating a parameter from 2 equations

The question given to me was actually of parametric differentiation, and the equations were: $$x = \dfrac{\sin^3 t}{\sqrt{\cos2t}}\ , \ \ \ \ y = \dfrac{\cos^3 t}{\sqrt{\cos2t}}$$ and we had to ...
0
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1answer
33 views

Need help with unit circle trig coordinates.

I'm in over my head and need some help with this question. Sorry if this is too simple for you but I'm really struggling. I can't for the life of me figure out how to write the angles A in terms ...
0
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1answer
29 views

Question about Flux and direction of normal

I am trying to do the following question; calculate the flux Suppose $$F(x,y,z)=(-x)i+(-y)j+(z^3)k$$ over the cone $z=\sqrt{x^2+y^2}$ between $z=1$ and $z=3$ with downward orientation My attempts: ...
2
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0answers
34 views

Can parametric equations graph all kinds of lines?

I saw this question which had a similar viewpoint, but was limited to straight lines and polynomials. Now we know that we can graph some pretty crazy stuff with parametric equations. For example: ...
0
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1answer
24 views

Parametric derivatives

Let $f(x) = \dfrac{2\sqrt{1+x^2}-5\sqrt{1-x^2}}{5\sqrt{1+x^2}+2\sqrt{1-x^2}}$. Hence, find $\frac{dy}{dz}$ when $y=\cot^{-1}(f(x))$ with respect to $z=\cos^{-1}{\sqrt{1-x^4}}$. To get this into a ...