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

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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
24 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
41 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!
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1answer
38 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 $$L=\int\limits_a^b\sqrt{1+y'}...
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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 $y(r,\...
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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
42 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 ...
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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
32 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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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 ...
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1answer
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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
39 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 $$\frac{dy}{dx}=\frac{-2b}{a}e^...
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3answers
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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
35 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
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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
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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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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 (...
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1answer
15 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
37 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 = 3t+...
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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 ...
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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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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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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
51 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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23 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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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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136 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
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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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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
26 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 ...
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57 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 ...
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2answers
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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 ...
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1answer
49 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 ...
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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 ...
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1answer
31 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: ...
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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: ...
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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 ...
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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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25 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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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 don'...
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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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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 $2t=\frac{\pi}{...
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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
41 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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36 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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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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33 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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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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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 $$S=\...