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

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2answers
85 views

Find points on curve $r=2\sin\theta$ where the tangent line is parallel to the ray $\theta = \pi/4$

I was thinking to convert to cartesian coordinates and then find when the slope of the tangent line is $1$, but I get a messy equation $2\cos^2\theta -2\sin^2\theta=4\sin^2\theta\cos\theta$ I was ...
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2answers
23 views

Velocity of a curve given by parametric equations

In standard Cartesian equations, $\frac{dy}{dx}$ is the velocity function because it's the derivative of position. $$\frac{dx}{dt} = \sin^{-1}\left(\frac{t}{1 + t}\right) ...
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1answer
40 views

Value of x of which a slope is undefined for a parametric graph.

For what values of $x$ is the slope undefined for the graph $$x=8-t^3$$ $$y=t^2-6t$$ The slope should be undefined when $\frac {dx}{dt}=0$. $$\frac {dx}{dt}=-3 t^2$$ $$-3t^2=0$$ $$t=0$$ When ...
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1answer
44 views

Comet Problem Using Parametric Equations

A comet has an elliptical orbit that is $144$ billion miles across the $x$-axis and $48$ billion miles across on the $72$ years to complete one revolution. If the center of the coordinate system is ...
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1answer
15 views

Curve parametrization verification: Plane and paraboloid

I wanto to find a parametric equation of the intersection between the plane $x+y-z=2$ and the paraboloid $z=(1-y)^2$ I proposed $y=t$, wich implies $z=(1-t)^2.$ So: $$\begin{align} ...
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3answers
55 views

Parametric representation of a plane cut of a sphere at y=5

The sphere is given by $x^2+y^2+z^2=36$ Parametric Form: $$x=6\sin t\cos u$$ $$y=6\sin t\sin u$$ $$z=6\cos t$$ If the sphere is 'cut' at $z=5$ this problem is trivial. ...
2
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3answers
59 views

Calculate u in terms of time such that a particle maintains a constant speed following a parametric equation

I have a parametric equation given by: $x=\cos(6u)$ $y=\sin(4u)$ And I understand that the speed of a particle at any given t is: $\sqrt{\left(\dfrac{dx}{du}\right)^2 + \left(\dfrac {dy}{du} ...
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0answers
21 views

Parametrization of sphere including constant inclination $(\theta, i)$ geodesics

Find parametrization of sphere with respect to $\theta$ = constant meridians and i = constant inclination geodesic circles passing through N-S axis and E-W axis respectively. The Earth does not rotate ...
4
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1answer
34 views

Easy question on the limits of an integral

So I would like to ask how exactly do we determine what limits to take when integrating both Cartesian and parametric equations. So let's say we have a graph of $y=x^2$. If we wanted to take the area ...
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1answer
19 views

Finding parameters for a quadrature formula

To compute the integral $\int_0^1f(x) dx$ numerical I want to use the following quadrature formula: $$Q(f)=\omega_0f(x_0)+\omega_1f(1)$$ The question is how one should choose $\omega_0,\omega_1 ...
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0answers
23 views

parameterization of a horizontal line

say I need to parameterize the boundary of a set in order to optimize. The equation is f(x,y) = 3 + x − y + xy and the boundary is the set inclosed by the line ...
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1answer
29 views

Reverse Parametric Equations

I have just learned about parametric equations. I have gotten the concept of turning the parametric equations to regular/ordinary equations, but am having trouble doing the reverse in this problem: ...
0
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1answer
20 views

How can I show that the set of images of a parameterized curve is smooth?

I know that in order for a set S to be smooth, it needs to be connected and that for each point $\vec{a} \in S$, there needs to be a neighborhood $N$ such that $S \cap N$ is a class $C^1$ function. ...
2
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5answers
81 views

Find the equation of tangent line to the curve $x=\cos(t) + \cos(2t)$, $y= \sin(t) + \sin(2t)$ at the point $(-1,1)$.

I get the equation for the slope as $\frac{\cos(t) + 2\cos(2t)}{-\sin(t) -2\sin(2t)}$ but I'm unsure how to solve for the value of $t$. I know I need to sub in $-1$ and $1$ for $x$ and $y$ but the ...
0
votes
1answer
63 views

Gradient function of a circle

The parametric equations of a circle $C$ are: \begin{align*} x&=2+\dfrac{13}{5\sqrt{2}}\cos t\\ y&=1+\dfrac{13}{5\sqrt{2}}\sin t \end{align*} for $t\in[0,2\pi]$. I am stuck on this part: Find ...
0
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0answers
17 views

Smallest interval for graph

Parametric equation of a graph is x = cos(4t) , y = sin(6t) What is the length of the smallest interval $I$ such that the graph of these equations for all $t\in I$ produces the entire graph ...
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1answer
44 views

Why does the vector field $(\sin (\theta), - \cos(\theta), 0)$ indicate sideways motion?

If I study a physical system, such as a car, and let it drive forward a little bit, say a distance $m$, then I can draw out the right triangle and find the car's position at $(m\cos \theta, ...
2
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0answers
43 views

Line integral - new parametric equation

We know that $$\int_\gamma V \cdot dr = \int_a^b V(r(t)) \cdot r'(t) dt$$ with $V$being our vector field and $r$being the parametric equation for the curve $\gamma$. Let now $\hat{r} = r \circ \phi$ ...
1
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2answers
35 views

Finding a local parameterization of a plane curve

I'm attempting to find a parameterization of $\frac{x_1^2}{a^2} + \frac{x_2^2}{b^2} = 1$. I find a tangent vector field: $X = \left( \frac{2x_2}{b^2}, -\frac{2x_1}{a^2} \right)$ (by taking the ...
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1answer
24 views

Understanding proof for Green's Theorem

In a proof for Green's theorem where we first assume that the area bounded by a closed $C^1$ curve $\gamma$ is of the following form: $$D = \{ (x,y) \ | \ x \in [a,b], \ \mu(x) \le y \le v(x) \}$$ ...
0
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1answer
33 views

finding parametric equations from a rectangular equation

Find the parametric equations for $x^2-4x+y^2-2y+5=2$, and graph. Hint: Complete some squares. I have completed squares and gotten $(x-2)^2+4=-(y+1)^2-2$ but I am confused with how to proceed. I know ...
1
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1answer
23 views

Parametrisize of vertecies (0,0) (a,0) (0,b)

Hi i have been given 3 vertecies. (0,0) (a,0) (0,b) The constants a and b are >=0. This forms a backwards triangle. The parametisation don't make sense to me, so basically what i am asking is for ...
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0answers
14 views

parametrization: t values that lead to the same point

Given a parametrization $r(t)$, we can have several $t$ values that correspond to the same point on the curve... what's the method of finding all of these when dealing with trigonometric functions? I ...
0
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1answer
72 views

Parametric equation of a circle in 3 space - odd result

I have the following problem involving parameterized circular equations, but am getting strange answers and wanted to check if my approach made any sense. In 3D space, the parametric equation of a ...
1
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1answer
56 views

Finding tangents to a cycloid

here's the question I don't really get the second part of the question.. it uses parametric curve equation to solve. A curve $\mathcal C$, a cycloid, is defined by $x=r(\theta-\sin\theta), ...
6
votes
2answers
253 views

Torsion and curvature of the curve $X(t) = (at, bt^2, ct^3)$

Hey all I am looking for help on a problem. I will post it, and than I will add what I have tried and my ideas etc. The question has been up now for a few days, I'm sure someone out there can help! I ...
0
votes
1answer
13 views

Parametrize a surface using cylindrical coord.

Hi! I am having trouble parametrizing this tower. Specifically the radius which has to be a function of the height $Z$ $$0<z<H, 0 ≤ r ≤ R(2 − z/H), \quad 0<\theta<2\pi$$ I do not ...
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1answer
41 views

Finding out if vectors are Parallel or Orthogonal in Parametric Form.

I have two Parametric Form Vectors. Is it possible in that form to work out if the vectors are Parallel, Orthogonal or neither. Or do I have to have it in standard vector form $ (a_1, a_2, a_3)$ and ...
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0answers
18 views

Higher dimension leads to different optimization?

It appears z= f(x,y) has a global max/min at another particular (x,y). Using only one independent variable $x$ at fixed $y$ i.e., for z = f(x) I get another max/min point for $x$ optimum point ...
0
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1answer
41 views

arclength parametrization intuition

I have a question about parametric curves. I have learnt about arc length re parametrization and I understand how do the problems , for example finding the length of the vector and integrating with ...
1
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0answers
26 views

Lissajous Curve

$$ \gamma(t)= (x(t),y(t))=(sin(2t),sin(3(t)) $$ Justify that we can reduce the domain of study to [0, $\pi/2$], by specifying the necessary symmetries to obtain the whole curve. I'm not really too ...
0
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2answers
15 views

A curve is described in polar coordinates . Find parametric equations for $x$ and $y$ and plot the curve.

A curve is described in polar coordinates by the equations $$ r = t; \theta = 3 \cos t; 0 ≤ t ≤ 10 $$ Find parametric equations for $x$ and $y.$ I cannot convert it into parametric form
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1answer
52 views

Find Parametric Equations for a line passing through point and intersecting line at 90 degrees

Let $P$ be the point $(3,1,-2)$ and $L$ be the line given by $x=-4+2t$, $y=2+2t$, $z=1+t$. Find parametric equations for the line passing through $P$ and intersecting $L$ at a right angle.
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2answers
27 views

Parametrization of a circular arc in terms of the angle between the tangent line and the $x$-axis

I'm struggling with this problem: "Find a parametrization of the first quadrant part of the circular arc $x^2 + y^2 = a^2$ in terms of the angle between the tangent line and the positive x-axis, ...
0
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1answer
17 views

Complex Integration parametrisation

I'm trying to integrate $\int_\gamma (z^2-2)dz$ where $\gamma$ is a spiral that loops 3 times and ends at (3,0) on the Argand diagram. I have found the parametric equations for this contour to be ...
0
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0answers
31 views

Mean curvature of polar parametric surface

For the purposes of modelling a fluid mechanics experiment, I'm dealing with a convex surface parametrized by the azimuth $\theta$ and an arc length $s$ along the surface. The points on the surface ...
0
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1answer
34 views

Solving the cubic $2t^{3} - 3t_{1}t^{2} + t_{1}^{3} = 0$

The question is: "4. Find the equation of the tangent and the equation of the normal to the curve $x = 3t^{2}$, $y = t^{3}$ at the point whose parameter is $t_{1}$. Find the parameter of the point at ...
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0answers
14 views

confirmation of curvature of paramteric curve

I just want to confirm the following calculation for curvature of a parametric curve: Given parametric curve $r(t) = (5 \sin t, 5 \sin t, 3 \cos t) $, I want to confirm that the curvature is given as ...
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2answers
33 views

Find the parametrization of the intersection of two surfaces.

I'm having trouble figuring this problem out since the $3x^2 + z^2$ is throwing me off. Especially because of the $z$. Find the parametrization for the curve of intersection between the cylinder ...
2
votes
1answer
17 views

Understanding Parameters

My textbook (New Tertiary Mathematics, Volume 1 Part 1, Pure Mathematics: The Core, by C Plumpton & P S W Macilwaine) introduces Parameters in the following manner: "The coordinates of a point on ...
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0answers
20 views

Prove that z(t) and z~(t) are admissible parametrizations of the same smooth curve

Prove that $$z(t)=t+it^2, 0\leq t \leq1 $$ and $$\tilde{z}(t)=tan\Gamma+itan\Gamma, 0 \leq \Gamma \leq \frac{\pi}{4}$$ are admissible parametrizatiions of the same smooth curve. Do the above ...
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0answers
3 views

Tangential Angle (Para)

Could anyone offer some insight into this question? I understand that where it's 0 on the angle because it's a minimum point from para differential. I also understand that at $t=0$ it's parallel to ...
1
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2answers
27 views

If $x=\cos t,y=\cos(2t+\pi/3)$ find an analytical relation between $x$ and $y$.

I'm having a bit of trouble figuring this out. At the moment this is the near solution I have: $$y=\frac12(2\cos^2 t-1)-\sqrt{3}\sin t\cos t.$$ I should be just about to solve it but find myself ...
1
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0answers
73 views

Trying to find the volume of a 3D torus shape that I made

After playing around with 3D parametric equations on my calculator (modifying the equations of a standard torus), I came across a shape that I like. The equations are: $$x=(2+\sin t)\cos u$$ ...
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0answers
14 views

Confused by Parameterizations and Coordinate Conversions

So I have a few questions regarding parameterizations and coordinate conversion. Ever since dealing with parametric equations last semester I have felt like I have never truly understood ...
0
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0answers
64 views

Parametric integration negative area?

I know there is a question very similar to mine already here Why does using an integral to calculate an area sometimes return a negative value when using a parametric equation? , but I am still a bit ...
1
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1answer
34 views

Is this differential equation correctly calculated as having as its solution this parametric family of curves?

Consider the following family of curves is parametric form: $$x(t) = a(\cos{t} + t\sin{t})$$ $$y(t) = a(\sin{t} + t\cos{t})$$ Where $a\in\mathbb{R^+}$ is a constant, and $t\geq0$. Find ...
0
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0answers
25 views

Parametrization of the intersection of the surfaces

How do I find a vector function that represents the curve of intersection of the sphere $(x+\sqrt 2/2)^2+(y+\sqrt 2/2)^2+(z-\pi/4)^2=4$ and the plane $x-y+\sqrt 2 z-\pi\sqrt 2/4=0$ ? Of course, I know ...
0
votes
1answer
27 views

Counterexample: Different curves

I am a TA in an introductory course to multivariable calculus. As defined in class two curves are said to be equal if their images are equal. Now a problem in their problemset was to prove that two ...
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0answers
20 views

Find the line tangent to the parametric curve $\left\langle t^3-1,t^4+1,t \right\rangle$

Firstly, this is a homework problem, so I would appreciate it if you might not just write the answer and rather, if I am wrong, provide suggestions only. I am given a parametric curve with the ...