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

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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 ...
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1answer
38 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 ...
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24 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 ...
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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
43 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
23 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, ...
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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 ...
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29 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 ...
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1answer
33 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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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
26 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 ...
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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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19 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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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 ...
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2answers
25 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 ...
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0answers
57 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
13 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 ...
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49 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 ...
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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 ...
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22 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 ...
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1answer
26 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 ...
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1answer
60 views

How to calculate area of curved surface in specific region

i don't know how to use this site. this question is my first. please see the image uploaded in this page.
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3answers
171 views

What function could describe this GIF animation?

I found this image on Beautiful Mathematical GIFs Will Mesmerize You and this GIF really caught my attention. From what I see, it's a 2D circle morphing into the 3D sphere. What function could ...
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1answer
15 views

Parametric representation of a solid trapezoid

Question: Define a parametrical representation of a solid trapezoid as shown in the following figure: I came up with a solution by combining representations of the left rectangle and the right ...
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3answers
39 views

Find a vector parametric equation $\vec r(t)$ for the line through the points $P\equiv(1,0,−4)$ and$ Q\equiv(3,−3,1)$

Find a vector parametric equation $\vec r(t)$ for the line through the points $P\equiv(1,0,−4)$ and $Q\equiv(3,−3,1)$ for each of the given conditions on the parameter $t$ If $\vec r(3)=P$ and $\vec ...
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0answers
38 views

Given the equation of a cylinder $ x^2+z^2=1,$ find the parametric and locus form of the curve of intersection with plane

Given the equation of a cylinder $x^2+z^2=1,$ describe the curve of intersection between the cylinder & the planes z=x & y=x in the parametric form & the form F(x,y,z)=0. I am so lost ...
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2answers
39 views

How do solve this equation with dot product?

$$[(2,-7)+t(2,10)]\cdot n_1=0$$ $$\text{Solving for $t$, we find }t=\dfrac5{12}$$ $n_1=(1,1)$ or $(-1,-1)$. How does $t=\dfrac5{12}$?
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Understanding the definition of “smoothly equivalent”

$\underline{Definition}:$ The two curves $C_1:z(t), a\leq t\leq b$ and $C_2:\omega(t), c\leq t\leq d$ are $smoothly\ equivalent$ if there exists a $1-1\ C^1$ mapping ...
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1answer
25 views

Determine a parameterization for the line which is tangent to the curve at t=2

(1) A curve is given by the function $$r(t)=(t^3 -3t^2 +2t +4)i + (13-5t)j +(t^2 -t-3)k$$ Determine a parameterization for the line which is tangent to the curve at $t=2$ I started by solving for ...
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0answers
17 views

Expressing Curvature of a Polar Function in Terms of its Derivatives

I could use a little guidance with this question: Consider the curve $r=f(\theta)$, where $f$ is any twice differentiable function. Determine an explicit formula for the curvature $\kappa$ in terms ...
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2answers
32 views

Finding arc length parametrization of a cycloid

Find an arc length parametrization of the cycloid with parametrization r(t)= . I took the derivative and found the speed to be sqrt(2(1-cost))but now I'm unsure how to integrate that to get s. How do ...
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0answers
40 views

Catenary equation in 3D

I have two points with coordinates A(x1,y1,z1) and B(x2,y2,z2). There is a third point which is lowest point of the catenary curve. I only know z-coordinate of this third point. I need to find ...
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1answer
27 views

Intersection of two parametric equations

This is a super basic question I'm sure but I can't figure it out and that's so frustrating. I must, in this homework problem (yes it is homework, so please do not give away the answer but rather make ...
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1answer
152 views

Equation of a parabola in 3D space

I have two points with coordinates A(x1,y1,z1) and B(x2,y2,z2). There is a third point which is vertex(lowest point) of the parabola. I only know z-coordinate of this point. I need to find coordinates ...
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2answers
36 views

Converting from Parametric to Cartesian?…

I've been working on converting from parametric equations into cartesian form, but can't figure out this problem? $$x=(t^2+1)/(t^2-1)$$ $$y=(2t)/(t^2-1)$$ How do I covert that to Cartesian? Any help ...
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0answers
17 views

Determining the curvature of $y=Asin(bx)$

I am asked to determine the curvature of $y=Asin(bx)$. Unfortunately, I don't think I am on the right track. So the curvature of a curve is: $\kappa = \frac{1}{\lvert \vec{V}\rvert}\lvert ...
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1answer
29 views

trigonometric parametrization

I am trying to figure out a pattern. I will start with examples. $$\text{Let } PD(\text{Set } A):= \text{Parametric Description of }A$$ $$ A:=\{(x,y)\in \mathbb R ^2|x^2+y^2 =1 \} $$ $$PD(A): ...
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1answer
35 views

Finding Arc Length Parametrization

Find an arc length parameterization of the line $y=6x+7$. Confused on how to start with x's and y's.
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3answers
39 views

Parametrization of the intersection of two given surfaces

Find a parametrization of the intersection between the two curves $z=x^2-y^2$ and $z=x^2+xy-1$. I figure I should set them equal to each other but I'm not sure where to go from there: $$x^2-y^2 = ...
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1answer
51 views

Convert vector parametric equation to general form

Given the equation of a plane $x$ is $$x(s,t)=(0,1,1)+s(1,0,1)+t(2,1,-1)$$ How can I convert this equation into the general form $$A(x-x_0)+B(y-y_0)+C(z-z_0)=0$$ Thank you.
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1answer
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Trajectory of a particle for $t\ge1$ given $r(t)$ for $0\le t \le 1$.

I have a question on the process for which to solving this question. It is a homework question, and I already have the answer, but I am not sure on the correct process to attaining that answer. The ...
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1answer
20 views

Intersection of two parametric lines

This is not a question on my homework, just one from the book I'm trying to figure out. They want me to find the intersection of these two lines: \begin{align} L_1:x=4t+2,y=3,z=-t+1,\\ ...
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0answers
19 views

Parametrisation of surface

Let $K= \{ (x,y,z) \in \mathbb{R}^3 : \sqrt{x^2+y^2} \leq z,\,\, x^2 + y^2 + z^2 = 1 \}$. I need a parametrisation of $K$ in order to calculate the flux of some function through $K$. I'm not sure ...
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0answers
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figuring out parametric equation of a moving dot of specific velocity along acurve

I currently need to model a dot moving along an arbitrary curve given it's velocity, initial point, and $y=f(x)$ form of equation. I vaguely remember from my high school teaching that it will possibly ...
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1answer
40 views

Two questions on parametric equations, vectors, and planes.

I have two questions regarding parametric equations that I am struggling with. Question 1: a) Give a parametric equation for the line passing through $(-1,-2,3)$ and $(1,5,-2)$. b) Give the ...
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1answer
39 views

use the Parameterization in u and v to write the term $x^2+y^2$

Given that : $u=xy$ $v=x^2-y^2$ we want to write the term $x^2+y^2 $ using only $u$ and $v$. how can we do this ? update: please reread my question I have edited it. I think it is clear now ...
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1answer
25 views

How to find the normal vector in a TNB problem

I have done this TNB problem multiple times; however, my online homework system keeps telling me my answer is incorrect. I was hoping someone would look at my work and tell me where I'm going wrong? ...
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1answer
19 views

What will the graph of this parametric equation look like?

What will the graph of this parametric equation look like? $$x = 2t$$ $$y = t + 5, \quad -2 ≤ t ≤ 3$$ Does "$-2 ≤ t ≤ 3$" represent the domain?
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20 views

Write an equation for the line through $A =(3, 1)$ and $B = (1, 2)$.

The line passes through $B$ and is parallel to $B - A$. So, the equation is $X = B + t(B - A)$. My question is: can we say that the following equations are correct as well? $X = B + t(A - B).$ $X ...