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

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Line integrals and parametrization

I've just learned about line integrals, and I need some help understanding an example problem in my textbook. The question is supposed to be really easy. Integrate $f(x,y,z)=x-3y+z$ over the line ...
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2answers
30 views

Find the area of the circle

Find the area of the circle defined by the parametric equations $x = \cos t$ and $y = \sin t$. I know this is circle defined by $x^2 +y^2 =1$ so i used $0 < t < 2\pi$ as my bounds, then ...
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0answers
25 views

Parameterizing an implicit curve

I have to parameterize this curve: $$F(x,y)=y-x^2+x-e^{-yx^2}=0$$ But I don´t know how to do it. thanks
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1answer
34 views

Help Needed Changing Parameter

Given that $r(t)=(4(\sin(t)−t\cos(t)),4(\sin(t)+t\sin(t)),(3/2)t^2)$ is a vector-value position function. Find the arc length function $s$. I need to change the parameter before deriving to calculate ...
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29 views

Analytical Models for Hysteresis of Complicated Systems

I’ve been working with a system that exhibits hysteresis and I’ve found that the more common models do not work for me. I am wondering if anyone is aware of other models that might be out there for ...
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54 views

How to find the parametric equation of $x^y=y^x$ without Lambert W function?

This is sort of a follow-up to my previous question. I've done basic conversions of parametric to to cartesian and back as part of my A-level, but never anything more advanced than a sin/cos ...
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2answers
50 views

Multivariable calculus - scalar field

I don't know how to solve this problem. Determine if $\mathbf{F}$ is or not the gradient of a scalar field. If it is find the corresponding potential function f. $\mathbf{F}(x,y,z)= 3y^4 ...
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25 views

How would I parametrise a straight line?

If I want to parameterise a straight line and I have the equation, eg $y=2x+1$ and I also have two co-ordinates it passes through, would it ok to use the co-ordinates to parameterise in terms of $t$?
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13 views

Finding the bounds of a parameterisation

So I need to understand how to find what in this case "t" runs between for the following co-ordinates so that I can find the work done - (1,0,0) to (5,6,8) Parameterisation - $x=1+t^2, y = 3t, z ...
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2answers
31 views

“Orthonormal” parameterization of solid sphere?

The standard parameterization of the solid sphere of radius $r$ centered at the origin in $3$-space is ...
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35 views

Faster way of finding critical points?

So I am looking at parametric vector function. $$ \begin{vmatrix} \cos (t) & -\sin (t) & 0 \\ \cos f(t) \sin (t) & \cos f(t) \cos (t) & -\sin f(t) \\ ...
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28 views

x(u,v), y(u,v), z(u,v) parametric equations for a special cycloid

I'm trying to find out a 3d parametric equations for a cycloid I know that a cycloid is a 2d curve it is generated by a point on a rolling circle. but my circle is rolling around another circle both ...
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1answer
19 views

What is the Implicitization Problem

Let $V$ be a subset of $k^n$ given parametrically as $x_1 =g_1(t_1,...,t_m) ...x_n=g_n(t_1,...,t_m)$. If the $g_i$ are polynomials (or rational functions) in the variables $t_j$, then $V$ will be an ...
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1answer
18 views

Sketching Parametrizations - how to get something more understandable?

So I have some parametric functions (of one variable) I'm trying to sketch. Generally I can do so by "reverse parametrizing" where I take $x(t)$ and make $t$ a function of $x$ and then substituting ...
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31 views

How to parameterise the curve $ x^2 = 4y, 3x^3 = 8z$?

As per title, I'm unsure how to parameterise the given curve? Are there different methods? I'm unsure about parameterisation in general, I just tend to remember specific formulas.
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12 views

Bezier Surface evalution

So the problem I'm having at the moment, is a thinking problem. I can draw a bezier surface (parametric surface) with 16 control points and if I evaluate S(u, v) I get a coordinate in the 3D space. ...
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34 views

Solving a complex exponential / logarithmic equation

I've found this interesting equation on the web: $$p-1 = (1 - e^{\alpha-\beta t})^{t+1}$$ which has to be solved for t, considering that the parametes: $\alpha, \beta, p$ are defined correctly. ...
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1answer
16 views

Is there a parametric form for a degenerate conic section?

With parametric form I mean a parametrization like $(\cos{t}, \sin{t})$ for a circle. A conic section has such a parametrization, but suppose it degenerates in 2 lines (ranges of points), is there a ...
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1answer
36 views

Parametric Equation of sine wave helically wrapped around a cylinder

I want a parametric equation of a sine wave at a small ramp angle wrapped around a cylindrical body (3D). The parametric equation below gets me close to what I'm looking for, but not quite since the ...
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2answers
37 views

Parametric equation of a curve in $R^3$

How to find the parametric equation of the curve in $R^3$, which is the intersection of the sphere of radius $a>0$ centred at the origin, and the plane $x+y+z=0$? I've tried to start looking for ...
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101 views

Why can't elliptic curves be parameterized with rational functions?

Background: For our abstract algebra class, we were asked to prove that $\mathbb{Q}(t, \sqrt{t^3 - t})$ is not purely transcendental. It clearly has transcendence degree $1$, so if it is purely ...
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1answer
32 views

Prove the normal will be at constant distance form origin in this parametric function?

Given a function, $x = a(cos \theta + \theta \sin\theta])$, $y = a(sin\theta - \theta\ cos\theta)$, $a \in R$ Prove that the normal drawn on each point is at constant distance form the origin? If ...
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1answer
35 views

Prove the continuity and differentiability of parametric integration

$$F(\alpha )=\int_{0}^{+ \infty } \frac{\cos x}{1+(x+\alpha )^{2} } dx$$ Prove the function F is continuous and differentiable on the interval $[0, +\infty )$
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22 views

Parametrization of a paraboloid part

Find the parametric equation of the surface $S$, where $S$ is the part of the paraboloid $z=x^2 + y^2 + 1$ bounded by the plane $z=2x+3$ My attempt The OXY projection of $S$ is $x^2 + y^2 + 1 = 2x + ...
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17 views

Parametrics Question Help Please [duplicate]

Would anyone be able to verify the answer of (iv) being $y=-a$? I assume since -a is a constant, the x value is irrelevant Thanks
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15 views

Calculate the convergence domain of parameter improper integral

$$\int^{+ \infty }_{1} x^{u} \frac{x + \sin x}{x - \sin x}dx$$ The answer is $u<-1$. I suppose we need to find the simplified equivalent form of it, but I stuck on my way.
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32 views

Parametric equations of manifold

I have am working for a linear algebra test and I realized that I don't know how to solve exercises with linear manifolds even the basic one. W : $ x+y-z+u=1 $ $ 2x+u=2 $ $ z -u=0 $ I don't ...
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4answers
40 views

Showing that these two lines are parallel.

$$ \dfrac{x - 1}{2} = 2 - y = 5 - z \quad \text{and} \quad \dfrac{4 - x}{4} = \dfrac{3 + y}{2} = \dfrac{5 + z}{2}. $$ I was given this math problem as homework, and I have spent the past hour ...
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1answer
21 views

Parametric form of curves?

Can someone tell me the steps to get the parametric form of a curve? For example: $x^{2\over 3}$ +$y^{2\over 3}$ =1
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59 views

Find the parametric equations of $x^2+y^2+\sin(4x)=4$ [closed]

Can someone find the parameters for $x^2+y^2+2\sin(4x)$. I'm trying to find a way to visualize a derivative for implicit functions that can be graphed on a free-online graphing calculator called ...
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2answers
24 views

Lengths of Plane Curves - Calculus 2: $\sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2}$

$$ \sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2} $$ I am having problems setting this up. Taking the derivative of $\sqrt{1-x^2}$. Leaves me with: $$ \frac{1}{2}\left(1-x^2 ...
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34 views

Parametrization for implicit function

$3y^2=x(1-x)^2$ By differentiation we can knwo that the sketch of this graph has one circle. I want to draw a graph in maple. Implicit plot does not work well So I will use parametric way. ...
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1answer
35 views

Surface described by parametric equations

If I've got the surface in $\mathbb{R}^3$ described by: $x(s,t)=s^2-t^2$, $y(s,t)=s+t$, $z(s,t)=s^2+3t$ for $(s,t)\in\mathbb{R}^2$, and I'm told this surface is the graph of a function $f(x,y)$, how ...
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1answer
18 views

using parametric equations to form a line equation from two points

hi in order to form a line equation from two points i have been told to do following and not to use any other ways. a(9,6) b(2,-1) x=9-7t y=6-7t cancel out the ts gives x-y=3 but when the signs ...
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3answers
76 views

Find all values of $a$ for which there are two real solutions of $x^3-2ax^2+a^2x-3=0$

Find all values of $a$ for which there are two real solutions of the equation. $$x^3-2ax^2+a^2x-3=0$$ Ans = $1.5\sqrt[3]{6}$ I tried to research the function by dint of derivative, but it didn't ...
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Relation between $\sin(t)$($\cos(t)$) and $\sin(at)$ ($\cos(at)$) when both are rational

This question relates to Parametric equations where sin(t) and cos(t) must be rational. Suppose it is given that $\cos(t)$ and $\sin(t)$ are both rational and also $\cos(at)$ and $\sin(at)$, where ...
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1answer
36 views

Calculating curvature of a curve on a the surface $x^2+y^2=1$. [closed]

Find a curve on the cylinder surface $x^2+y^2=1$ in $\mathbb R^3$ such that its curvature is equal to $\frac1{100}$ at each point of this curve. Does this easily generalize to different surfaces?
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2answers
69 views

Parametric equations where sin(t) and cos(t) must be rational

Suppose there are parametric equations $$ x(t) = at - h\sin(t) $$ $$ y(t) = a - h\cos(t) $$ and it is required that both $\sin(t)$ and $\cos(t)$ should be rational. What the values of $t$ should be ...
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2answers
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Paramaterizing a path $C$ along a parabola $y=2x^2$

I am doing a line integral where the path $C$ is defined as the arc of the parabola $y=2x^2$ from the points $(-1,2)$ to $(2,8)$. Is there a "catch all" approach or method that can be applied here? ...
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1answer
27 views

What is the non-piecewise curve that resembles the following roller coaster track?

I want to create an animation about roller coaster. One track I want to use looks like the following figure. I am looking for the simplest non-piecewise parametric equation for both $x(t)$ and ...
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How to determine the coordinate of roller coaster's wheels?

I want to create an animation about roller coaster. For a simple track, for example, a circle, I can determine the position of the center of its wheel easily. However, for any parametric curve, I ...
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1answer
28 views

Help find the equation of two planes

I have the question Consider the line L through the distinct points A = (a,b,c) and D = (d,e,f) Find the equations of the two planes which intersect at right angles along L MY ATTEMPTED SOLUTION I ...
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1answer
42 views

When and why must we parameterise $f(x, y) = …$ with variables besides $x, y$?

For 10C, my choice of parameterisation $\mathbf{r} (x,y) = ( x, y, z(x, y))$ fails to effect the right answer, but that of user ellya does function. Yet for 9C, the parameterisation $\mathbf{r} (x,y) ...
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35 views

Graphing a Parametric Equation

I need to graph and show the work for this problem. The graph needs to include arrows on the curve to show the direction of motion and I need to label the t-values graphed. $$c(t)=(2+4t, 3+2t)$$ So ...
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3answers
61 views

Finding the speed of a particle (parametric math)

I have to find the speed (as a function of $t$) of a particle whose position at time $t$ seconds is represtented by $$c(t)=(\sin t+t, \cos t+t)$$ How would I go about finding the maximum speed? ...
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1answer
30 views

Find the length of the curve

I need to find the length of the curve $$c(t)=(3e^{t}-3, 4e^{t}+7)$$ for $$0\le t \le 1$$ If I understand correctly, I need to take the derivative of the y part of that coordinate over the ...
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Find the parametric equation to the curve

Find the parametric equation for the curve. $$x^{2}+y^{2}=10$$ I haven't learned parametric equations fully yet, so I wanted to check with you guys and see if you can confirm if I'm doing this ...
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1answer
32 views

Express the parametric equation in form of y=f(x)

I need to express the parametric equation in the form of $y=f(x)$ by eliminating the parameter. I haven't learned how to do this yet, I've attempted to read a few pages though but they didn't help me ...
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3answers
31 views

Parameter “m” for which $P(x)=4(m+1)x^3+(m-3)x+1-m$ has a root with multiplicity two…

Can you please help me solve this parametric problem. So, we have to find all the values of real parameter $m$ for which the following equation has a solution with multiplicity ...
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1answer
26 views

Find a right angle triangle in with 3 vertices and one parameter

Given three coordinates, which could be $A=(7,3)$, $B=(2,4)$, $C=(k,-2)$ I want to find the values of $k$ that make a right angle diagram out of the three points. So I initially was thinking to find ...