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

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2
votes
0answers
19 views

Cartesian/Parametric 3d equation of a cheese twist?

Hi I'm looking for the equation of a cheese twist in 3d (either parametric or cartesian)... Can be multiple planes but was wondering if anyone had any idea to execute something like this? Thanks e.g. ...
3
votes
4answers
139 views

Parametric to implicit form of a curve

"Find the implicit form of the curve defined by parametric equations $x = t+1,y=\frac{1}{t^{2}}$" How can I clear $t$ to arrive at the implicit equation?
0
votes
1answer
13 views

Parametric equations of a line

"Find the parametric equations of a line that passes through point $(1,1,0)$, parallel to plane $2x+3y+z=7$ and perpendicular to the line $\frac{x-1}{-2}= \frac{y}{3}=-z-2$" I don't know where to ...
1
vote
1answer
763 views

Find the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$.

Find a vector equation and parametric equation of the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$. Find two points on the line that are ...
1
vote
1answer
54 views

How to convert the parametric equation into implicit form?

This problem is generated from another Green's theorem related question of mine. The original equation of the plane curve is not in rational parametric form. In order to calculate the symbolic ...
-2
votes
1answer
19 views

Conic-Sections, Ellipse, Parametrics, Normals and Tangents [on hold]

Find the equation of the tangent and normal to the ellipse defined parametrically by: $x=5\cos\theta$ and $y=3\sin\theta$ at the point where $\theta=\pi/4$.
1
vote
1answer
13 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
0
votes
1answer
380 views

Find the area of the surface obtained by rotating the curve of parametric equations

Rotate about the $x$ axis $x = 2t-2/3t^3$ $y = 2t^2$ $0 \leq t \leq 1$ I did the integral of $\sqrt{(2-2t^2)^2+(4t)^2}$ and got $(2x(x^2+3))/3$ and then I did the integral of $2\pi 2t^2 ...
3
votes
1answer
37 views

Conversion between trig functions and hyperbolic trig functions

Using trig identities we can see that $\sin^2 x + \cos^2 x = \tanh^2 x + \text{sech}^2 x = 1$ , and so the parametric graph $(\cos t, \sin t)$ is similar to $(\text{sech} t, \tanh t)$. The first ...
1
vote
1answer
23 views

Slope of a Parametrized Curve

Say that we have the parametrized curve $x=e^{3t}, y=te^{-t}$. What would be the slope of this at the point $(1,0)$ and also on which points on the curve would the curve be horizontal? What I have ...
1
vote
2answers
22 views

Given two curves, find parametric curve

I am given two graphs x versus t and y versus t and I have to determine the parametric curve. The two graphs I am given: Parametric curve (that is the right answer): So the solutions say that: ...
0
votes
0answers
28 views

Is it possible to turn the parametric equation of a line in 3 dimensions into the general equation?

I Know it is impossible to do so since the parametric equation for a plane is the intersection of $2$ planes.For example: $x$ $=$ $\frac{-5}{4t}+\frac{1}{4}$; $y=\frac{3}{4t}+\frac{5}{4}$; $z=t$ ...
0
votes
0answers
10 views

Surface integral: Cone cut by a cylinder

Ok I've got this exercise from Apostol I'm trying to do: "The cylinder $x²+y²=2x$ cuts out a portion of a surface S from the upper nappe of the cone x²+y²=z². Compute the value of the integral: ...
4
votes
2answers
92 views

Show three ways that $f(z)=\frac{\overline{z}}{z-1}$ is not analytic

I need to show the complex function $$f(z)=\frac{\overline{z}}{z-1}$$ is not analytic in three ways; using Cauchy's equations, geometrically, and by integrating over the circle of radius 2. I used ...
3
votes
1answer
58 views

Show that $Y^2-X^3\mid f$ if $f$ vanishes on the curve $C: (t^2,t^3)$, and determine what property of a field $k$ will ensure that the result holds.

Let $\phi: \mathbb{R^1}\rightarrow \mathbb{R^2}$ be the map given by $t \mapsto (t^2,t^3)$; prove directly that any polynomial $f\in \mathbb{R}[X,Y]$ vanishing on the image $C=\phi(\mathbb{R^1})$ is ...
0
votes
1answer
2k views

Find a parameterization for the circle of radius 2 in the xy-plane, centered at the origin, clockwise

Find a parameterization for the circle of radius $2$ in the $xy$-plane, centered at the origin, clockwise. I know to use $2\cos(t)$ and $-2\sin(t)$ but I'm not sure what to do after that
0
votes
2answers
295 views

Line integral around intersection of sphere and plane

The unit sphere is intersected by the plane x + y = 1. Find the line integral of F = around the intersection. $\int\int\nabla$x$F\cdot$ n dA the unit normal vector is easily found by looking at the ...
1
vote
1answer
32 views

Parametric equations of perpendicular lines

I'm having problems with this: Find the parametric equation of the line that passes through the point $(-1, 4, 5)$ and is perpendicular to the line: $$x = -2 + t$$ $$y = 1 - t$$ $$z = 1 + 2t$$
0
votes
1answer
23 views

Parametric equations - locus at midpoint

Consider the parametric equations $x=-2t^2$ and $y=4t$ The normal at any point, P, cuts the x-axis at Q. Find the Cartesian equation of the locus of the midpoint, M, of PQ. Can anyone help get me ...
1
vote
1answer
35 views

Counting the integer soultions to this parametric inequality

hello I am looking for an efficient way, hopefully a formula or a somewhat tight upper bound, for the number of integer solutions to the following let $k$ be a fixed integer and $\lambda \ge 1$ and ...
0
votes
2answers
28 views

Evaluate $\int_C z^2 e^{1/z} \cosh(1/z)\,dz$, where $C$ is any simple-closed curve, oriented counterclockwise, and containing 0 in its interior.

Evaluate $\int_C z^2 e^{1/z} \cosh(1/z)\,dz$, where $C$ is any simple-closed curve, oriented counterclockwise, and containing 0 in its interior. my works I'm stuck in next step
1
vote
2answers
19 views

Parametric equation question showing minimum value of d^2

for the equation $d^2 = (1-a)^2t^2 + 18(1-a)t +117$ Show that when $a = 2$, the minimum value of $d^2$ is attained when $t=9$. I set $a=2$ to get $d^2 = t^2 - 18t + 117$ should i now just run it ...
0
votes
2answers
16 views

Rearranging this equation

This is based on a parametric equation problem. We have two ships A and B at $(-2,at +1)$ and $(4, t+10)$ respectively. I need to show that $d^2 = (1-a)^2t^2 +18(1-a) t +117$ using the distance ...
1
vote
2answers
67 views

How to show that the curve $ (x,y,z) = \langle \cos t, \sin t, c\sin t\rangle $ is an ellipse?

Show that the curve $$(x,y,z) = \langle \cos t, \sin t, c\sin t\rangle $$ is an ellipse in the plane it lies on. $$x^2 + y^2 = (\sin t)^2 + (\cos t)^2 = 1$$ $$x^2 + (z/c)^2 = (\sin t)^2 + (\cos ...
0
votes
2answers
34 views

What's the parametric equation for the plane through a point (x,y,z) perpendicular to (a,b,c)?

Find the parametric vector and Cartesian equations for the following planes: a. The plane thru point $(2,1,-2)$ perpendicular to vector $(-1,1,2)$. b. The plane thru the three points $(2,2,-2)$, ...
4
votes
2answers
28 views

Curl of a vector field.

Let S be a piecewise smooth oriented surface in $\mathbb{R}^3$ with positive oriented piecewise smooth boundary curve $\Gamma:=\partial S$ and $\Gamma : X=\gamma(t), t\in [a,b]$ a rectifiable ...
0
votes
2answers
20 views

2 lines passing Q and R meets at the mid-point,

Consider the straight line whose parametric equation is $$(x, y) = (1, 1)+ t(12,−1)$$ Show that the above line and a line passing Q and R meets at the mid-point. $Q = (5, 5)$ and $R = (9,−4)$ How ...
2
votes
3answers
63 views

Find the shortest distance between the point and a parabola

Find the shortest distance between the point $(p,0)$, where $p> 0$, and the parabola $y^2=4ax$, where $a>0$, in the different cases that arise according to the value of $p/a$. [You may wish ...
0
votes
0answers
25 views

Find the parametric equations of the plane given the following pieces of information

Find parametric equations of the plane that passes through the point $$P (- 2, 1, 7)$$ and is perpendicular to the line whose parametric equations are $$x = 4 + 2t , y=-2+3t, z =-5t$$ Here is my ...
0
votes
1answer
21 views

finding a vector valued function for the intersection of two shapes

I have a problem for my cal 3 class to find a vector valued function for the intersection of these two equations. $4x^2+4y^2+z^2=16$ and $x=z^2$ so i know that the first equation is a ellipsoid ...
5
votes
1answer
62 views

what is the parametric form for “mystery curve”?

Mystery curve found here looks like this : Was given by the complex formula : $$e^{it} – \frac{e^{6it}}{2} + i \frac{e^{-14it}}{3} $$ Is the parametric form simpler or the polar form would be ...
2
votes
2answers
2k views

Finding a vector parametric equation given P and Q equations?

Find a vector parametric equation $r⃗(t)$ for the line through the points $P=(3,5,4)$ and $Q=(1,4,7)$ for each of the given conditions on the parameter $t$. If $r⃗ (0)=(3,5,4)$ and $r⃗ (7)=(1,4,7)$, ...
0
votes
1answer
31 views

The area of surface obtained by rotating a rectifiable curve

Let $\Gamma :X=\gamma(t),a\leq t\leq b$ be a rectifiable parameterized curve in the $(x,z)$-plane of $R^3$, which means $\gamma:[a,b]\to R^3$ is a $C^1$-mapping with $\gamma(t)=(x(t),0,z(t))^T$ and ...
1
vote
1answer
32 views

How to parameterize a straight line?

Why does the straight line from $(x_1,+y_1,+z_1)$ to $(x_2,+y_2,+z_2)$ become $r(\vec t)=(1-t)(x_1,+y_1,+z_1)+t(x_2,+y_2,+z_2)$ for $0 \leq t \leq 1$?
2
votes
0answers
30 views

Parametrization of surfaces for vector integration

I'm having some trouble calculating vector fields through surfaces. After attempting a few and being dissapointed with a wrong answer multiple times I figured I must be doing something wrong in the ...
0
votes
1answer
1k views

Parametrization for the curve $y = 7 - x^4$ that passes through the point $(0, 7, -3) $when t = 0 and is parallel to the xy-plane

Can you help me? So far I have turned $y = 7-x^4$ into $\langle1, 1, 0\rangle$ and used it to make the equation $L = (0, 7, -3) + t(1, 1, 0)$. I know this is wrong, but I just don't know what, and I ...
0
votes
0answers
22 views

Prove for criterion that two curve families are orthogonal on a surface in 3D

Let $E, F, G$ be the coefficients of the first fundamental form of a regular surface $R = R(u, v).$ Let $f(u, v) = c$ and $g(u, v) = d$ be two families of regular curves defined in the ...
0
votes
1answer
7 views

Derivative of the magnitude of a parametric function

I am trying to show that $d/dt$ $|r(t)|^2 = r(t)*r'(t)$, where $r(t)= <x(t), y(t), z(t)>$ and $r(t) \neq 0$. I first tried using the fact that $|r(t)|^2 = (x(t))^2+(y(t))^2+(z(t))^2$ and then ...
0
votes
2answers
31 views

Parametrization of an intersection cylinder ellipsoid

I'm trying to parametrize the surface given by the equations : $$\frac{x^2}{2}+\frac{y^2}{2}+z^2=1$$ and $x^2+y^2=y$. I found this function : $f:[0,1] \times [0,2\pi] \to \mathbb{R}^3$, $$(r,x) ...
1
vote
3answers
37 views

Parametrization of $x^2+y^2-ay=0$

I am to find the circulation of $$y^2 dx + x^2 dy$$ along the (counterclockwise) path $$\Gamma : x^2+y^2-ay = 0$$ both with and without using Green's theorem. Apparently, $\Gamma$ is supposed to ...
1
vote
1answer
47 views

Find parametric line between two 2D line segments that is an exact distance from a point

Given two 2D line segments, $\overline{ab}$ and $\overline{cd}$, and a point $p$, I would like to find a scalar value $t$ such that the line segment between $\overline{ab}(t)$ and $\overline{cd}(t)$ ...
0
votes
1answer
22 views

In parametric equations , how can the resulting equation after eliminating $T$ , consist of points not on the original set of equations?

I'm doing my Math level $2$ SAT subject test and there is a problem in the book that says "The resulting equation of eliminating $t$ may consist of points not on the graph of the original set of ...
2
votes
2answers
45 views

Find the plot of $y=1+\cos t$, $x=\sin^2t$.

I'm trying to find the plot for the following : $$y=1+\cos t, x=\sin^2t$$ I'm trying to get ride off variable $t$. This is what I done for some reason is incorrect : ...
1
vote
0answers
66 views

3D surface intersections

I tried to look at 3D Hypersurface intersections of 4D this way based on four Mathematica (circular) trigonometric parametrization combination selections. No hyperbolic functions are directly ...
0
votes
1answer
14 views

Finding the coordinate at time $t$ of a line determined by the points $(x1,y1), (x2,y2)$

I have the problem here, I create a program that clipping a line with the input (x1,y1,x2,y2). but the algorithm only explain until I get ...
7
votes
0answers
88 views

Very difficult surface integral

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}), \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
0
votes
0answers
9 views

Injective parametrization of a curve. ( piecemeal $C^1$)

$\gamma:$[0,1]$\to R^2$ is an injective parametrization of a curve $\Gamma$, which is piecemeal $C^1$ and the length of the curve is $L(\Gamma_k)<\infty$. 1.1.: Show that for every $n\in N$ there ...
0
votes
2answers
28 views

Converting parametric equations with trigonometric functions into Cartesian form

Ahoy, I am having trouble with a computer-based assignment and the question is as follows: $$x = 2\cos^5 t, \quad y = 2 \sin^5 t$$ Write these in Cartesian form, $F(x,y) = c$. I understand ...
1
vote
3answers
59 views

Consider the parametric curve given by: $x=3\cos(t)$, $y=t^{3/2}$.

The question asks to find the equation of the tangent to this curve at the point $t=\pi/4$. I've determined $$\frac{dy}{dx} =(\frac{dy}{dt})/(\frac{dx}{dt}) = -0.222$$ Have I got the right idea? ...
1
vote
3answers
40 views

Parametric Curves and Tangents

I am struggling with a question regard parametric curves and finding tangents to them but something is going wrong somewhere in the process and I cannot figure out why. The question asks: consider ...