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Questions tagged [parametrization]

For questions on parametrization, the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation.

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Prove that there are infinitely many triples of integers $(x, y, z)$ s.t. $x^2+y^2=(m^2+n^2)^{z}$ with positive integers $m, n$ and with $z$ even

To prove that there are infinitely many triples of integers $(x, y, z)$ such that $x^2+y^2=(m^2+n^2)^{z}$ with positive integers $m, n$ and with $z$ even, I tried to apply the parametric method, with ...
Thomas's user avatar
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Can an arbitrary path in $\mathbb{R}^3$ be parametrized as $f(t)\hat{u}$ for a basis vector of some arbitrary coordinate system?

This question comes from a recent question in Physics SE I answered, link. This has sparked a question I don't have the background to answer. Consider some arbitrary path in $\mathbb{R}^3$. Some of ...
agaminon's user avatar
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Radius of curvature of parametric equation for a helix

The equations i have for the $x, y, z$ coordinates for a helix are as follows $$x = R\cos(\omega t)$$ $$y = R\sin(\omega t)$$ $$z = vt$$ Where v is the velocity of the particle moving on the path of ...
koiboi's user avatar
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4 votes
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Directly parameterize to calculate two integrals

Directly parameterize to calculate the integrals: (a) $\int_{K} \sin(\bar{z}) dz$, where $K$ is the line segment from the point $i$ to the point $3i$. (b) $\int_{K} \sqrt{z} dz$, where $K = \{ z \in \...
ukm2030's user avatar
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Curves on a trousers space. [closed]

How does one go about defining curves on a trousers space? I want to define two curves evolving cyclically around a cylinder and then at some time let one of the curves evolve on the other cylinder. ...
manyworlds's user avatar
5 votes
1 answer
71 views

Calculate the integral using direct parameterization

Calculate the integral using direct parameterization: (a) $\int_K \overline{z} \, dz$, where $K$ is the line segment from the point $2i$ to the point $2 - 4i$. (b) $\int_K \sqrt{z} \, dz$, where $K = \...
general123's user avatar
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How to parametrize trefoil knot with arbitrarily shaped cross section

I've been trying to add one or two dimensions to the following threadlike trefoil knot $$\mathbf r(t): \left\{\begin{aligned} &x=\sin(t)+2\sin(2t)\\ &y=\cos(t)-2\cos(2t)\\ &z=-\sin(3t) \...
Conreu's user avatar
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Parametrization of Diophantine Equation

I know you can parametrize all the rational solutions to $x^2 + y^2 = 1$ as $(x, y) = (\frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2} )$ where t is rational. The way we do this is by showing that $(x, y) = (0, ...
Ravikanth Athipatla's user avatar
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Can't confirm the descent direction at a point

Consider the problem \begin{equation} \underset{\mathbb{R}^2}{\text{min}} f (x) = \frac12x_1^4 + 2x_2^4 + x_1^2 - x_1 x_2 + x_2^2 . \end{equation} Suppose that the function $f$ is minimized starting ...
Superunknown's user avatar
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2 votes
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How can I modify a surface to satisfy two distance conditions?

I have two variables, $\phi$ and $\theta$, and I'm trying to create a smooth surface such that the following rules are met for the distance between on the surface, $D$ \begin{align*} 1)& \: D[(\...
David G.'s user avatar
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A functional that maps a differentiable function $g:[0,1]\to[0,1]$ to a closed loop in two dimensions

How do I describe a functional $f$ that maps a differentiable function $g:[0,1]\to[0,1]$ to a smooth $h: [0,1]\to[0,1]^2$ such that $h(0)=h(1)=g(0)$ with the remaining constraints described in English ...
fool's user avatar
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How do you parameterize simultaneous solutions to equations with expressions like "$ x +2 \left\lfloor\frac{x}{3}\right\rfloor + 1 - [3 \mid x]$"?

Let all functions be integer functions herein. I.e. $\Bbb{Z}\to\Bbb{Z}$ or $\Bbb{N}\to\Bbb{Z}$ where appropriate. I found this jewel of floor functions. So that made me wonder whether, we can solve ...
SeekingAMathGeekGirlfriend's user avatar
2 votes
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Parameterization by arc length: the concept

Can someone let me know if my understanding of parameterization by arc length is correct? If we have a regular parameterized differentiable curve $\alpha: I \rightarrow \mathbb{R}^3$, it is ...
DC2974's user avatar
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Finding or constructing Archimedes spirals with/from parametric lengths

I'm using Desmos, and have already combed through this site not finding anything close to what I need, nor have the equations and modifications I have tried been of help. Desmos Trial by Combat I need ...
CryptoMynd's user avatar
1 vote
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Why can't I find the intersection point between this parametric plane and line?

We have the plane defined in normal form by: \begin{equation} x + y - 2z - 2 = 0 \end{equation} To convert this to parametric form using the orthogonal vectors $\mathbf{u} = (1, 1, 1)$, $\mathbf{v} =...
willaayy's user avatar
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Parametric equations of the normal to a curve $\gamma(t)$ at a point $\gamma(t_0)$

Suppose we have some curve $\gamma$ which can be parametrized with time parameter $t\in(a,b)$. Let's take $\gamma(t)=(x(t),y(t))$ for simplicity. We are restricted to $C^\infty(\mathbb{R})$ class ...
user1299519's user avatar
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Let C be the path in $\Bbb R^2$ from from $(0,0)$ to $(2,1)$ defined by the equation $x^4-6xy^3-4y^2=0$. Find $\int_C (10x^4-2xy^3)dx -3x^2y^2dy.$ [duplicate]

Let C be the path in $\Bbb R^2$ from from $(0,0)$ to $(2,1)$ defined by the equation $x^4-6xy^3-4y^2=0$. Find $\int_C (10x^4-2xy^3 ) dx -3x^2y^2dy$. I have no idea how to do this line integral. In our ...
Thomas Finley's user avatar
1 vote
1 answer
74 views

Parameterization of the intersection between a cone and a paraboloid

I'm solving some exercises on arc length of intersection of surfaces and I cannot conclude the following: Calculate the arc length of the curve give by the intersection of the surfaces $z^2 = 2x^2 + ...
Greg's user avatar
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Proof that Torus is a 2-dim submanifold of $\mathbb{R}^3$ using parametrization

I want to show that the torus $T:=\{(x, y, z) \in\mathbb{R}^3 \mid (\sqrt{x^2+y^2}-R)^2+z^2=r^2\}\subset \mathbb{R}^3, 0<r<R<\infty$ is a 2-dimensional submanifold showing that $\forall p\in ...
Lu1998's user avatar
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Is there a straightforward way to triangulate this tetrahedrally-symmetric convex surface according to these criteria?

I have a tetrahedrally-symmetric surface of constant width defined in spherical coordinates by the support function $$ h(θ, φ) = \frac{S}{16} ⋅ \left(\sin(θ)^3 ⋅ \cos(3 ⋅ φ) + \frac{5 ⋅ \cos(θ)^3 - 3 ⋅...
Lawton's user avatar
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1 vote
2 answers
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How can i parametrize rationally an hyperbola with the form $-aX^2 + bY^2 =1$?

I know that a rational parametrization of an hyperbola with the form $$-(\frac{x}{c})^2 + (\frac{y}{d})^2 =1$$ is: $$(c·\frac{t^2-1}{2t}, d·\frac{1+t^2}{2t}) $$ The problem is that if I transform my ...
Marc's user avatar
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2 votes
1 answer
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Parameters behind non-symmetric Lissajous loop?

I'm trying to guess what kind of two signals can create this kind of Lissajous curve: However I can't figure out what are the parameters that break the symmetry of the curve. The relative phase ...
ValientProcess's user avatar
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Computing a surface integral

I have to verify that the line integral and the surface integral of Stokes' Theorem are equal for the vector field $\boldsymbol{\mathrm{F}}(x,y,z)=(x,y,z)$ and the portion of the surface $S$ defined ...
limber's user avatar
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Confusion about orientation of line integral

Consider a continuous curve $\gamma$ in $\mathbb{R}^3$ parameterized by $\mathbf{r}(t)$ as $t:a \rightarrow b$. Now, it is my understanding that line integrals $$ \int_\gamma \phi \ ds $$ for a scalar ...
Incubu121's user avatar
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2 answers
42 views

Curve is traveled clockwise or anti-clockwise

Given the curve $$ \vec{\mathbf{r}}(t) = \bigl(t - \tfrac{1}{3}t^3 \bigr) \, \vec{\mathbf{i}} + (4 - t^2) \, \vec{\mathbf{j}}, $$ how can I tell whether it's traveled clockwise or counterclockwise? ...
Emmannuelle_Legolas's user avatar
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0 answers
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How can i find the way to escape robot?

A scout robot is trapped at the center of a square with side length 1km. The scout robot can move at a speed of 3km/h. A guard robot is located at a vertex of the square, and patrols the boundary of ...
sorksnrnro's user avatar
2 votes
1 answer
43 views

How can I find a positively oriented parametrization?

I need to find a parametrization of the surface defined by $x^{2}+y^{2}+z^{2}=9$, $x+y+z\geq 1$. The orientation must be compatible with the orientation of its boundary, that is, if the loop lies in ...
limber's user avatar
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1 answer
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If $\psi(u,v) = (u, u^2 + v^2, v)$ and $\alpha(t)=(ct - b, 5t^2 - ct + c, t + a)$ is a curve with $\alpha$ in $\text{img}(\psi)$, then $a + b + c$ is

If $\psi(u,v) = (u, u^2 + v^2, v)$ with $(u,v) \in \mathbb{R}^2$ and $\alpha(t) = (ct - b, 5t^2 - ct + c, t + a)$ is a curve with $\alpha \subset \text{img}(\psi)$, then $a + b + c$ is .... Is anyone ...
Angelo's user avatar
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3 votes
2 answers
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Paramterizing the surface on the intersection of $x+z=a$ and interior of $x^2+y^2+z^2=a^2$

So I am trying to verify Stokes' theorem for $\vec{F}=y\hat{i}+z\hat{j}+x\hat{k}$ where the curve $C$ is on the intersection of $x+z=a$ and $x^2+y^2+z^2=a^2$. Solving these equations yields the curve $...
MathArt's user avatar
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1 answer
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Derivation of the formula for parametrized surface area element

The surface integral is given by $$\int_{S} f \,dS = \iint_{D} f(\mathbf{\sigma}(u, v)) \left\|{\partial \mathbf{\sigma} \over \partial u}\times {\partial \mathbf{\sigma} \over \partial v}\right\| \,...
Sasikuttan's user avatar
1 vote
1 answer
70 views

Calculating Electric Flux Through a Closed Surface

I'm trying to solve a problem involving the calculation of electric flux through a closed surface, but it's my first time attempting such a problem and I could use some guidance. Any help would be ...
Athanasios Paraskevopoulos's user avatar
1 vote
0 answers
20 views

Low-rank decomposition of matrices with all-one diagonal and small off-diagonal

Recently, I encountered a paper utilizing the concept of Kissing numbers from the perspective of matrices. Specifically, the authors presented the Kissing numbers as follows. Def: For a given $m \in \...
Vezen BU's user avatar
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3 votes
3 answers
101 views

Finding the tangent line to curve to the ellipse $(x-3)^2+\frac{(y-4)^2}{4}=1$ through the origin

I am told to find the two tangent lines to the ellipse that pass through the origin, but have been stuck for far too long with my approach, hence am thinking that my approach may be flawed. Here is ...
Jason Xu's user avatar
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Parameterizing intersection curve of paraboloids

I'm at a loss for ideas on what to do. I'm given two paraboloids and their equations: $z_1 = x^2 + 1$ and $z_2 = 5 - y^2$. I know their intersection is when $z_1 = z_2$. This gives me $x^2 + y^2 = 4$, ...
Emmannuelle_Legolas's user avatar
1 vote
0 answers
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Parameterization of area between $y=x+\cos x$ and $y=x+\sin x$

In the $xy$-plane a function is given: $f(x,y)=x+y$. Let $A$ be the area that is in the $xy$-plane and is encapsulated by $x=0$, $x=\frac{\pi}{4}$, $y=x+\cos(x)$ and $y=x+\sin(x)$. a) Make a ...
Zert44's user avatar
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2 answers
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Why do we say this is a reparametrization? ("Analysis on Manifolds" by James R. Munkres.)

I am reading "Analysis on Manifolds" by James R. Munkres. Definition. Let $k\leq n$. Let $A$ be open in $\mathbb{R}^k$, and let $\alpha:A\to\mathbb{R}^n$ be a map of class $C^r (r\geq 1)$. ...
佐武五郎's user avatar
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1 vote
0 answers
50 views

Sets on $\mathbb{R}^2$ that are continuously parametrizable

I'm studying multivariable calculus, and I was wondering: Is there a way to characterize all sets in $\mathbb{R}^2$ that are continuously parametrizable? What I mean with a continuously parametrizable ...
Iván G M's user avatar
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2 votes
1 answer
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Parametrization of the rational $x$ where both $\sqrt{1+x^2}$ and $\sqrt{9+x^2}$ are rational

I could parametrize the rational $x$ where $\sqrt{1+x^2}$ is rational as: $y^2-x^2=1,\ y=\frac{a^2+b^2}{a^2-b^2},\ x=\frac{2ab}{a^2-b^2}$. $\sqrt{9+x^2}$ is similar. However, I cannot find $x$ where &...
aleph0's user avatar
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How do I find constants so that the curvature of an oscillating funtion is between them?

I am currently struggling with how to answer this question: "Consider the planar curve $y = a\sin(bx)$ where $a, b$ are non-zero real numbers. Find constants $A$ and $B$ so that $A \le k \le B$ ...
Jarvis's user avatar
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Is there a curve such that it's parametrization r(t) has κ → ∞ as t → ∞?

I thought this question was easy but I can't think of a proper example. I've tried (e^t, e^-t, 1), (sin(t), cos(t), 1), and many many different variations of these. The closest I got to was (t, e^t, t+...
Jarvis's user avatar
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1 vote
1 answer
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How to get the max of $(aa^*-bb^*)^2 - [(ab^*)^2 - (a^*b)^2]^2$?

If I have two complex numbers $ a,b \in \mathbb{C} $,and $ |a|^2 + |b|^2 = 1 $,I need to get the max of \begin{aligned} (aa^*-bb^*)^2 - [(ab^*)^2 - (a^*b)^2]^2 \end{aligned} In the answer book, the ...
liZ's user avatar
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0 answers
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Stokes Theorem for the intersection of $z=x^2+y^2$ and $z=x+2$ and parameterizing the portion of the plane inside of $z=x^2+y^2$

I have been trying to solve the surface integral $$\int \int_S curl(\vec{F}) \cdot \vec{dS}$$ using stokes theorem for the vector field $$F(x,y,z)=\left(xz,yz,xy\right)$$ and where $S$ is the ...
aygx's user avatar
  • 200
1 vote
3 answers
64 views

Implicit equation of revolution $(x^2+y^2)^2=x$

I have a curve given as $(x^2+y^2)^2=x$ in the plane $z=0$. We then rotate this curve around the x axis and then must find a parametrization for the surface as well as find the tangent plane in the ...
pavcheck's user avatar
0 votes
1 answer
37 views

Unambiguous definition of smoothness in the parameterization of a curve

Given the parameterization of a curve in $\mathbb{R}^n$, $$\boldsymbol \gamma (t) = (x_1(t), x_2(t), \ldots, x_n(t))$$ I can not find a univocal definition of smoothness. This answer requires existing ...
BowPark's user avatar
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22 views

If $x_2(t)/x_1(t)=y_2(t)/y_1(t)$, then is $\vec Y(t)=(y_1(t),y_2(t))$ a reparametrization of $\vec X(t)=(x_1(t),x_2(t))$?

The definition of a parametrized curve $\vec X:I\rightarrow \mathbb{R}^2$ was given by a continuous function $g:J\rightarrow I$, then $\vec \xi =\vec X \circ g:J\rightarrow \mathbb{R}^2$ a ...
ShoutOutAndCalculate's user avatar
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0 answers
18 views

how to parameterize the boundary of a a parabolic cylinder

Let $\Sigma$ be the surface of equation $x^2+y+z=1$ with $y,z \geq 0$. Let $F(x,y,z)=(6y,1-z^2,3x)$ be a vector field. I have to compute the circuitry of $F$ in $\gamma=\partial \Sigma$ without the ...
Mario's user avatar
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1 answer
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Equation of an ellipsoidal capsule

I'm trying to calculate the equation of an ellipsoidal capsule. It is a problem similar to what was presented here but with a twist: this time the capsule is made of two ellipsoid of axes length a, b ...
Vincent's user avatar
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0 answers
28 views

Stitching together parametric tubes?

Say I have two parametric tubes of the form $C_i(u,v) = (x(u, v), y(u, v), z(u,v))$. The most common case would be sweeping surfaces along a path, but the sweep can have a scaling factor. These ...
Makogan's user avatar
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Exercise parameterization of a surface

Let $\varphi: (0, +\infty) \times (0, 2\pi) \to \mathbb{R}^3$ with $\varphi(r, \theta) = (r \cos(\theta), r \sin(\theta), \theta)$, and let $S$ be the image of $\varphi$. (a) Show that $S$ is a ...
Andreadel1988's user avatar
1 vote
1 answer
79 views

Diffeomorphism between open surfaces

Is it true that for every pair of surfaces $S_1, S_2$ in $\mathbb{R}^3$, there exist two non-empty open sets $W_1 \subseteq S_1$ and $W_2 \subseteq S_2$ that are diffeomorphic? Here is my attempt: Let ...
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