Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
1answer
19 views

Breaking a path, in 2 integrals

Let $\overrightarrow V, \Gamma$, be a vectorial field and a path such that: $$\overrightarrow V=-(x^2+y^2)\overrightarrow i-(x^2-y^2)\overrightarrow j$$ and $$\Gamma=\{(x,y)\in\mathbb{...
4
votes
1answer
28 views

Compute $\int_{\Gamma}\omega$ where $\omega=(y-2z)dx+(x-z)dy+(2x-y)dz$

Compute $\int_{\Gamma}\omega$ where $\omega=(y-2z)dx+(x-z)dy+(2x-y)dz$ and $\Gamma$ is the intersection between: $x^2+y^2+z^2=r^2$ and $x-y+z=0$ My attempt: $\Gamma$ is some kind of ellipse in the ...
1
vote
1answer
18 views

Calculate $\int_{\Gamma} \omega$ when $\omega =z(z-y)dx+xzdy-xydz$, $\Gamma=\Gamma_1 \cup \Gamma_2 \cup \Gamma_3$

Calculate $\int_{\Gamma} \omega$ when $\omega =z(z-y)dx+xzdy-xydz$ $\Gamma=\Gamma_1 \cup \Gamma_2 \cup \Gamma_3$ $$x^2+y^2=(z-1)^2$$ $x\geq0, y\geq0,z\geq0$ $\Gamma_{1,2,3}$ are ...
0
votes
1answer
20 views

Calculate $\int_{\Gamma}\omega$ when $\omega = x^2yzdx+xy^2zdy+xyz^2dz$ and $\Gamma$ is the intersection of surfaces $x=1$ and $y^2+z^2=1$

Calculate $\int_{\Gamma}\omega$, when: $\omega = x^2yzdx+xy^2zdy+xyz^2dz$ and $\Gamma$ is the intersection of the surfaces: $x=1$, $y^2+z^2=1$. Can you help me with that? At first, I thought of ...
0
votes
1answer
28 views

Rectangle paramtrization

Let $\Gamma=ABCD$, where: $$A(0,0),B(a,0),C(a,\alpha),D(0,\alpha)$$ $$\omega=Pdx+Qdy=e^{-x^2+y^2}\cos(2xy)dx+e^{-x^2+y^2}\sin(2xy)$$ with $\alpha\in\mathbb{R}$, and $a$ they don't say ...
-1
votes
3answers
96 views

Parametrization for the figure '8' curve? [closed]

Is there a parametrization for the figure '8' curve, which is self-intersected?
0
votes
1answer
22 views

Thought process behind vector valued functions and parameterisation

Can someone please confirm whether my intuitive notions behind what vector-valued functions and parameterisation is correct. Below are some questions. Are vector-valued functions like functions of a ...
1
vote
0answers
67 views

How to determine an integral of a differential form?

Let $$ \eta = x^2 \, dy \wedge dz + yx\,dz \wedge dx + z^3 \, dx \wedge dy $$ Can you show me how to calculate : $$ \int_{\Phi} \eta, $$ where $ \Phi $ is supposed to be the parametrization of ...
0
votes
1answer
31 views

Parameterization of a curve

A general question: Does every (continuous) curve in space necessary have a parametrization? (even if we cannot find it or express it?)
0
votes
1answer
68 views

Non-singular elliptic curve parametrization

It is known that some singular elliptic curves can be expressed with parametric equations. For example : $y^2=x^3$ can be parametrized with $x=t^2$ and $y=t^3$ $y^2=x^3+x^2$ can be parametrized with $...
10
votes
1answer
457 views

Parametric equation for a space curve

With reference to the following image: the blue curve has trivially a parametrization: $$(x, y, z) = (\cos\theta, \, \sin\theta, \, 0) \; \; \; \text{with} \; \theta \in [0,\,2\pi)$$ I would like ...
1
vote
2answers
198 views

Calculation of Christoffel symbol for unit sphere

We use the following parameterisation for the unit sphere: $\sigma(\theta,\phi)=(\cos\theta\cos\phi,\cos\theta\sin\phi,\sin\theta)$. I have calculated the Christoffel symbols to be $\Gamma^1_{11}=\...
0
votes
3answers
186 views

Parametric form of ellipse $x^2 + 16y^2 = 4$

The given ellipse is $-$ $x^2 +16y^2 = 4$ I know the standard method of changing the equation of ellipse in parametric form. Instead of complicated calculation, can we simply put $x= 2 \cos \theta , ...
2
votes
0answers
93 views

Cubic Bézier curve arc length parametrization reversal: find t given a length

I am following this paper Approximate Arc Length Parametrization, M. Walter & A. Fournier, 1996 and have succesfully implemented the direct solution, as in finding the length $s(t)$ given $t$. ...
1
vote
1answer
19 views

Simple Curve Integral using Parametrization

I know the answer but cannot for the love of everything figure out how the book got the answer and the solution is only the answer nothing step-by step at all. $$ F(D) = F*dr $$calculate curve ...
2
votes
2answers
46 views

Finding limits of integration using spherical coordinates

I would like to integrate some function $f:\mathbb{R}^3\to\mathbb{R}$ over $C_1\cap C_2$ where $$C_1=\{(x,y,z)\in\mathbb{R}^3:x^2+4y^2+9z^2\leq1\}$$ $$C_2=\{(x,y,z)\in\mathbb{R}^3:x^2+4y^2+9z^2\leq 6z\...
0
votes
1answer
19 views

Parametrisation of a curve notation

If I have some curve $C$ and the parameterisation is the bijective map $P:[a,b] \rightarrow C$ What do $a$ and $b$ represent? I though that they would be the coordinates of the start and end of the ...
1
vote
1answer
77 views

When to parameterize to find equation of tangent plane and normal line?

I'm working through some problems asking to find the equation of a tangent plane and the normal line to the surface. I notice that some example questions parameterize the curve before solving and ...
0
votes
2answers
55 views

How to find the vector equation of the line segment.

Let $C$ be the line segment from $(0,0)$ to $(2,2)$, and let $f(x,y)=x^2+y$. Write down a vector equation $r(t)$ of the line segment, that is, find a parametrization of $C$. The answer given is $r(t)...
2
votes
2answers
45 views

How to prove that $ | \Phi_u \times \Phi_v | = \sqrt{ \det g} $?

Let be $ \Phi $ a parametrization of a surface $ \in \mathbb{R}^3 $ and $g$ the metric tensor. As the title says...how to show that How to prove that $ | \Phi_u \times \Phi_v | = \sqrt{\det g} $ ? I ...
3
votes
3answers
558 views

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles.

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles. $\frac{dr}{d\theta}=-a\sin\theta$ for the first curve and $\frac{dr}{d\theta}=a\sin\theta$ for the second ...
0
votes
0answers
48 views

Doubt parameterization

Good afternoon guys, I'm studying some math and I came across a problem. It is necessary, from a graph obtained through experiments, to carry out the parameterization of this one. Honestly, I did not ...
3
votes
1answer
57 views

Cauchy Integral Theorem with $f(z)=e^{z^2}$

I have $z(t)=t(1-t)e^t + \cos(2 \pi \cdot t^3)i$ with $0 \le t \le 1$ and need to evaluate the line integral of $e^{z^2}$. I know that the endpoints are $z(0)=z(1)=0+i$, so the line is a closed ...
1
vote
2answers
46 views

Find $\frac{\partial y}{\partial z}$ of the surface $g(s,t)=(s^2+2t,s+t,e^{st})$ near $g(1, 1) = (3, 2, e)$.

Consider the surface given by $g(s, t) = (s^2 + 2t, s + t, e^{st})$. Think of $y$ as a function of $x$ and $z$. Find $\dfrac{\partial y}{\partial z}(3,e)$ near $g(1, 1) = (3, 2, e)$. really ...
2
votes
4answers
75 views

Parametrization of a circle (extra credit on calculus final)

This was an extra credit question on my Calculus final. Parametrize the circle lying in the plane with normal vector $(1, 1, -2)$ with center at $$\Big(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{...
0
votes
1answer
40 views

Solve the integral $ \int{y^2dl}$ where L:$x^2+y^2+z^2=a^2;$ $x+y+z=0$

Solve the integral $ \int{y^2dl}$ where L:$x^2+y^2+z^2=a^2;$ $x+y+z=0$. I tried to apply a 2 parametrizations: 1) $x= \sqrt{\frac{2t^2}{-3t-\frac{a^2}{2}}}$ $y= \sqrt{-\frac{3}{2}t-\frac{a^2}{4}}$ $...
0
votes
0answers
12 views

Doubt about the procedure of parametrization

I can't understand how parametrics equations are found. For example, I realize that the parametrization of the curve given by the intersection of the plane $\ 2x+2y+z=2$ and $z=x^2+y^2$ is: $x=-1+\...
0
votes
1answer
27 views

Does parametrizing a function of three independent variables reduce the function to one independent variable?

For example, $f(x,y,z)= x^2 + 2y^2 + 3z^2$ Now, if we parametrize the independent variables $(x,y,z)$ in terms of $u$ and it happens to be $x=u, y=u-1, z=u+1$ (just an assumption), then the ...
3
votes
1answer
65 views

Parameterization of uncommon curves

Preface: Out of all the Stack Exchange networks with which I am familiar, this seems the most appropriate for the following question. Please let me know if there's a more suitable network for similar ...
1
vote
1answer
28 views

Proof of equivalence well defined function

This is the definition: Definition 1: A function $f:D\subseteq R\to R^n$ is said to be continuously differentiable of a $C^1$ function, if f is differentiable and the first derivative of f is ...
0
votes
0answers
25 views

Intersection curve: Parameterization, length.

I'm stuck on this problem and i'm not really sure how to proceed to resolve it: Consider the curve $C$ of intersection between the plane: $z=2-x$ and the cylinder: $x^2+y^2=1$ $a)$ ...
0
votes
1answer
35 views

Parametrization of distance to non-unit circle/sphere with non-centered origin

I attempt to parametrize the distance $z(\theta;r,z_0)$ from the origin $(x,y)=(0,0)$ of my coordinate system to arbitrary points $(x,y)$ on a circle, as a function of the variable $\theta$ (angle), ...
0
votes
1answer
135 views

How to evaluate a parameterized surface integral?

Suppose you have to evaluate the surface integral $$\int\int_S (x^2+y^2+4)\space dS$$ where $S$ is the surface parameterized by $\textbf{r} = <2uv, u^2-v^2, u^2+v^2>$ with $u^2+v^2 \le 16.$ I ...
1
vote
1answer
236 views

show that the parameterization of a torus is ____________ [closed]

I need help on a homework problem. a) A torus of revolution (doughnut) is obtained by rotating a circle C in the xz-plane about the z-axis in space. See accompanying figure. If C has a radius r>0 ...
0
votes
2answers
174 views

How Do You Find an Equation of the Tangent Plane for a Torus?

I've parameterized the torus given by $z^2 + (r - 2)^2$ = 1 using: x = (2 + $\cos\theta$)$\cos\alpha$, y = (2 + $\cos\theta$)$\sin\alpha$, z = $\sin\theta$. I'm really stuck on how to find the ...
0
votes
1answer
52 views

Parametrization of a circle at (-1,-8) with Radius 9

Question: The circle centered at $(−1,−8)$ with radius $9$ can be parametrized in many ways, this still happens even if we impose the extra constraint that the circle must be traversed in the counter-...
3
votes
2answers
234 views

How to calculate the surface area of parametric surface?

Suppose you have the surface $z=3xy$ and you want to find the area that lies within the cylinder $x^2+y^2\leq 1$. My homework is forcing me to use the parameterization $$\textbf{r}_1(s,t)= <s\...
0
votes
1answer
63 views

Parameterize the sphere of radius $5$ in $\mathbb{R}^3$ so that in all its points, the normal vector points outwards.

Parameterize the sphere of radius $5$ in $\mathbb{R}^3$ so that in all its points, the normal vector points outwards. I've thought about doing the following but I do not know if I'm doing things ...
0
votes
0answers
26 views

Parametrization and Projections

I have a question over this problem that I've encountered. I don't know how to solve it. Let $\ell$ be the line parametrized as $(t, 2t+1, 3t+2)$ and let $P$ be the plane with equation $x+y+z = 1.$ (...
1
vote
0answers
34 views

Question on judging a regular surface in differential geometry

This is the proposition and proof saying that if we know that S is a regular surface and x is a candidate of parametrization which satisfies 1)differentiable 2)surjective differential map 3)continuous ...
0
votes
1answer
38 views

intersection of 2 surfaces for line integral

I am asked to find the curve of intersection of the following: $x+y=2$ and $x^2+y^2+z^2=2(x+y)$. I suppose (1) is a cylinder and unable to find what (2) is, however I think their intersection will ...
0
votes
1answer
37 views

How do I complete this arc length problem?

Let $c(t) = (t^3, t^2, 2t)$ and $f(x,y,z)=(x^2-y^2, 2xy, z^2).$ (a) Find $(f\circ c)(t)$ (b) Find a parametrization for the tangent line to the curve $f\circ c$ at $t=1$. I know how to find the ...
0
votes
1answer
30 views

How to evaluate $ \int_c(2x+y)\,ds$ where c is defined by $x^2+y^2=25$ from the point $(3,4)$ to $(4,3)$ why it gives me $-15$?

If I parametrize the curve with $\begin{cases}x=5\cos t \\ y=5\sin t\end{cases}$ it gives me $-15$. Why if I parametrize the curve with $\begin{cases}x=t \\ y=\sqrt {25-t^2}\end{cases}$ the correct ...
0
votes
0answers
13 views

Characteristic of a matrix with gramian determinant

What does the gramian matrix got anything to do with the characteristic of a matrix? I have to prove that $\psi :\mathbb{R} \to \mathbb{R^2},t\to (acos(t),bsin(t))$ is an immersion, and one way that ...
0
votes
1answer
25 views

$x$ and $y$ parameterised in terms of the variable $t$.

If I have a function $$y=\frac 1 3 x+ \frac{11}{3}$$ The bounds are from $(-2,3)$ to $(1,4)$ 1) How do I put $x$ and $y$ in terms of a third variable $t$ 2) what will be the bounds of ...
0
votes
1answer
118 views

Parameterization of the portion of a cylinder between two planes

I have to parametrize the lateral section of the cylinder $\frac{x^2}{4} + \frac{y^2}{9}=1$ between the planes $z = 1-x$ and $z = 0$. I have $r(u,v) = (2\cos{uv},3\sin{uv},\frac{3v}{2} + \frac{3}{2})$...
0
votes
1answer
378 views

Verifying Stokes' Theorem for an upper hemisphere

There is a hemisphere, radius $1$, centred at $(0,0,0)$, where the vector field is $$\vec F = \Big(x^3+\frac{z^4}{4}\Big) \hat i + 4x \hat j + (xz^3+z^2) \hat k$$ Verify Stokes' theorem for this ...
0
votes
0answers
46 views

What is parameterization in general?

Consider the set of all infinite arithmetic sequences of real numbers: $$\forall f,d\in\mathbb R\ (f,f+d,f+2d,\cdots,f+id,\cdots)$$ Most people would say that the set of all of these sequences is ...
0
votes
0answers
28 views

Why do two vector fields determine a paramaterisation of a surface

I am trying to understand the following proof about how two vector fields determine a paramaterisation of a surface. What I do not understand is why the inverse of the constructed $\phi$ is the ...
5
votes
1answer
215 views

Finding the volume of a pseudosphere that has been parametrised in $\theta$ and $t$

I've got a problem calculating the volume of the top half of a pseudosphere. The pseudosphere is parametrised by $$\Phi(t,\theta) = \Big ( \frac{\cos(\theta)}{\cosh(t)}, \frac{\sin(\theta)}{\...