Parametrization Definition and 90 Threads

Hello, I'm looking for an appropriate rotation representation for the following situation. I have two (always non-zero) vectors, v1, v2, that may or may not be parallel. The rotation relating the two vectors is obviously non-unique having one degree of freedom, parametrized by p. So my...

Homework Statement 2. Consider a stationary circular spool of thread of radius R. Assume the end of the thread is initially located at (0; R). While keeping the thread taut, the thread is unwound in a clockwise direction. (a) Parameterize the path described by the end of the thread as r(t) =...

Homework Statement Hi, Refer to: http://press.princeton.edu/books/maor/chapter_7.pdf ( Page 2 & 3) How do we derive the x-coordinate to be (R-r)cosθ + r cos[(R-r)/r]θ Homework Equations Let 'r' & 'R' be radius of small & big circles respectively; Let the angle by which a point on the...

Homework Statement Compute the line integral of the scalar function. f(x,y,z) = xe^{z^2}, piecewise linear path from (0,0,1) to (0,2,0) to (1,1,1) Homework Equations The Attempt at a Solution In this problem, all I need is a parametrization. First I drew the line from (0,0,1) to...

Homework Statement Find the outward flux of F = <x + z, y + z, xy> through the surface of the paraboloid z = x^2 + y^2, 0 ≤ z ≤ 4, including its top disk. Homework Equations double integral (-P(∂f/∂x) - Q(∂f/∂y) + R)dA where the vector F(x,y) = <P, Q, R> and where z = f(x,y) <-- f(x,y) is the...

Homework Statement Let \Gamma be the square whose sides have length 5, are parallel to the real and imaginary axis, and the center of the square is i. Compute the integral of the following function over \Gamma in the counter-clockwise direction using parametrization. Show all work...

I am struggling to understand the concept of natural parametrization of pdf of exponential family. Say that we have a function with the following pdf: f(x;\theta)=exp\left[\sum_{j=1}^k A_j(\theta)B_j(x)+C(x)+D(\theta)\right] where A and D are functions of \theta alone and B and C are functions...

Homework Statement I'm doing a line integral and can't seem to figure out the parametrization of this curve: x^2+y^2+z=2\pi Homework Equations Looking to get it to the form: \textbf{c}(r,t)=(x(r,t),y(r,t),z(r,t)) (I don't even know if this is right though).The Attempt at a Solution Trying to...

I understand that the key to parametrization is to realize that the goal of this method is to describe the location of all points on a geometric object, a curve, a surface, or a region. However, I am looking for a general rule for parameterization. How would one know which parametrization to use...

What is the minimum number of parameters needed to uniquely specify a point in a curved line?

Homework Statement Evaluate the surface integral \iint_S y \; dS S is the part of the sphere x^2 + y^2 + z^2 = 1 that lies above the cone z=\sqrt{x^2 + y^2}The Attempt at a Solution I know to use spherical coord so I did r = <\rho cos\theta sin\phi, \rho sin\theta sin\phi, ?> The book did...

Homework Statement Let b > a > 0. Consider the map F : [0, 1] X [0, 1] -> R3 defined by F(s, t) = ((b+a cos(2PIt)) cos(2PIs), (b+a cos(2PIt)) sin(2PIs), a sin(2PIt)). This is the parametrization of a Torus. Show F is a quotient map onto it's image. Homework Equations Proving that any subset...

How can we find a parametrization for SU(3) in terms of angles?

Homework Statement Parametrize a circle of radius r on a sphere of radius R>r by arclength. Homework Equations Circle Equation: (cos [theta], sin[theta], 0) The Attempt at a Solution I don't know if the professor is tricking us, but isn't the parametrization just Circle...

I'm having problems understanding surface parametrization from differential geometry. We are given two general forms for parametrization: \alpha(u,v) = (u,v,0) and x(u,v)=(u,v,f(u,v)) This is one I'm especially stuck on: y=Cosh(x) about the x-axis \alpha(u,v)=(u, Cosh[v],0)...

Homework Statement can someone help me how to parametrizise this z^2 = x^2 + y^2 Homework Equations I am doing Surface integral, i get the rest i just need to know how to parametrisize this in a simplier way The Attempt at a Solution x=x y=y z=(x^2 + y^2)^(1/2)

all elements of su(2) can be written as \exp(iH) with H being a traceless hermitian matrix thus H can be written as the sum of \sigma_x,\sigma_y,\sigma_z H=\theta (n_x \sigma_x + n_y \sigma_y+ n_z \sigma_z). Here (n_x,n_y,n_z) is a unit vector in R^3. we can take \theta in the...

Homework Statement Find an arc length parametrization of the curve r(t) = <e^t(cos t), -e^t(sin t)>, 0 =< t =< pi/2, which has the same orientation and has r(0) as a reference point. Homework Equations s = int[0,t] (||r'(t)||) The Attempt at a Solution So I found the derivative of r(t), and...

Homework Statement Consider the parametrization of torus given by: x=x(ø,ß)=(3+cos(ø))cos(ß) y=y(ø,ß)=(3+cos(ø))sin(ß) z=z(ø,ß)=sin(ø), for 0≤ø,ß≤2π What is the radius of the circle that runs through the center of the tube, and what is the radius of the tube, measured from the...

What does the arc length parametrization mean?

Second attempt here to get an answer, I am really lost on this. Im reading "A first course in String Theory" by Zwiebach and it says that when applying a general \tau gauge parametrization in the form of n_\mu X^\mu = \lambda \tau we can take the vector n_\mu so that for open strings...

Im reading "A first course in String Theory" by Zwiebach and it says that when applying a gauge parametrization in the form of n_\mu X^\mu = \lambda \tau we can take the vector n_\mu so that for open strings connected to branes (fixed end points), n^\mu \mathcal{P}^\tau _\mu is conserved...

I am reading Differential Topology by Guillemin and Pollack. Definition: X in RN is a k-dimensional manifold if it is locally diffeomorphic to Rk. Suppose U is an open subset of Rk and V is a neighborhood of a point x in X. A diffeomorphism f:U->V is called a parametrization of the...

Show that the circle that is in the intersection of the plane x+y+z=0 and the sphere x2+y2+z2=1 can be expressed as: x(\vartheta) = (cos(\vartheta)-(3)1/2sin(\vartheta)) / (61/2)y(\vartheta) = (cos(\vartheta)+(3)1/2sin(\vartheta)) / (61/2)z(\vartheta) = -(2cos(\vartheta)) / (61/2) I'm really...

We saw in the thread https://www.physicsforums.com/showthread.php?t=238464" that arc length that is usually defined by taking an arbitrary parametrisation of the curve as l(\gamma)=\int_{0}^{1} {|\dot\gamma(t)|} dt can be defined also by avoiding parametrization, introducing the notion of...

Find a vector parametrization for: y^2+2x^2-2x=10 My attempted solution is to say that x(t)=t and y(t)= +-sqrt(-2t^2+2t+10) but I don't think it's correct to have the +- and I might need to use polar coordinates instead. I'm just not sure of the function with the extra x in it.

could someone explain to me what exactly parametrization of a function in more than parameter means? so i know that for f(x)=x^2 there are two parameters, x x^2 but how does that lead to a circle being Sin(x) Cos(y)? what does this actually mean?? i.e. in the first example, i get...

Hi, I'm preparing a little exposition of curvature and torsion for my Calculus class and so I need to include some simple proofs for the things I'll use to define curvature. So I'm looking for a proof of the formula for the curvature of an arbitrarily parametrized curve that doens't use the...

Curves are functions from an interval of the real numbers to a differentiable manifold. Given a metric on the manifold, arc length is a property of the image of the curves, not of the curves itself. In other word, it is independent of the parametrization of the curve. In the case of the...

I have a complex analysis final exam on Wednesday, and I am studying the section on complex integration. I am having trouble seeing how to parametrize an equation. "\Gamma is the line segment from -4 to i" In the homework solutions our TA said, "Parametrize \Gamma by z = -4 +t(i+4), 0<t<1"...

Homework Statement Find a vector function that represents the curve of the intersection of two surfaces. Homework Equations z^2=x^2+y^2 with plane z=1+y The Attempt at a Solution So shouldn't it be r(t)=<cos(t), sin(t), 1+sin(t)> since x=cos(t), y=sin(t), and z= 1+sin(t)? The...

How can I parametrize the straigth line C from (2,-1,3) to (4,2,-1)? In the xy-plane I simply use the eq. y-y(0)=m(x-x(0)) to find the parametrization, but what should I do when we have 3 dimensions?

Homework Statement Our prof talked about arc length as a parameter today and I understand how to do problems associated with it, however I do not fully understand why we do it. Homework Equations In our text, the only relevant reading says: "A curve in the plane or in space can be...

Implicitly Defined Parametrization I'm having difficulties with the following question, and having checked through my working several times I just can't find a problem...problem is, so far in the book implicit and parametric differentiation have been covered independently of each other and this...

Do different parametrizations of the same curve in R^3 result in identical tangent vectors at a given point on the same curve? Example may be helpful.

My problem is finding a surface parametrization of the surface x^2-y^2=1, where x>0, -1<=y<=1 and 0<=z<=1. I know that x and y in x^2-y^2=1, can be represented as cosh(u) and sinh(u), but I'm not sure what to do for the z part. Any quick help?

The problem is find a parametrization of the surface x^3 + 3xy +z^2 = 2, z > 0, and use it to find the tangent plane at the point x=1, y=1/3, z=0. How is this possible when z > 0? I found a parametrization but when I plug the point in the x and the y places are undefined.

I know this is dumb question but for some reason I have not been able to get the right answer to the following problem: \int_{c} 2xyzdx+x^2 zdy+x^2 ydz where C is a curve connecting (1, 1, 1) to (1, 2, 4). My parametrization is (1, 1+t, 1+3t). My limits are the problem...I think. By...

I was wondering about the different methods by which one could "parametrize" a Moebius Strip. I asked someone about this a while ago, and they said that since the center of a Moebius Strip (z=0) is a circle, you can begin with the parametric equations for that and draw vectors out to other...

Hi all! I'm having some problems with parametrization. I read somewhere that you should locate circles, ellipses, hyperboloides, paraboloides etc and use these elements to express a parametric function. But someone must have figured out how to do it! The way I see it, there's nothing logical...