Parametrization Definition and 84 Threads

  1. R

    Parametrization of the path described by the end of a thread

    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) =...
  2. A

    Parametrization of Hypocycloid

    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...
  3. D

    How do I get the parametrization?

    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...
  4. W

    Flux of a Paraboloid without Parametrization

    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...
  5. W

    Natural parametrization of pdfs

    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...
  6. D

    Parametrization of a Corkscrew Curve on a Paraboloid

    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...
  7. G

    I understand that the key to parametrization is to realize that the

    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...
  8. TrickyDicky

    Curve Parametrization: Minimum Parameters for Unique Point Specification

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

    Surface integral parametrization

    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...
  10. Y

    Finding a Parametrization for SU(3) in Terms of Angles

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

    Parametrization of a circle on a sphere

    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...
  12. C

    What is the surface parametrization for rotating y=Cosh(x) about the x-axis?

    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)...
  13. S

    How Can I Simplify Parametrization for the Equation z² = x² + y²?

    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)
  14. W

    Parametrization of su(2) group

    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...
  15. K

    How Do You Find an Arc Length Parametrization for a Given Curve?

    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...
  16. J

    Understanding Torus Parameterization

    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...
  17. K

    Concept: Arc Length Parametrization

    What does the arc length parametrization mean?
  18. L

    Parametrization vs. coordinate system

    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...
  19. F

    Parametrization - circle defined by plane intersection sphere

    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...
  20. M

    Defining the integral of 1-forms without parametrization

    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...
  21. K

    Find a vector parametrization for: y^2+2x^2-2x=10

    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.
  22. S

    What is parametrization of a function with more than one parameter?

    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...
  23. M

    Definition of arc length on manifolds without parametrization

    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...
  24. C

    Parametrization in Complex Integration

    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"...
  25. W

    What Is the Correct Parametrization for the Intersection of Two Surfaces?

    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...
  26. K

    Parametrization of straigth line in space

    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?
  27. V

    Parametrization as Arc Length: Why Do We Need It?

    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...
  28. G

    Implicitly Deifned Parametrization

    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...
  29. A

    Parametrization in R^3: Comparing Tangent Vectors of Different Curves

    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.
  30. T

    What are the parameters needed for surface parametrization of x^2-y^2=1?

    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?
  31. T

    How to Find a Tangent Plane on a Surface with Positive Z Values?

    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.
  32. T

    Line integral and parametrization

    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...
  33. A

    Parametrization of a Moebius Strip

    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...
  34. L

    How Do You Solve Complex Parametrization Problems in Mathematics?

    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...
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