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Vector calculus

  1. May 23, 2010 #1
    Im afraid my use of LaTeX code, sucks. My apologies to anyone friendly enough to help!

    The problem:

    A string takes a path shown by the equation below:

    r(t)=(t,3t^2,6t^3) Where the RHS is a verticle vector (didnt know how to code this!)

    and 0 <= t <= 1

    The mass per unit length of the wire at a point (x, y, z) is given by
    p(x,y,z) = xyz^2

    Im having trouble answering these last two questions..

    c) Find the entire length of the string?
    d) Find the mass of the string?

    Thanks for your help!
  2. jcsd
  3. May 23, 2010 #2
    For c) you're being asked for the arc length. Do you know the expression that gives you arclength for a parameterized curve? It's not too bad to remember/derive: if you think of t as time, then the distance (not displacement) that the particle goes at time dt is the speed at the time, which is |r'(t)|, times dt. Therefore the total distance will be the integral from t = 0 to 1 of |r'(t)|dt.
  4. May 25, 2010 #3
    Hi there, thanks for your help!

    Is the expression you refer to the arc length formula?

    you said "therefore the total distance will be the integral from t = 0 to 1 of |r'(t)|dt."
    does that mean the integral from 0 to 1 of the absolutle vaule of r(t) differentiated?

    How do you integrate vectors like this?

  5. May 26, 2010 #4
    By |r'(t)| I meant the norm (magnitude) of the vector r'(t). So if r(t) = (x(t), y(t), z(t)) then r'(t) = (x'(t), y'(t), z'(t)) so that |r'(t)| = sqrt(x'(t)^2 + y'(t)^2 + z'(t)^2).
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