1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integral of unit tangent vector equals arc length?

  1. Feb 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Let c(t) be a path and T the unit tangent vector. What is [tex] \int_c \mathbf{T} \cdot d\mathbf{s} [/tex]


    2. Relevant equations

    The unit tangent vector of c(t) is c'(t) over the magnitude of c'(t) :
    [tex] \mathbf{T} = \frac{c'(t)}{||c'(t)||} [/tex]

    The length of c(t) can be represented by :
    [tex] \int_c ||c'(t)|| \; dt [/tex]

    3. The attempt at a solution
    [tex] \int_c \mathbf{T} \cdot d\mathbf{s} = \int_c \frac{c'(t)}{||c'(t)||} ... [/tex] d-something. dt I suppose.

    But this is clearly not quite the arc length integral. So what am I missing? (the book says the answer is the length of c)
     
  2. jcsd
  3. Feb 18, 2012 #2

    I like Serena

    User Avatar
    Homework Helper

    Hi Arcana!

    I guess you need that ##\mathbf{s} = \mathbf{c}(t)##.
    That is, ##\mathbf{s}## is a point on the curve that is parametrized by ##\mathbf{c}(t)##.
    What do you think ##\textrm{d}\mathbf{s}## is?
     
  4. Feb 18, 2012 #3
    ds=c'(t)dt ?
    But then I still have 1/magnitude, so still not quite there.
     
  5. Feb 18, 2012 #4

    I like Serena

    User Avatar
    Homework Helper

    Yep.

    How so?
    What do you get if you substitute ds=c'(t)dt?
     
  6. Feb 18, 2012 #5
    (c'(t))^2 / llc'(t)ll dt
     
  7. Feb 18, 2012 #6

    I like Serena

    User Avatar
    Homework Helper

    Yes.
    And what is ##\mathbf{c}'(t) \cdot \mathbf{c}'(t)##?
     
  8. Feb 18, 2012 #7
    is it 1 ? no

    it's the same thing as we have under the square root (calculating....)
    okay Im stuck
     
  9. Feb 18, 2012 #8

    I like Serena

    User Avatar
    Homework Helper

    No.
    Suppose ##\mathbf{v}## is a vector.
    Did you know that ##\mathbf{v} \cdot \mathbf{v} = ||\mathbf{v}||^2##?
     
  10. Feb 18, 2012 #9
    oh. excellent observation. I see it now. thanks :)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integral of unit tangent vector equals arc length?
  1. Arc length units (Replies: 1)

Loading...