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Time Derivatives of Unit Vectors

  1. Dec 14, 2011 #1
    1. The problem statement, all variables and given/known data

    The Hyperbolic coordinate system is given by: u=2xy and v=x^2 - y^2

    a.) Find the unit vectors u and v in terms of u,v,x(hat),y(hat)
    b.)Find the time derivatives of u(hat) and v(hat), your answers will have du/dt and dv/dt in them

    2. Relevant equations

    None really


    3. The attempt at a solution

    I found the unit vectors by

    u(hat) = (du/dx)i + (du/dy)j
    v(hat) = (dv/dx)i + (dv/dy)j

    then I substituted in for x and y to give equations in terms of v and u

    u(hat) = (sqrt(2))sqrt(sqrt(u^2 + v^2) - v))x(hat) + (sqrt(2))sqrt(sqrt(u^2 + v^2) +v))y(hat)

    v(hat) = (sqrt(2))sqrt(sqrt(u^2 + v^2) + v))x(hat) + (-sqrt(2))sqrt(sqrt(u^2 + v^2) - v))y(hat)

    I just don't understand how to take the time derivatives of these unit vectors, any help would be greatly appreciated.

    Thanks
     
  2. jcsd
  3. Dec 14, 2011 #2

    SammyS

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    Your unit vectors do not appear to have the correct magnitude, which should be 1.

    For [itex]\hat{u}[/itex] I get: [itex]\displaystyle\hat{u}=\frac{\displaystyle
    \frac{\partial u}{\partial x}\hat{i}+\frac{\partial u}{\partial y}\hat{j}}{\displaystyle\sqrt{\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2}}=\frac{y\hat{i}+x\hat{j}}{\sqrt{x^2+y^2}}[/itex]
     
  4. Dec 14, 2011 #3
    Thank you for the correction sammy, I'm not very good at vector calculus, and my book is hard to get useful information from.

    so, I have

    u(hat) = (2yi + 2xj)/(sqrt(4x^2 + 4y^2))

    and

    v(hat) = (2xi -2yj)/(sqrt(4x^2 + 4y^2))

    I still do not understand how to take the time derivatives though, thanks
     
    Last edited: Dec 14, 2011
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