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Jacobian matrices

  1. Jun 29, 2014 #1
    [tex]x=t^2-s^2, y=ts,u=x,v=-y[/tex]

    a) compute derivative matrices


    [tex]\vec{D}f(x,y) = \left[\begin{array}{cc}2t&-2s\\s&t\end{array}\right][/tex]

    [tex]\vec{D}f(u,v) = \left[\begin{array}{cc}1&0\\0&-1\end{array}\right][/tex]

    b) express (u,v) in terms of (t,s)

    [tex]f(u(x,y),v(x,y) = (t^2-s^2,-(ts))[/tex]

    c) Evaluate [tex]\vec{D}(u,v)[/tex]

    [tex]\vec{D}(u,v) = \left[\begin{array}{cc}1&0\\0&-1\end{array}\right] \left[\begin{array}{cc}2t&-2s\\s&t\end{array}\right][/tex]

    [tex]= \left[\begin{array}{cc}2t&-2s\\-s&-t\end{array}\right][/tex]

    d) verify if chain rule holds


    need help with this last part, also need to know if I even did the rest correctly
     
    Last edited: Jun 29, 2014
  2. jcsd
  3. Jun 29, 2014 #2

    Zondrina

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    Homework Helper

    Part a) looks okay, but the notation is a little rough. Something more like:

    [tex]
    J_{s, t}(x,y) = \begin{pmatrix}
    x_s & x_t \\
    y_s & y_t
    \end{pmatrix}, \quad

    J_{x, y}(u,v) = \begin{pmatrix}
    u_x & u_y \\
    v_x & v_y
    \end{pmatrix}
    [/tex]

    ##J_{s, t}(x,y)## and ##J_{x, y}(u,v)## are the Jacobian matricies.

    I'm sure you meant ##u = t^2 - s^2## for part b).

    For part c), I'm sure what is intended is you find the derivative matrix ##J_{s, t}(u,v)## after expressing ##u## and ##v## as functions of ##s## and ##t##.
     
  4. Jun 29, 2014 #3
    I fixed the error in (b) and yeah thats what I want for (c), I'm using the notation that is used in my book/what my professor uses

    as for part (d) how do I verify this
     
  5. Jun 29, 2014 #4

    Zondrina

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    For part d), if ##f## is a function of ##u## and ##v##, which are functions of ##x## and ##y##, which are functions of ##s## and ##t##, what is the partial derivative of ##f## with respect to ##u##? How about the partial with respect to ##v##?
     
  6. Jun 29, 2014 #5
    [tex] (\frac{\partial f}{\partial u})_{t,s} = 2t,-2s \,\,\, (\frac{\partial f}{\partial v})_{t,s} = -s,-t [/tex]

    I know this is kind of an abuse of notation but I've been typing LaTeX all day
     
    Last edited: Jun 29, 2014
  7. Jun 29, 2014 #6

    Zondrina

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    Homework Helper

    Looks good.
     
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