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Math Help - Jacobian of a function

  1. #1
    MHF Contributor arbolis's Avatar
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    Jacobian of a function

    Show that the following transformation doesn't change volumes.
    \begin{cases} x_1=u_1 \\ x_2=u_1+u_2 \\  ... \\  x_n=u_1+...+u_n   \end{cases}.
    My attempt : I believe I must show that the Jacobian of the transformation is worth 1. I'm having a hard time finding the Jacobian though. I've no clue.
    f(x_i)=\sum _{j=1}^{i} u_i.

    \begin{bmatrix} \frac{\partial f(x_1)}{\partial u_1}, \frac{\partial f(x_2)}{\partial u_1}, ... , \frac{\partial f(x_n)}{\partial u_1}      \\    \frac{\partial f(x_1)}{\partial u_2},    \frac{\partial f(x_2)}{\partial u_2}, ... ,    \frac{\partial f(x_n)}{\partial u_2}  \\ ......................... \\    \frac{\partial f(x_1)}{\partial u_n}, \frac{\partial f(x_2)}{\partial u_n}, ... , \frac{\partial f(x_n)}{\partial u_n}    \end{bmatrix}
    I know it's wrong, but it's my attempt.
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  2. #2
    Moo
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    Hi,

    But that's the way !

    However, you can note that the Jacobian matrix is just :

    \begin{pmatrix}<br />
1 & 1 & 1 & \ldots & 1  \\<br />
0 & 1 & 1 & \ldots & 1  \\<br />
0 & 0 & \ddots  & \ddots & \vdots   \\<br />
\vdots & \vdots & \vdots & \ddots & 1\\<br />
  0 & 0 &  0 & \ldots & 1<br />
\end{pmatrix}

    Since it's an upper triangular matrix, its determinant is just the product of the terms that are in the diagonal, namely 1.
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  3. #3
    MHF Contributor arbolis's Avatar
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    I think I made a mistake with the Jacobian matrix, it should read the transpose of the matrix I put. The determinant won't change though.
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