Results 1 to 3 of 3

Thread: Jacobian of a function

  1. #1
    MHF Contributor arbolis's Avatar
    Joined
    Apr 2008
    From
    Teyateyaneng
    Posts
    1,000
    Awards
    1

    Jacobian of a function

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

    $\displaystyle \begin{bmatrix} \frac{\partial f(x_1)}{\partial u_1}, \frac{\partial f(x_2)}{\partial u_1}, ... , \frac{\partial f(x_n)}{\partial u_1} \\ \frac{\partial f(x_1)}{\partial u_2}, \frac{\partial f(x_2)}{\partial u_2}, ... , \frac{\partial f(x_n)}{\partial u_2} \\ ......................... \\ \frac{\partial f(x_1)}{\partial u_n}, \frac{\partial f(x_2)}{\partial u_n}, ... , \frac{\partial f(x_n)}{\partial u_n} \end{bmatrix}$
    I know it's wrong, but it's my attempt.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hi,

    But that's the way !

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

    $\displaystyle \begin{pmatrix}
    1 & 1 & 1 & \ldots & 1 \\
    0 & 1 & 1 & \ldots & 1 \\
    0 & 0 & \ddots & \ddots & \vdots \\
    \vdots & \vdots & \vdots & \ddots & 1\\
    0 & 0 & 0 & \ldots & 1
    \end{pmatrix}$

    Since it's an upper triangular matrix, its determinant is just the product of the terms that are in the diagonal, namely 1.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor arbolis's Avatar
    Joined
    Apr 2008
    From
    Teyateyaneng
    Posts
    1,000
    Awards
    1
    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. joint probability density function, using the Jacobian
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: Nov 5th 2010, 11:19 AM
  2. Jacobian
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Oct 21st 2009, 07:54 PM
  3. Jacobian of a composed function+chain rule
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jul 27th 2009, 09:37 AM
  4. Help with Jacobian
    Posted in the Calculus Forum
    Replies: 0
    Last Post: Oct 18th 2008, 10:22 AM
  5. jacobian
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Oct 24th 2007, 11:10 AM

Search Tags


/mathhelpforum @mathhelpforum