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Change of variables and discrete derivatives

  1. Mar 25, 2015 #1
    Hey

    I am trying to evaluate d/dx, d/dy and d/dz of a wavefunction defined on a grid. I have the wavefunction defined on equally spaced points along three axes a=x+y-z, b=x-y+z and c=-x+y+z. I can therefore construct the derivative matrices d/da, d/db and d/dc using finite differences but I don't know how to convert these to d/dx, d/dy and d/dz.

    I have been trying to use some sort of chain rule

    d/dx=d/da*da/dx+d/db*db/dx+d/dc*dc/dx ... but I do not know how to express da/dx, db/dx and dc/dx using matrices.

    Any help would be greatly appreciated
    Thanks
     
  2. jcsd
  3. Mar 25, 2015 #2
    1. The problem statement, all variables and given/known data

    Hey

    I am trying to evaluate d/dx, d/dy and d/dz of a wavefunction defined on a grid. I have the wavefunction defined on equally spaced points along three axes a=x+y-z, b=x-y+z and c=-x+y+z. I can therefore construct the derivative matrices d/da, d/db and d/dc using finite differences but I don't know how to convert these to d/dx, d/dy and d/dz.

    I have been trying to use some sort of chain rule

    d/dx=d/da*da/dx+d/db*db/dx+d/dc*dc/dx ... but I do not know how to express da/dx, db/dx and dc/dx using matrices.

    Any help would be greatly appreciated
    Thanks

    2. Relevant equations
    a=x+y-z,
    b=x-y+z
    and c=-x+y+z
    x=½(a+b)
    y=½(a+c)
    z=½(c+b)
    3. The attempt at a solution

    I have been trying to use some sort of chain rule

    d/dx=d/da*da/dx+d/db*db/dx+d/dc*dc/dx ... but I do not know how to express da/dx, db/dx and dc/dx using matrices.

    I tried using [(d/da*x)]^(-1) but it does not work

    any help would be greatly appreciated.
    Thanks
     
  4. Mar 25, 2015 #3

    Stephen Tashi

    User Avatar
    Science Advisor

    If the problem is to approximate the partial derivatives of a smooth function from the values of the function on a grid, you should look up the topic of "stencils" in numerical analysis. Search on keywords like "stencil, derivatives, 3D".
     
  5. Mar 25, 2015 #4
    My problem is somewhat different...

    I have the partial derivatives with respect to a, b and c but would like partial derivatives with respect to x, y and z.

    I have used a stencil (I think) in order to get the partials with respect to a, b and c.
     
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