Change of variables and discrete derivatives

[pericles]

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

 

[pericles]

Homework Statement

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

Homework 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)

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

 

[Stephen Tashi]

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".

 

[pericles]

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".

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.