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The derivative approximation is simple in a single direction:

df/dx(i,j,k)= [f(i+1,j,k) - f(i-1,j,k)] / 2

And I know that in the second order:

d2f/dxdy(i,j,k)= [f(i+1,j+1,k) - f(i+1,j,k) - f(i,j+1,k) + f(i-1,j-1,k)] / 4

The final item I need is the third order approximation, and I'm not sure how to scale the first two into a third variable.

d3f/dxdydz(i,j,k)= ?

Can anyone shed some light on this?

Thanks in advance!