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## 1.3.4 Second derivatives in 2 dimensions

By again fixing one of the independent variables - , say - and considering only , we obtain, in terms of the Stirling (centered) approximation,

Analogous (and less accurate) formulae are valid within the NGF- and NGB-approximations, respectively.

How about mixed derivatives?

Mixed derivatives

Approximating use the same kind of approximation with respect to both the - and the -direction. (This may not hold if and have a different character, e.g. one space and one time variable.)

Stirling:

And now, fow for the curvature of :

Curvature of a function f(x,y)

To find the local curvature at the grid point we have to apply the nabla operator twice.(*) There are two ways:
Either difference'' along the grid axes,

or apply diagonal differencing'', writing

 Axial vs. diagonal differencing

(*) Note that the nabla operator mentioned here is not to be mixed up with the backward difference for which we use the same symbol.

Next: 1.4 Sample Applications Up: 1.3 Difference Quotients Previous: 1.3.3 First derivatives in
Franz J. Vesely Oct 2005