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Hypatio

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I want to find a simpler surface function G(x,y,z) which approximates F(x,y,z) within a region close to the point (x0,y0,z0).

Can I write a second-order accurate equation for G if I know F(x0,y0,z0) and can compute the derivatives at the point using finite-differences. What does that function look like? What derivatives are needed?

I want to do this because the function F(x,y,z) is very complicated, but I want to compute an approximate result many times at positions which only change slowly.