Determine the quadratic approximation surface at the point (0,0) on the surface of
Start by defining the "quadratic surface" by the standard form and we'll find values for the coefficients such that . Since this is a system of six equations and six unknowns, we are guaranteed a unique answer.
Derivatives of z:
Derivatives of q:
So, is a quadratic function that shares its six derivatives with . Notice that this is always possible, as long as all six derivatives exist and are defined at the point of interest.