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What is the best second order approximation using Taylor's theorem for vector value functions? Specifically in the case when $f: \mathbb{R}^d \to \mathbb{R}$ and then $g= \nabla f$ then the first order approximation by Taylor's theorem is $ g(x) = g(a)+(x-a) \cdot \nabla g(a) + ... = g(a) + (x-a) \cdot H(f)(a)+... $ what would be the third term in this approximation?

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  • $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ – amd Jan 24 at 6:05
  • $\begingroup$ @amd those are scalar valued functions... $\endgroup$ – ADA Jan 24 at 12:03
  • $\begingroup$ Apply individually to each component. $\endgroup$ – amd Jan 24 at 14:14
  • $\begingroup$ @amd So the Hessian of a Hessian is going to be what, a matrix with matrices as entries? $\endgroup$ – ADA Jan 24 at 19:12
  • $\begingroup$ More or less. All of those higher-order derivatives are tensors. $\endgroup$ – amd Jan 27 at 5:01

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