# Quadratic approximation of vector valued functions

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?

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