Multivariable Taylor expansion
= Multivariable Taylor expansion
{title2=$f(a+h)=f(a)+Df(a)h+\tfrac12h^TH_f(a)h+O(\|h\|^3)$}
For a scalar function with continuous third derivatives near $a$, apply the one-dimensional <Taylor theorem> to $g(t)=f(a+th)$. The <chain rule> gives $g'(0)=Df(a)h$ and $g''(0)=h^TH_f(a)h$, proving the displayed formula. Here $H_f$ is the <Hessian matrix>. The finite polynomial part is the <Multivariate Taylor polynomial>; the remainder controls its accuracy.