 Multivariable Taylor expansion

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 $g^{\prime }(0)=Df(a)h$ and $g^{\prime \prime }(0)=h^T{H}_f(a)h$, 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.