Assume a vector-valued function, for example ${\bf f}=(f_1, f_2)$, where

$$f_1(x,y)= x^2+3xy$$ $$f_2(x,y)= 2xy+y^2$$

(here f is column vector, x, y are variables)

Assume that each $f_i$ is a polynomial with degree at most 2, and thus we can write (in vector form) that:

$${\bf f}({\bf x}) = Q({\bf x},{\bf x})+L{\bf x}+{\bf c}$$ where $\bf f$ and $\bf x$ are both vectors, and $Q({\bf x},{\bf x})$ is the "quadratic" part，$L{\bf x}$ is the linear part, and c is a constant vector.

It seems to me that $f({\bf x}+1)= f(1)+f'(1){\bf x}+Q({\bf x},{\bf x})$ where $f'({\bf x})$ is the Jacobian matrix.

It looks like Taylor expansion, but I do not know exactly how to prove, except for stragihtforward calculation. Does anyone help me out?

-