In Boyce and Diprima's ODE's and BVP's (10th edition page 522), it says that for the nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y) \qquad\qquad\qquad (10),$$
"The system (10) is locally linear in the neighborhood of a critical point $(x_0,y_0)$ whenever the functions $F$ and $G$ have continuous partial derivatives up to order two. To show this , we use Taylor expansions about the point $(x_0,y_0)$ to write $F(x,y)$ and $G(x,y)$ in the form $$F(x,y)=F(x_0,y_0)+F_x(x_0,y_0)(x-x_0)+F_y(x_0,y_0)(y-y_0)+\eta_1(x,y)\\ G(x,y)=G(x_0,y_0)+G_x(x_0,y_0)(x-x_0)+G_y(x_0,y_0)(y-y_0)+\eta_2(x,y)$$ where $\eta_1(x,y)/[(x-x_0)^2+(y-y_0)^2]^{1/2}\to 0$ as $(x,y)\to(x_0,y_0)$, and similarly for $\eta_2$."
Why does it say "whenever the functions $F$ and $G$ have continuous partial derivatives up to order two" instead of "whenever the functions $F$ and $G$ have continuous partial derivatives up to order one"? $\eta_1(x,y)$ and $\eta_2(x,y)$ contain the nonlinear parts and the second order partial derivatives, right? And, Taylor's Theorem says the function being k times differentiable is sufficient for an approximation by a kth degree Taylor polynomial.