How do I find the second order derivative of a function on $\mathbb{R}^n$

Question

I am studying systems of differential equations, i.e., $$f(\vec{x})=\dot{\vec{x}} = A\vec{x}$$ for some matrix $A_{n\times n}$, a vector valued function $x(t)=x $ and a function $f:\mathbb{R}^n \to \mathbb{R}^n$. Now, if $f$ is differentiable, then its derivative is given as $$Df= \left[\frac{\partial{f_i}}{\partial{x_j}}\right]$$ which is an $n\times n$ matrix. Now, I am studying from Perko's DE's and Dynamical Systems and the second order derivative of $f$ is given exactly as follows. Assuming $f:E \subseteq\mathbb{R}^n\to \mathbb{R}^n$, $$D^2f(x_0)(x,y) = \displaystyle\sum_{j_1,j_2=1}^{n} \frac{\partial^2{f(x_0)}}{\partial{x_{j_1}\partial{x_{j_2}}}}x_{j_1}y_{j_2}$$ for some $(x,y) \in E \times E$. Now, I want to the find $D^2f(x_0)$ for $$f(x) = \left(\begin{matrix} x_1+x_1x_2^2 \\ -x_2+x_2^2+x_1^2 \end{matrix}\right)$$ and $x_0 = (0,1)$ but I do not know how to use the formula. Is the sum gives a column or row of the desired derivative matrix? How do I proceed for this question and for higher order derivatives?

Answer 1

For a real-valued function $\mathbb{R}^n \to \mathbb{R}$, $$\displaystyle\sum_{j_1,j_2=1}^{n} \frac{\partial^2{f(x_0)}}{\partial{x_{j_1}\partial{x_{j_2}}}}x_{j_1}y_{j_2}$$ is the same as: $$\sum_{i=1}^n \sum_{j=1}^n H_{ij}x_i y_j$$ which is just $x^THy$ where $H$ is the Hessian of $f$ at $x_0$. For a vector-valued function is gets a bit tricky when it comes to interpreting $\frac{\partial^2{f(x_0)}}{\partial{x_{j_1}\partial{x_{j_2}}}}$. It is ambiguous, because you might think this is a matrix. It is in fact a vector where the i-th component is the second derivative (to $x_{j_1}$ and $x_{j_2}$) of the i-th component of your objective. An easy way to compute the full expression is by writing the Hessian of the coordinates of your function: $$H_1 = \begin{pmatrix}0 & 2x_2 \\ 2x_2 & 2x_1\end{pmatrix}$$ $$H_2 = \begin{pmatrix}2 & 0 \\ 0 & 2\end{pmatrix},$$ and writing $D^2f(x_0)(x,y)$ as a vector in $\mathbb{R}^2$: $$\begin{pmatrix}x^T H_1 y \\ x^T H_2 y\end{pmatrix}.$$