Second order derivatives of a function. We have to use the Hessian to calculate the second order derivatives of a function. While that is okay if the function is mapped from $\mathbb{R}^n$ to $\mathbb{R}$, how does one proceed if it is mapped from $\mathbb{R}^n \longrightarrow \mathbb{R}^m$, where $m > 1$? How does one take the derivative of f with respect to each variable when f is itself in more than one dimension?
For example, if there is a function f : $\mathbb{R}^2 \longrightarrow \mathbb{R}^3$ with $(x,y)$ mapped to $(x^2 + y, y^3, \cos(y))$, how does one calculate the Hessian?
Thank you

I guess what you are looking for is the following:
Let $\vec{f}(x,y) = \begin{pmatrix} u(x,y) \\ v(x,y) \\ w(x,y)\end{pmatrix}$. Its "Hessian" is simply the "vector-valued matrix"
$$ H_{\vec{f}} = \begin{pmatrix}
\begin{pmatrix} u_{xx} \\ v_{xx} \\ w_{xx}\end{pmatrix} & 
\begin{pmatrix} u_{xy} \\ v_{xy} \\ w_{xy}\end{pmatrix} \\
\begin{pmatrix} u_{yx} \\ v_{yx} \\ w_{yx}\end{pmatrix} &
\begin{pmatrix} u_{yy} \\ v_{yy} \\ w_{yy}\end{pmatrix}
\end{pmatrix} $$
where $ u_{xy} = \frac{\partial^2}{\partial x\partial y} u$ etc. Note that each entry of the "matrix" is now a vector in its own right. 
