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