Third and higher derivatives of n-dimensional funtctions I was thinking about Taylor expansions of an N-dimensional function and thereby wondered:
How does third and higher derivates look like?
I mean: the function itself maps onto $\mathbb{R}$ , the first derivate onto $\mathbb{R}^N$, the second $ \mathbb{R}^N$x$~\mathbb{R}^N$ but how can I envision the next and further ones?
Are they just higher dimensional matrices?
Thanks, 
Xi Tong
 A: Yes, they are... such things are called tensors. 


*

*First order derivative  is an object that takes one vector $u$ and returns a number (in a linear way), namely the directional derivative along $u$ at a given point. Such an object  can be encoded as a row vector. 

*Second order derivative   takes two vectors $u,v$ and returns a number (in a   way that is linear in each argument separately). Namely, it shows what you'll get if you take first derivative in direction $u$ (at every point), and then take derivative of the result in direction $v$ (at a given point). Such an object can be encoded as a matrix.  

*Third order derivative   takes three vectors $u,v,w$ and returns a number (in a   way that is linear in each argument separately). Namely, it shows what you'll get if you take first derivative in direction $u$ (at every point),   then take derivative of the result in direction $v$ (at every point), and finally the derivative in direction $w$ (at a given point). Such an object can be encoded as a 3-dimensional array of numbers, using three indices like $a_{ijk}$.

