Take the tour ×  Possible Duplicate: Does a “cubic” matrix
exist?  There is some theory about spatial matrices ?, for
example matrices of order $3\times 3\times 3
,A=(a_{ijk}),i,j,k=1,2,3$ where $A$ would be a cube in space
with entries inside the cube.  enter image description here  For
example if $f:\mathbb{R}^3\longrightarrow \mathbb{R}$ , $f\in
C^3$  $f(a+h)=f(a)+f'(a).h+\frac{1}{2}f''(a).h^2+\frac{1}{6}f'''
(a).h^3+r(h)$ , $\lim_{h \to 0}{\frac{r(h)}{||h||^3}}=0$ where
$f'''(a)$ would be a spatial matrix of order $3\times3$
(application symmetric trilinear).  $f'''(a)=(\frac{\partial^3
f}{x_ix_jx_k}),i,j,k=1,2,3$ $f'''(a):\mathbb{R}^3 \times
\mathbb{R}^3 \times \mathbb{R}^3 \longrightarrow \mathbb{R}$
$f'''(a)(x,y,z)=\langle (x^t.f'''(a)).y,z\rangle$ Where
$x^t.f'''(a)$ would be the multiplication of each plane matrix
of the spatial matrix by the vector $x$ resulting in a plane
matrix.  share|improve this question Please take a look at this
meta question on the reasons for accepting answers. –  joriki
Jul