Is the third order term of a Taylor approximation a covariant tensor of rank 3 restricted to (h,h,h)? Recently reading Peter Lax's linear algebra text, he had a very concise way of writing the Taylor approximation up to second order: $f(x+h) = f(x) + l(h) + \frac{1}{2}q(h) + \|h\|^2 \epsilon(\|h\|)$, where $l$ is a linear functional, $q$ is a quadratic form, and $\epsilon \to 0$ as $h\to 0$. 
How would we describe the third-order term of the approximation as a function of $h$? It couldn't be represented as a matrix, in the way that a bilinear form can; we'd need a 3d array I think. It seems we could express it as a homogeneous polynomial of degree 3 in $h_1, \dotsc, h_n$.  Am I right that it is also a covariant tensor of rank 3, restricted so that it takes arguments $(h,h,h)$, so it is of the form $\left(\sum\limits_{I\in \{1,\dotsc,n\}^3}a_I\varepsilon^*_I \right)(h,h,h)$? Is there a name for such a thing?
 A: Yes, this is the multivariate Taylor expansion. To derive the terms (of whatever order your heart desires) you simply feed a line $a+th$ into a multivariate function $f$ to obtain a single-variable function $g(t)=f(a+th)$ to which the single-variate Taylor expansion applies. After that, systematic application of the multivariate chain-rule reveals the expansion you seek. Just set $t=1$ and you obtain $f(a+h)=f(a)+f'(a) \cdot h + \cdots $. There are a variety of cute ways to express the formula. The one I have in my notes for functions of two variables $x=x_1$ and $y=x_2$ centered at $(a_1,a_2)$
$$ f(x, y) = \sum_{n=0}^{\infty} \sum_{i_1=0}^{n}\sum_{i_2=0}^{n} \cdots \sum_{i_n=0}^{n} \frac{1}{n!}
\frac{\partial^{(n)}f(a_1,a_2)}{\partial x_{i_1}\partial x_{i_2} \cdots \partial x_{i_n}}
(x_{i_1} -a_{i_1})(x_{i_2} -a_{i_2})\cdots (x_{i_n} -a_{i_n}) $$
The formula for functions of three or more variables is similar. I know of no particular name for the third order piece. Intuitively, it becomes important when the expansion is trivial up to second order and yet it is nontrivial. In such a case it is the dominant term (locally). Of course, I have no particular sense of what the graph of such a thing resembles. In contrast, the second order term is nicely captured by the theory of quadratic forms. 
