What are the analogs of quadratic forms of degree $k>2$? Let $f:{\Bbb R}^n\to{\Bbb R}$ be a smooth function. Its differential of $k$-th degree in a point $a\in{\Bbb R}^n$ can be defined as a map 
$$
d^k f(a):{\Bbb R}^n\to{\Bbb R},\qquad d^kf(a)(p)=\frac{d^k}{d t^k}f(a+tp)\Big|_{t=0},\qquad p\in{\Bbb R}^n.
$$
For $k\le 2$ differentials are well-known objects:


*

*For $k=1$ the differential $df(a):{\Bbb R}^n\to{\Bbb R}$ is a linear form (or linear functional) on ${\Bbb R}^n$.

*For $k=2$ the differential $d^2f(a):{\Bbb R}^n\to{\Bbb R}$ is a quadratic form on ${\Bbb R}^n$.
Question:

What kind of objects are differentials of degree $k>2$?

There must be analogs of quadratic forms of degree $k>2$, where are they described?
 A: I guess the classical name is "algebraic form of degree $k$." The next step after quadratic forms is $k = 3$ when you get cubic forms, then quartic forms, and so forth. Algebraic forms of degree $k$ on a finite-dimensional vector space $V$ can be identified with homogeneous polynomials of degree $k$ on $V$, or equivalently with elements of the symmetric power $S^k(V^{\ast})$.
There is a bijection between algebraic forms in this sense and symmetric multilinear forms which concretely is given by polarization and which abstractly reflects a natural isomorphism
$$S^k(V^{\ast}) \cong S^k(V)^{\ast}$$
(say, in characteristic zero). 
The higher derivatives of a smooth function $f : V \to \mathbb{R}$ are just the coefficients of its Taylor polynomials. Taylor polynomials, like all polynomials, break up into a sum of homogeneous polynomials, one in each degree. The degree zero part is $f(0)$, the degree one part encodes the first derivatives, etc. (Here I am always taking derivatives at $0$ WLOG.) 
For example, in two variables the Taylor expansion takes the form
$$f(x, y) = \sum_n^{\infty} \sum_{i + j = n} f_{ij} x^i y^j$$
where
$$f_{ij} = \frac{1}{i! j!} \frac{\partial^i f}{\partial x^i \partial y^j} \Big|_{x=y=0}.$$
So e.g. the constant part is $f_{00}$, the linear part is $f_{10} x + f_{01} y$, the quadratic part is $f_{20} x^2 + f_{11} xy + f_{02} y^2$, the cubic part is $f_{30} x^3 + \dots$ and hopefully you get the idea. And of course you can isolate the degree $k$ part by looking at the coefficient of $t^k$ in $f(tx, ty)$. 
