The general question:
Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order $d$ tensors on $V$ and the space of order $d$ homogeneous polynomials in $n$ indeterminates over $k$.
I am wondering whether this identification is ever of much use in the study of tensors and tensor fields. In particular, I am interested in ways in which this identification might simplify problems in differential geometry, but uses in other fields would be interesting also.
A potential example of what I'm asking about:
This question is inspired by the following observation I made, which I would also like to confirm is valid:
Let $\omega \in \mathrm{Sym}^1(V)$ be some unknown. Suppose we have a map $$\phi \colon \mathrm{Sym}^1(V) \longrightarrow k$$ along with a few known elements $\eta, \theta, \nu_1, \nu_2 \in \mathrm{Sym}^1(V)$. Additionally, we have the following system of equations: $$ \begin{align} \phi(\omega) \cdot \eta + \phi(\eta) \cdot \omega &= \nu_1 \\ \phi(\omega) \cdot \theta + \phi(\theta) \cdot \omega &= \nu_2 \end{align} $$ Our goal is to solve for $\omega$. In light of the above, we can identify $\mathrm{Sym}^1(V)$ with $k[x_1, \dots, x_n]_1 \subset k[x_1, \dots, x_n]$, form the field of fractions $k(x_1, \dots, x_n)$ and solve the above system for $\omega$ using linear algebra. Doing so, we will arrive at an expression of the form: $$ \left(\phi(\theta)\eta - \phi(\eta)\theta\right)\omega = \theta \cdot \nu_1 + \eta \cdot \nu_2 $$ where all the elements are now considered to be in $k(x_1, \dots, x_n)$, so all the products make sense. Dividing, we can obtain an expression for $\omega$ expressed entirely in terms of objects we know. We can now evaluate this expression on the appropriate vectors to obtain an expression for $\omega$ in terms of a basis for $\mathrm{Sym}^1(V)$.