Let $R$ be a commutative ring and $M$ be an $R$-module.
Then, we can construct the tensor algebra $T(M):=\oplus_{n\in \mathbb{N}} M^{\otimes n}$ which is likely a free object. That is, for any $R$-algebra $A$ and an $R$-module morphism $\phi:M\rightarrow A$, $\phi$ can be uniquely extended to an $R$-algebra morphism from $T(M)$ to $A$.
Similarly, we can construct the symmetric algebra $S(M)$ and the exterior algebra $\bigwedge(M)$, and they have universal mapping properties. Moreover, $S^n(M)$ and $\bigwedge^n(N)$ have universal properties too.
Unlike I know free groups are directly related to van kampen theorem (hence it is extremely important to calculate fundamental groups), I don't get why I should learn symmmetric and exterior algebras. Especially, symmetric algebra.
Would someone sketch (with some details) how they are used? Especially how they are used in differential geometry?
Of course I have read wikipedia article and I know that exterior algebra can be used to calculate determinants and is a generalization of cross product and etc. However, I don't get how exactly so..