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Tensor fields

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Multivectors

Multilinear forms of order $k$ over a real vector space $V$ are multilinear antisymmetric functions $$\varphi =\varphi ^k: V^k \rightarrow {\bf R}$$ that give us oriented lengths, area, volumes, etc. They constitute a vector space $$\Lambda = \Lambda^k(V).$$ Forms of all orders constitute a graded vector space: $$\Lambda = \Lambda^0 \times \Lambda^1 \times \Lambda^2 \times \ldots.$$ This vector space has its operations acquired from $\Lambda^0,\Lambda^1, \Lambda^2 , \ldots$. It also an extra operation, the wedge product: $$\wedge \colon \Lambda^k \times \Lambda^m \rightarrow \Lambda^{k+m},$$ which makes it a graded linear operator which is graded commutative. The two operations $$+, \wedge \colon \Lambda \rightarrow \Lambda $$ make this space into a graded ring.

When the forms are continuously parametrized by location on a smooth manifold $M$: $$\varphi =\varphi ^k: M \times V^k \rightarrow {\bf R},$$ they are called differential forms.

One may also consider a vector space $V$ over any other field $F$. Then the forms of order $k$ over $V$ are multilinear antisymmetric functions $$\varphi =\varphi ^k: V^k \rightarrow F.$$ If $F$ isn't a field but just a ring, we are dealing with a module $V$. When $F={\bf Z}$, forms of order $k$ over $V$ are multilinear antisymmetric functions $$\varphi =\varphi ^k: ({\bf Z}^n)^k \rightarrow {\bf Z}.$$

When such forms are parametrized by location on a cubical complex $K$: $$\varphi =\varphi ^k: K^{(0)} \times({\bf Z}^n)^k \rightarrow {\bf Z},$$ where $K^0$ is the set of vertices of $K$, they are called discrete differential forms. The target space can be ${\bf R}$ as well.

Even more generally, the differential forms are functions defined on the tangent bundle.