Density On A Manifold - Motivation (Densities in Vector Spaces)

Motivation (Densities in Vector Spaces)

In general, there does not exist a natural concept of a "volume" for a parallelotype generated by vectors v₁,...,v_n in a n-dimensional vector space V. However, if one wishes to define a function μ:V×...×V→R that assigns a volume for any such parallelotype, it should satisfy the following properties:

• If any of the vectors v_k is multiplied by λ∈R, the volume should be multiplied by |λ|.
• If any linear combination of the vectors v₁,...,v_j-1,v_j+1,...,v_n is added to the vector v_j, the volume should stay invariant.

These conditions are equivalent to the statement that μ is given by a translation-invariant measure on V, and they can be rephrased as

Any such mapping μ:V×...×V→R is called a density on the vector space V. The set Vol(V) of all densities on V forms a one-dimensional vector space, and any n-form ω on V defines a density |ω| on V by