Thinking about how to prove the multilinearity of the volume of a parallelepiped definition I like this sketched approach:
The two dimensional case is a “cute” problem involving rearranging triangles and ordinary areas (or you solve this case in any other way you want). The general case then follows from linearity of integrals (you get the higher dimensional cases by integrating the two dimensional case appropriately).
Thinking about how to prove the multilinearity of the volume of a parallelepiped definition I like this sketched approach:
The two dimensional case is a “cute” problem involving rearranging triangles and ordinary areas (or you solve this case in any other way you want). The general case then follows from linearity of integrals (you get the higher dimensional cases by integrating the two dimensional case appropriately).