# Quantization of basis vectors

1. Nov 3, 2013

### mathnerd15

$$|\oint_{0}^{x} dV|=i, |\oint_{0}^{y} dV|=j, |\oint_{0}^{z} dV|=k$$

also I want to express in terms of Dirac delta
$$\oint_{-\infty}^{\infty} \delta(x)dVx= i$$

can you quantize basis vectors say in R3 further, I kind of want to take the integral from 0 to infinity.

is there anything significant about a Basis being linearly independent and spanning the whole vector space as in orthogonal i,j,k so that you can write all vectors in the space as unique linear combinations or maybe it's a better quality to be linearly dependent and not all (bi-ci)=0, spanning the vector in different ways?

and can vectors be curved instead with curvature theta maybe in radial coordinates, would this be an interesting quality, say we defined vector operations and calculus on curved vectors?
like when you are calculating the curl of a field isn't that difficult to represent with straight vectors? maybe this has been done already in modern geometry differential geometry?

Last edited: Nov 3, 2013