A real vector can be thought of as an oriented line segment. Linear algebra and multivariable calculus can be taken pretty far just by considering these types of objects (obviously there are generalizations to other vector spaces, but let's not worry about those).
Using these oriented line segments, we can do several important things. We can
- describe oriented subspaces of $\Bbb R^n$
- build a calculus on them such that derivatives of paths tell you not only the tangent space but also "which way a particle would move on this path"
- in integration, it allows us to find oriented areas/ volumes/ etc
- Lots and lots more examples exist
To me, though, the idea of orientation is just something extra and possibly unnecessary. $\Bbb R^n$ isn't naturally endowed with an orientation -- we have to add one. So could we devise a similar set of objects without the orientation? So you'd just have a line segment, the derivative of a path would just give you the tangent space at a point, and integrals would just give you areas/ volumes/ etc.
As I understand it, Descartes described geometry with just line segments. And later Hermann Grassmann's father thought that a natural product of two line segments is just an area segment (like a bivector but without the orientation).
This question partly comes from me thinking it would be nice if we could take integrals and not have to worry about the part of of the function above or beneath the $x$-axis -- we'd just get an area. And it'd be nice if we didn't have to add an orientation onto a space to use differential area/ volume/ etc elements. And it'd be nice if we could use elementary calculus methods to integrate on non-orientable spaces.
So my questions are: Could we develop a consistent system with these properties? Would it be at all useful (maybe it'd be more complicated and completely subsumed by vector math)? Has it already been done and I just haven't heard about it (if so, could you provide a source)?