Where will you find mathematics of n spacial dimensions?

coeilsmicah

For example, finding the properties of an n dimensional figure. Is this called something in math, or do I just refer to it as 'geometry in multiple dimensions'? What subjects can I find this topic under?

chiro

Hey coeilsmicah and welcome to the forums.

Mathematics currently usually works in an abstract number of dimensions. For example R^n is the geometry that looks like normal 3D space but in n-dimensions where every component of the vector (in n-dimensions) can be changed without affecting any of the others.

You can read up on differential geometry which looks at general situations where geometry is curved (i.e. not like the one above): in other words, the geometry has a dependency.

As an example consider y = x + 2: y depends on x so it's not like changing x won't change y: it will change y. But consider x = 2, y = 1: we change x but y doesn't change.

The situation where we can change any element and it doesn't change any other, the main results of looking at these spaces can be found in linear algebra for the fixed dimension theory (i.e. n is finite) and for the infinite-dimensional theory (yes it exists and it's used for the theory of quantum mechanics) it's called Hilbert-Space theory.

Also you will need to learn vector calculus before differential geometry.

The general theory of geometric objects is known as manifold theory which encompasses a lot of differential geometry.

The differential geometry can be understood when you have taken enough calculus and some linear algebra and the idea used in tensor theory is to use the main concepts of geometry (distance and angle) and see how these things change between different co-ordinate systems: this way you can look at how deforming a co-ordinate system (i.e. treating like a play-doh thing where you can squish it and stretch it) changes its properties of distance and angle.

This is a highly simplified description, but hopefully it will help you.

mathwonk

look up higher dimensional geometry, or just start with linear algebra in n dimensions.

coeilsmicah

chiro and mathwonk,
Thank you guys for the kind replies, they really cleared up a lot. I'll be sure to shift my efforts to lean more toward studying calculus.