n-dimensional vector analysis

According to the original question, n is a count of the number of dimensions, eg 1 dimension for a ruler, 2 dimensions for a sheet of paper (or the surface of the Earth), 3 dimensions for space (or the volume of the Earth), 4 dimensions for Einsteins's spacetime, etc.

n will be a whole number (unless you are describing some

fractal system).

The units of n are the number of dimensions for whatever space you are discussing.

this says the distance from (a) to ''nowhere'' is (n) any number, zero point source

This notation seems to be representing "d" as a distance vector from point "a" to another point. But this point cannot be nowhere, or the vector will be meaningless. It must be to

*somewhere*.

But it is a bad choice of notation to call this second point "n", because you started by defining n as the number of dimensions.

So call the end point of the distance vector something else, like "b": "d" is a distance vector from point "a" to another point, "b".

The position of "a" and "b" is generally defined relative to some arbitrary zero point, eg Greenwich, the equator, the Sun, or the center of town.

The length of "d" will in general be a real number (unless you

*construct* "d" to be an exact integer in length). The units of d will be some units of length, eg Angstroms, furlongs, Astronomical Units, cubits, miles or meters (personally, I prefer meters).

to ''nowhere'' is ... the ''blackness'' of space in any direction or in the ''dark''.

You've lost me here.

If point "b" is in the middle of the Sun, and you started at point "a" and then moved distance "d", you would not be in the blackness of space, but in an uncomfortably bright location.

The distance d does not determine the brightness. If you start off at a different point (call it "c") then move distance "d", you will end up at a different point which may happen to be dark - or it may be bright.

As I understand it, the intensity of illumination is an example of a

scalar field, which varies as you move through the n dimensions. Some points (like in the middle of the Sun) have a very high intensity of light, while others (like half way to Alpha Centauri) have a much lower level of illumination. You would not call intensity of illumination a vector.

What would you like to prove or analyse with you new multi-dimensional tools?