[talanum1 (OP)]

If physical space is the domain of forces what makes up the ranges?

[Kryptid]

It's difficult to understand what you are asking. Are you asking what the ranges of the fundamental forces are?

[talanum1 (OP)]

Are you asking what the ranges of the fundamental forces are?

No, Mathematically what is the output spaces of the forces?

[Bored chemist]

Forces are not a mathematical function.

[Kryptid]

No, Mathematically what is the output spaces of the forces?

I'm not sure that question makes sense.

[evan_au]

Mathematically what is the output spaces of the forces?

The question is not clear, but I'll try to read between the lines.

Simplification: I will ignore Einstein's relativity & spacetime, and just talk about 3 dimensions of space+1 dimension of time.

General Answer: The output space of forces is 3 dimensional for a static model.
- The output space of forces is 3 dimensions of space + 1 dimension of time, for a dynamic model.
- For a static model, the input space is 3 dimensions of position in space, plus other miscellaneous information like mass or charge for each object
- For a dynamic model, the input space also needs 3 dimensions of velocity for each object
- The intermediate calculations may involve additional information like projecting positions or velocities ahead in time, or probability densities (for quantum objects)

Example 1: Simplified Solar System
Here the dominant force is gravity.
- There (say) 10 objects, each with their own mass (m₁, m₂,m₃... m₁₀).
- Each with their own 3-Dimensional initial position in space (X,Y,Z),
- They have zero electrical charge, and very little magnetic field at these distances,

Static Model: If you just want to know the forces at an instant in time, this is enough information.
- the magnitude of the force between each pair of objects are given by Newton's gravity: F=Gm₁m₂/r²
- where r is the distance between these two objects, and G is Newton's gravitational constant
- Each force can be broken up into 3 dimensions: (F_x, F_y, F_z)
So the input space is 3-dimensional positions plus the zero-dimensional mass; the output space is the 3-dimensional force between each pair of objects.
- With 10 objects, there are 10x9 = 90 forces to be tracked (because the force is symmetrical, half are the negative of the other half)

Dynamic Model: If you also want to calculate the forces at a future point in time, you also need:
- the initial velocity of each object through 3-dimensional space (v_x, v_y, v_z)
- Because the relative positions are always changing, all the forces change over time.
- For this model, the input space is still 3-dimensional, but has additional velocity information for each object
- The output space is also 3-dimensional forces, at a particular point in time
- But internally, you also need to keep track of the new 3-dimensional positions and velocities and time, even if you don't provide this in the output.
- As you need the new position and new velocity to predict the forces at the next point in time.

Example 2: Simplified Hydrogen Atom
In this case, the dominant force is electrostatic attraction.
- The electron and proton each have an electrical charge, ±e
- you can't provide a specific point in space for the electron, as it is a quantum object (you can do a little better for the proton)
- The probability of finding the electron at a certain position in space is given by the Schroedinger equation
- This gives a 3-dimensional probability of finding the electron in a particular position
- For each possible position of the electron, you could calculate a classical electrostatic force between the electron and proton, but in general, this is not a very useful thing to do
- Similar quantum effects make the calculations for the strong and weak nuclear forces even more complex. But they are still probability distributions.

PS: It is easy to lose track of all these individual numbers, so in practice it is best to handle them in a computer as orderly arrays and vectors.
- The array showing the magnitudes of the forces between 10 objects would typically be represented as a square array of 10 rows and 10 columns, which is a 2-dimensional array.
- To make it more confusing, this might sometimes be expressed as "the dimensions of the array are 10x10"
- even though the forces operate in 3 dimensions

[Bored chemist]

I think this is the sort of thing he's catastrophically failing to grasp.
https://www.varsitytutors.com/hotmath/hotmath_help/topics/domain-and-range-of-rational-functions#:~:text=Call%20now-,Domain%20and%20Range%20of%20Rational%20Functions,q(x)%E2%89%A00%20.