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Once we study vector calculus, we are all set to describe physical fields such as velocity fields , electromagnetic fields etc. The study makes sense to me mathematically but I can not imagine how we would measure such quantities. How exactly do you continuously assign a number to each point in space? The amount of measurements would be way too much.

As a note, though I started with mathematics I expect an answer which describes the experimental procedures which are done.

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I'm sure you are familiar with both ways, so I'll just point out the obvious (hoping not to insult your intelligence). There are many approaches, but I'll limit it to the few that I use more often. Namely:

  1. least squares
  2. Kernel density estimation
  • 1 bonus for completely deterministic system

I'll give it a try with a 1D distribution function, but you can easily extend to more dimensions.

Least Squares

So assume you have the following set of data.

enter image description here

Obviously there is trend there (and I will assume it follows a second order from my theoretical model). Then if I fit my data with a polynomial of second order I would get the following distribution.

enter image description here

Kernel density estimation/histogram

The next case is when you have a lot of data which you gather and you want to get a distribution. The following is from a dataset with Earthquake Magnitudes above 4 Richter (quakes) around Fiji since 1964.

enter image description here

If you are interested in the density distribution, then you would create either a histogram or a kernel density estimator (kde), and you would plot the Earthquake magnitudes on the horizontal axis and probability/probability density on the vertical.

enter image description here

Bonus: Interpolation

This is probably not likely the one that you would expect to use often when doing an experiment. Still, it has its uses. To use it you should get everytime the same results for X. The calculation is also trivial and has many variation (linear interpolation, log interpolation etc).

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  • $\begingroup$ So, if I were to break it down.. you are saying to sample points and find the best fit polynomial between the points? $\endgroup$
    – user28616
    Commented Sep 30, 2020 at 20:21
  • $\begingroup$ you start with a theory. The theory assumes some function of variables. Then you conduct an experiment, measure the value at some points, and based on the model you have you fit the data according to the function the theory imposes. The point is to validate the theory. $\endgroup$
    – NMech
    Commented Sep 30, 2020 at 20:38
  • $\begingroup$ I just hope I understood your initial question correctly. What I understood about your main question, was that you were worried that when measuring a field you'd have infinite number of points to measure and exhausting that would make it impossible. $\endgroup$
    – NMech
    Commented Oct 1, 2020 at 7:53

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