Since I was tiny and not bald as now, in my old physics classes in exercises some simplifications that those children who were not able to abstract enough, put them not only very nervous but it was one of the issues were made fundamental physics because they did not like.
Surely many as I said before, you have heard this example: “Let a cow round of radius R and mass m”. And surely ye have said, “cows are not round, but have a very complex way.”
And you are right, the cows are very complex so with what, physically, if you want to make a calculation on it (not the density, for example) calculations can complicate everything we want if, in the first instance, have to adapt to the different shapes that form the cow and, second, knowing that the cow is not uniformly dense (has different densities because inside has different components).
So, surely you thought that calculations are performed may be not very real.
If we wanted to be “realistic” should take different parts of the cow, reduce them to something whose form (and therefore volume) know mathematically calculate the differential equation of the mass of each of the pieces (so you do not have the same density) and integrate in order to know a “more accurate” estimate. One thing that if the traditional way (spherical cow) is simple, the complicated form need a computing cluster for work.
There are two important points here to see how this is done and why. One is a simple form of physical work and another is a mere mathematical help.
Physically limits is necessary. When any theory or hypothesis arises, at first it arises generically. In our example (cow), one would think as I have done in the end, reality and would try to raise the equations to solve the problem. The equations that arise are a total which is observed in everything that can influence our development and gets into the garlic.
That is, in our example would come from outside pressure (with their variances over time and its difference depending on the position), by Archimedes thrust suffering cow, electromagnetic radiation received by the cow (if influences) … that I have besides the calculation of the shape and mass as described above. Which would give us an equation “king size” that only the bravest would want to solve.
In a second step, the above equation, to develop the terms influencing little or nothing would be eliminated. Known “despicable”. An arduous task because there is great discussion there is to see and know that spare and not because the term “influence” or “despise” is very relative. And it is that, many of the terms described above are relative to other terms or equations may even cancel each other (eg pressure and thrust).
This is the limits of the equation. That is, the city is to take the theory or complete hypothesis, calculate all possible (in my career they called the delta motion) and limit equations to what actually happens (if, I repeat limit).
Normally, limitations, theory or hypothesis evolves, there are new, they are simplified and improved. This is usually, usually because when “inserted” all equations are singularities observed. That is, when played with all the equations you can see where the equations are removed and therefore where there are singularities (eg the old division by zero or places, areas where results are infinite). For example, string theory was born the theory of quantum gravity loop to try to resolve the singularity when the big bang. And I tell you this because I want you to see the relationship between “cow” and more modern things.
And mathematics?. Well, physics and mathematics are closely related. In fact, mathematics is the language of the universe and all self-respecting physicist who has to handle them by heart as mathematics, physicists interpret and draw new conclusions that are part of new theories.
And what has this to do with the cow ?. In this example (and others, of course), mathematics, there are things called conformal transformations (conformal map). Mathematically (and simplified) is a conformal transformation function which preserves angles. That is, if we know that a function is as a machine having an inlet and an outlet making some modifications, we can have functions that transform the (two-dimensional, three-dimensional) space from one form to another.
Conformal transformations have their properties apart from that, mathematically, preserve angles. I will not enter if they are holomorphic, differentiable and others, but retaining properties within and retained when reversed, ie when the machine backwards functions used. Come on, that the results are the same when done on a site (before transformation) and the other (after transform) and, moreover, if we do things later, worth the same as before.
Then think of, it is easier to get the volume of a cow, get all your pieces or by a conformal transformation realize that it is a sphere of radius R, calculate, for example, density?.
Man, the easier it is to think that the cow is a sphere because we used a conformal transformation knowing that the density thus calculated is as good as taking the cow pieces. Whereupon so that we will make life difficult if mathematically simplifying get the same result.