# Application of Calculus in Physics

Why do we apply Calculus in Physics when most of the quantities are not continuous and are not symmetrical at all levels of magnification? Aren't most, if not all, forms of Matter and Energy discrete? We talk about differential elements of energy, charge, fields, current, liquids and so many other quantities which don't or can't exist and we derive results from such assumptions. How is this right?

• Two things: the first is that the calculus you see in introductory physics is a model that only works for macroscopic systems, i.e. it breaks down under certain 'levels of magnification'. That's why we have classical and quantum mechanics. Second, energy is not discrete, i.e. the energy of a photon depends on its frequency, which last time I checked is continuous.
– Greg
Jun 3 '13 at 12:43
• I disagree with the assertion that "most of the quantities are not continuous." There are a large number of continuous quantities in classical, quantum, and relativistic physics. As Greg said, energy is just one such quantity, but so are position/length/volume, and time. Wave equations are also calculated using calculus. Jun 3 '13 at 12:49
• It would be equally valid to ask why we use the real number system to do physics, since it contains features such as the distinction between rational and irrational numbers that can never be relevant for physical measurements. The answer is that we're making models, and we want the models to be convenient to work with.
– user4552
Jun 3 '13 at 14:49

While it is true that some parts of physics are discrete; matter, energy, charge, etc. There are many others that are continuous. Distances, time, temperature, probability, and angles are just a few examples of continuous quantities. In addition, even though things such as Energy are "discrete" for one system, the smallest unit of Energy is not the same for everything. When you have systems of many different components that all have different smallest units of energy, your possible values for energy become continuous or a close approximation of it. Furthermore, even if we consider a single type of particle with a set smallest (ground state) energy, at the macroscopic scale and/or at extremely high energy levels, the distance between each energy state approaches 0, which means it is for all intents and purposes continuous.

But don't let me give you the impression that we physicists just go throwing calculus around willy-nilly. We only use it when there is a reason too. If we know something is discrete, we can't just integrate over it. If we have a discrete function (let's say of energy) we might try something like this:

$$\sum_{i=0}^{N-1} f(E_i) (E_{i+1}-E_i)$$

And this is how it will stay. However, if (as in most cases) we find that N approaches $\infty$ and $(E_{i+1}-E_i)=\Delta E_i$ approaches $0$, then our sum becomes:

$$\int_{E_1}^{E_2}f(E) dE$$

But back on topic, we never "assume" values are continuous. We always are able to state why we use calculus; why we know things are continuous. I understand that from a non-physicist perspective, much of the math we do and things we claim can be frustrating and seemingly outrageous, egregious, and/or ignorant at times. Believe me, sometimes I look at other physicists work and I think the same thing. But what you should do is learn how they reached that result, go through their work if you can. More often than not, you'll find yourself agreeing with the way we do things. Calculus is a very useful tool that we find applies and correctly explains many aspects of how things work. While it may seem contradictory to an outsider that we claim some things are sometimes discrete and then we claim there are differentials of them, we still do have very good reasons for all of it. And, we would be willing to explain specific examples. But in the end, can you really argue with the results?

First, I'm not aware that 'all forms that matter and energy', whatever that means, are discrete.

To begin with, Energy is a conserved quantity that comes from time translation invariance of physical systems, it's not 'something' nor can 'it' have any 'form', if you refer to things like kinectic vs potential energy, you can say it's a convenient division, though the most important concept is total energy. Anyways, we can speak of dispersion relation of physical systems, i.e., the relation between velocity and kinectic energy, as far as most physical theories go, for free particles, they are exactly continuous.

Besides, many (read MOST) quantities that I'm aware of are continuous, at least most of the time, in the majority of physical systems. Think on position, time and velocity, can you think in any good reason for any of them to be discrete?

Finally, when we speak of physical systems, we must also speak of the models we use to understand them. Most of the time we want to model things that for us are because:

1) Either they are continuous by default, that's the case of kinematic quantities mentioned above. There are situations that they may turn not to be, but you don't assume beforehand that they aren't.

2) They seem continuous because we want to see them in a scale that we can't really distinguish the one unit of that thing. It's the case of particle number when we talk of a classical gas.

When we speak of quantum mechanical systems, we don't don't force that neither quantity is discrete a priori. We allow everything to be 'compatible' and calculate the specific case that we want. When we try to deal with bound states, we find out that in that case, we have a discrete energy spectrum.

Still on quantum mechanics, we don't have any problem to speak of wave functions in terms of 'quasi-continuous' fields.

In any case, even on quantum mechanics, the discretization of energy for bound states comes from solving an PDE, and thus, utterly, comes form differential calculus.

About the validity of treating something as a continuous quantity or not. When you create a theory in physics, you need a mathematical structure that will represent your physical system and a 'dictionary' that will translate the predictions of your maths into things that you can verify on your laboratory.

The first step is (or should be) to create a mathematical framework that is consistent, i.e., everything is 'correctly defined' and there is no (mathematical) paradox lurking behind the scenes.

If you pass this step, you need to confront your predictions with experiment. Most of the successfull physical theories have, at least, some regime where they agree with many experiments, and other regimes where they fail, either miserably or not. And in a third case, there is always a class of experiments that they simply don't know how to deal with.

Examples:

1) Classical Mechanics of Particles and Macroscopic Bodies: Most of your daily objects a fairly well described by this kind of approach. This kind of reasoning fails as soon as you go to very small scales (< 1 micrometer), very large scales (> planetary) or very fast speeds (> 1% of c).

2) General Relativity: As far as I know, it's the best theory of gravitation that we have today. It describes fantastically well gravity in planetary and solar system scale. There is no verifiable prediction in very small (someone correct me with I'm wrong), i.e., ~ micrometer scales and below, and struggles to stay by itself in very large scales, galactic and cosmological scales, where you need to include extra ingredients like dark matter and dark energy to fit the observational data.

3) Quantum Mechanics : Works very well in ~ nm scale, and have good prediction in upper scales, though not everything on the quantum to classical transition is completely clear. It simply doesn't have any prediction of gravity (or any other interaction btw) and so, you can't use QM to predict, for example, the behavior of the planets. Still, if you feed it with coulumb interaction, you will have a good agreement in atomic and molecular scales.

Finally and most important, all these theories use calculus, and many other mathematical tools, intrinsically and throughout all the treatment that they do.

So, it's not because some quantities are not continuous that we can't apply calculus to them, it's just we can't naively apply it.

edit: correcting typos