Application of calculus to physics

I am currently a 3rd year undergraduate electronic engineering student. I have completed a course in dynamics, calculus I, calculus II and calculus III.

I am very interested in mathematics and physics, however there is some clarification that I need regarding these matters as I am confused as to why we study functions, calculus and physics.

I know that mathematics is required in order to describe physics, but in many cases we require a function that describes the phenomena at hand. The major question I have is, how do we know what these functions are? I know that we make use of differential equations in order to obtain a function that represents the desired system, but this is not always the case.

For the sake of clarity I will provide a few examples that I have in mind:

(a) Take for instance Maxwell's equations, in order to apply Faraday's law we need the function describing the electric field, for Gauss' law we need a function for the magnetic flux density &/or electric displacement field etc.

(b) For Green's theorem, if we wanted to determine the flux of a vector force field within a given region we would first need the vector function of this force field.

(c) In the subject of sequences and series, how do we determine an explicit formula for the pattern describing these discrete systems, if there exists any at all and why would we need this?

In real life how do we find these functions on our own? In a calculus and physics class, we are given these functions and are required to compute the quantities at question using the appropriate theorems/formulas.

The thought of these subjects requiring us to poses a function, which we in real life do not know is causing me discomfort as I am uneasy with the idea that we first need to obtain a mathematical object, which is non trivial - we can't see a magnetic field nor can we see the function that governs it - in order to make use of these beautiful theorems. This is discouraging me from wanting to pursue further studies in the field of mathematics and physics. Where's the machine which generates these function that I need? :(

• This is not so much electromagnetism, as the tags say, but rather a general question about how to find connections in physics. – Steeven Sep 7 '16 at 13:09
• It takes experience with several different types of problems to know how to mathematically model them. Even in the case of having experience, there are some phenomena that can be modeled equally effectively with more than one approach. – David White Sep 7 '16 at 23:40

You are very right; expressions, formulae and laws are not given. We can derive many equations from others, but the fundamental ones - like the laws of nature - are not derived, but found or formulated.

This is the bread of a physicist.

They look out into the world and try to see patterns. When something could look like a pattern, it is tried in numerous experiments. If all results show the same pattern - then they believe this pattern to be true.

Consider Newton's 2nd law for example. $$\sum \vec F=m\vec a$$

Someone (Newton) got the idea or observed that when something changes it's velocity (accelerates), it requires an "effort". This "effort" could we then call force and give a symbol $F$. By doing many experiments, there seems to appear a pattern - a connection between acceleration and force. If you push on an object, it accelerates, but if you push double as hard, it accelerates double as much!

Okay, so they are proportional, $\vec F \propto \vec a$. The proportionality constant is then something that causes more or less acceleration for the same force. It is some kind of a "resistance" against acceleration. Let's call that inertia or mass.

Newton didn't think exactly like this; he actually made his law as the version $\sum \vec F=\frac{d\vec p}{d t}$. But I hope this illustrates the simple one-step-at-a-time way of thinking when finding connections and patterns that will end out as laws and formulae.

Same method goes for other formulae, like for instance Coulombs law: $$F=k\frac{q_1q_2}{r^2}$$ He had to first find out that there are two kinds of charges. That would require many tests, and eventually he would believe that there are no others, because all charges he could find behaved like one of the two. He could then call them positive and negative, just for having names for them.

He could see that they attract or repel and he would then, as Newton, have to find out how large this attracting or repelling force is compared to the two charges. He might eventually have found out that the force is proportional to each of the charges but reversely proportional to the square of the distance. And then he is close to writing it as math.

Math is here just a tool that makes it much easier to talk about the physical world. The tools that mathematicians find/develop can namely then be used on the physical formulae, when these are written as mathematical expressions.

In nature we can observe phenomena and measure quantities: I can measure the magnetic field of an object at different points in space and if I wait a bit also in time. With such an experiment I get a discrete set of points and now we want to understand, explain and eventually predict this distribution.

In order to do so I need a theory for the given phenomena. For electro magnetism that would be Maxwell's theorie and his equations for electro magnetism. He developed this formulars in order to describe the phenomenon of electro magnetism which we can observe in the experiment. The goal of a theory is to describe nature as we see it using as much symmetries and as less degrees of freedom as possible. We know today that in order to describe elector magnetism as compact as possible that the magnetic field (3 components) and the electric field (3 components) are not the best way to do so, because this six components are not indipendent. They may be accessable in the experiment but behind this 6 components lies a deeper meaning: We can descibe this 6 components using only 4 indipendent components (scalar+vector potential) using maxwells equations. So introducing more abstract objects or if you want to call them functions makes describing nature, as we see it, easier.

The aim of theoretical physics is to describe nature mathematically as "easy"/compact as possible making extensive use of the obvious and at first sight hidden symmetries of nature. So this sometimes means introducing new quantities which are not directly accessable in experiment. But it makes solving problems easier and sometimes it is even the only way to solve some problems.

I can give you another nice, but mathematical, example that sometimes using more abstract objects is even a necessity to solve "real life" problems. Imagine a polynomial of order 3: This is an object I can imagine in nature; I can draw its graph on a real sheet of paper. Now I want to find its roots. On a sheet of paper in "nature" I can find those just by looking on my drawn graph. But now if I want to calculate them using math I will come across a problem: It is impossible to find the roots of an abitrary polynomial of order 3 using the real numbers. During my calculations I will come across $...\sqrt{-1}...$ and now I have a problem. I do not know what to do with that this expression $\sqrt{-1}$ is not something I can see and construct on a sheet of paper. The only way to sovle my problem is to find a mathematical formalism to deal with such an expression. I need to introduce complex numbers to solve a problem based on real numbers: My polynomial is defined on the real numbers and has real roots but in order to find those I need to work with complex numbers.

There are countless examples of that in physics to. Quantum mechanics (QM) would be one for example: In QM we calculate most of the time using abstract objects which can not be measured in nature but at the end we can use those objects to predict nature and our experiments.