Happiness said:
Math in the sense of more than just calculation skills or techniques. Should a physicist be able to proof math theorems?
For example,
1. prove the chain rule in single-variable and multi-variable calculus
2. prove that any partition of a set is associated with exactly one equivalence relation and vice versa
3. prove that the well-ordering principle, the principle of finite induction and the principle of complete induction are equivalent
4. prove that ##\sqrt2## is an irrational number
(Roughly speaking, I've arranged the examples according to how closely related I think they are to physics.)
Hello Happiness, I am a physics major. I shall attempt to tackle your question for each numbered item on the list, and then in general.
For #1:
As a physics major, I tutor both the introductory Physics I class for engineers at my university (through my school's SI program), and the theoretical-based Calculus 1 courses. During one of my sessions in the calculus review sessions, I made a specific point to prove the Chain Rule because I found that much of the difficulty students displayed with not understanding what was going on, such as during differentiation rules and when performing maneuvers like "u-substitutions," stemmed from a poor grasp of the nested structure created by the chain rule. During these calculus review sessions, I occasionally make "plugs" into how the chain rule relates to known formulas regarding kinematic equations or harmonic motion, in order to prime them for their physics sequence (which the engineers typically take the semester after Calc 1 if they are on the standard track).
During my SI session with the physics classes, I find that a great deal of struggle with them understanding the "Why?" of what we are doing (and thereby their struggle with solving problems) corresponds to a lack of perception for how the physical ideas are being communicated through the mathematical substrate upon which these ideas are being communicated. The disconnect between what they do understand and what they are expected to understand seems to lie in how well we explain the derivations (in class, as well as in the textbook), and a great many of the derivations in physics involve the chain rule somewhere/somehow.
Since so many of the various formulas used in physics to solve introductory mechanics problems "pop-out" of Newton's Second Law, in order to have students understand how all of this makes sense/is even possible, they need both a good sense of algebraic shuffling (which they should already have) and a good sense for how the chain rule links together the hierarchy of interesting quantities obtained through summations (integrations) and "rates of accumulation/change" (differentiations). For example, in order for a student to more fully understand precisely
why a string acquires the amount of tension it does when you stretch it, or
why a block slows down as much as it does when friction is present, showing them how to calculate the integral that breaks-down the situation, rather than just showing them the memorizable-algebraic expression, goes a long way to produce and cement such understanding. Likewise, in order to understand
why a string's tension changes at a certain rate in proportional to its stretching, and
why the block moving across friction decelerates at the rate that it does, it is vastly helpful to have a great mental picture of the exact definition of a derivative. Then, when we start to add new levels of complexity to our physical problems, various parameters may be changing all at once (the spring force, the force due to air resistance, the force due to friction, etc. may all be changing at the same time!), and without a proper mental picture of the simultaneous action of partial derivatives, and how we describe those partial derivatives mathematically, it is difficult to really know what you are looking at. The total derivative of a system, which takes into account the action of all of the partial derivatives, is obtained using the chain rule.
And there is also the ever-important propagation of error, which requires an understanding of the chain rule to understand where the "error" is coming from.
#2:
When we study wave motion, we describe a set of frequencies built off of a "fundamental" frequency, which all together describe the possible standing waves characteristic of each of the normal modes. This is a denumerable set which can be partitioned into subsets, based upon qualities of interest. If we consider the musical chromatic scale, then each note represents an equivalence class, where the elements of the class are the doublings/halves of the "root note." When you describe constructive interference, you can obtain maxima by specifying integer multiples of a phase difference of 2 Pi, which can then be further broken down into the set of even integer multiples and odd integer multiples. Likewise, for destructive interference you can obtain minima by specifying
odd integer multiples of a phase difference of Pi. In quantum mechanics, we similarly use integer multiples to describe the class of energy levels. These are all examples of how sets are obtained in physics, and how from this follows that such sets can be partitioned. Proving theorems regarding equivalence relations and partitions of sets (or anything else about sets) helps prime you to be able to better understand both where these analyses came from, and how to elaborate upon them in the future for novel physical problems which might benefit from similar methods.
#3:
We want to know that our formula for constructive interference applies for both the cases where n=1, and for each n=k, where arbitrary k is an element of the integers such that k is greater than or equal to 2, as k approaches infinity. So we prove the case for arbitrary k+1 to show this. However, we are lucky that our recursion relationship is simple for these scenarios of interference. With more complicated recursion relationships, you may find that you need the strong principle of induction, which allows you to assume more information, and still acquire a rigorous proof. Understanding which type of induction you need, based on how complicated your recursion is, can help you know the best method to tackle your physical problem, and show that your result holds for all cases. The Well-Ordering Principle can be also used in "reverse" to perform a proof by minimum counter-example, which can sometimes be easier than regular induction.
#4
If you can understand how to compute one irrational number, you are better able to compute another irrational number. If you know how to prove why an irrational number is irrational (as opposed to rational), you are better able to understand why we would care to compute irrational numbers in the first place. Once you both know how to compute irrational numbers, and also care about their significance, you are better able to understand the fundamental constants of nature.
The fundamental constants appear everywhere in your calculations in physical problems. Best to appreciate what they stand for, and how we got them put there.
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In General:
In my university, the best "hard math" students graduating each year with mathematics degrees also tend to be physics double-majors.
As a physicist, it is never a bad thing to be able to speak the language of science (math) as fluently as possible. Like a good poet with the English language, we can only reorganize Math to tell a new story once we know that Math very very very well.
Math is our sandbox.
As I said: "This because of many reasons: (...) sometimes, to learn the actual math is to embark in a journey of pure math that will distract you from physics, your actual career, for which your boss pays you!"
. With god, you never know, he works in mysterious ways, they say.
