Is Understanding the Derivation of Formulas Crucial in Math and Physics?

[erok81]

Sorry about that title, I had zero ideas what to put there.

How important is it to learn the backgrounds/how the formulas came about in math and physics? Obviously you need a general understanding, but does one really need to know exactly how these formulas came about/work? Since that doesn't make too make much sense, here are two examples.

Math: We are doing centers of mass in calculus II. The teacher spent about 1/3 of the class explaining the formulas before he even got to writing them on the board. Taking averages of the f(x) + g(x) doing this and that until eventually arriving at the actual formula.

Physics: Conservation of momentum. We spent around 20 minutes going through derivatives, expanding stuff, solving for different things, etc etc until arriving at (for example) m1vi + m2vi = m1vf + m2vf.

It is much easier for me just to remember the formulas. I suck at memorizing but formulas I can remember without much work.

Is this going to hurt me in the long run by doing this or is it ok to vaguely remember how the formulas are derived but just stick to the actual formulas?


p.s. If this in the wrong spot, sorry. It fits about four sections but not completely in any.

 

[dx]

Is this going to hurt me in the long run by doing this or is it ok to vaguely remember how the formulas are derived but just stick to the actual formulas?

Depends. What do you want to do in the long run? You certainly can't be a physicist or a mathematician by memorizing formulas.

 

[pallidin]

Yeah, and I can only relate to you my experience as a certified computer tech.
By understanding the intimate foundations of "how things work in a computer" I find myself vastly more prepared to inspect/diagnose/correct a problem.

Not sure if this type of educational approach is necessary in the fields you mentioned, but I would assume it would be helpful, and, if you desired to be a "theorist" or "anaylist" in those fields I would assume it to be essential.

Just my thoughts...

 

[erok81]

Better example!

Well I understand how they work and when he does them on the board I have no problem following along and can sometimes predict the end result. But on my own I couldn't do it.

However, another example looking at my notes regarding the above physics example. He took the kinematic equation vf^2=vi^2 +2as and Newton's second law, f=ma, then finally arrived at KE=1/2mv^2 only using those two formulas.

That is a perfect example. I understand how he did that, but it took an entire pages of notes to get there. I am quite certain if I was only provided Newton's Law and that single kinematics equation, I could not end up with the KE formula.

So what you are saying, if one wants to be a physicist or mathematician, they should be able to calculate all of those steps in the above example?

 

[dx]

So what you are saying, if one wants to be a physicist or mathematician, they should be able to calculate all of those steps in the above example?

Definitely. Any result in basic Newtonian mechanics you should be able to prove for yourself. Of course you won't be able to do it so easily when you are learning it for the first time, but that should be the goal if you intend to be a physicist. It is not about remembering each of the steps of a derivation, but to understand the logic and reasoning well enough that, given any result, you should be able to reconstruct its derivation. If you can't see how to do this, you don't understand the result.

 

[pallidin]

Another thing... stay connected through reputable forums!
Whereas in my computer work I use Experts-Exchange for difficult computer problems, with math and physics I would recommend where you are here right now!

The one thing I love about both EE and PF is the caliber of professionalism. Sure, there are "crackpots" in any forum, but I find these two to be rich in very good info.

Regarding math and physics, my opinion is that PF is your destination for seriously competent info.
My experience with PF is that if you take the time to word your question as best you can you will get an answer! Sure, you might have to clarify your question, but they are here to help on a pro-level!

Continue your studies, and PLEASE, keep asking questions.

 

[Amanheis]

Is this going to hurt me in the long run by doing this or is it ok to vaguely remember how the formulas are derived but just stick to the actual formulas?

Short answer: Honestly, I have to say "no".

Long answer: When I was in high school we were allowed to have a little book with formulas, which makes sense since I don't think you have to know every single formula by heart. But the downside was, I was able to do pretty well just by looking at what kind of formula fits the problem, sometimes just by matching the variables in the problem, mixing them around a bit, do some simple algebra until I got x=...

Now I am a physics grad student and I'm still doing pretty well. Of course now I have a rough idea of how those formulas came to be, and if you gave me a couple minutes by myself I could probably come up with some sort of derivation for most, if not all of them.
But I often had to look up the Maxwell equations or the Schroedinger equation... was the left hand side now -i*hbar*dPsi/dt or was there a +? Was it 16*pi*G in the Einstein equation or 8*pi*G? Stuff like that I have a hard time to remember.
And after all, in practice, all you need most of the time is the actual formula, not its derivation. Doesn't mean you shouldn't care about the derivation, but vaguely remembering how it goes should be enough if you ask me. But only if you then are passionate and/or competent enough that you actually can come up with the derivation if you think about it long enough! It also gets easier after some time, if you have to go through hundreds of exams, some of them even oral where the above method doesn't work at all and you can't use any kind of cheat sheet, and eventually having to teach all that crap to undergrads.

 

[erok81]

Thanks for the advice everyone. I'll start paying more attention to those and then trying to work them after class. Maybe it will be a good skill to have in case one day I don't have a formula and will have to put one together.

And like Amanheis said, usually if you have the formulas on a piece of paper with you, usually it's just a matter of finding one with the right varibles, solving for your variable, and plugging stuff in. Not the hardest thing to do...at least at my level.

So here is my next question (see what you've done pallidin ) are there ever any classes or homework sets you come along during your education to actually learn and practice this stuff? That would help me a ton as right now we do it once in class and move on. Therefore it's up to me to pull out my notebook and practice that one derivation. If I had a few simliar sets of those to practice, it would go a long way.

 

[Amanheis]

Yes, it is very common in college that the teacher is leaving the derivation of some formula out, along with the words "this is going to be an exercise on your next set of homework". Of course this is usually not going to happen with the big, central results of a theory, but some minor corollary maybe. And often you are given some guideline on how to derive the formula.

 

[minger]

Is this going to hurt me in the long run by doing this or is it ok to vaguely remember how the formulas are derived but just stick to the actual formulas?

As mentioned, it really depends on what you're doing. My girlfriend is a middle school math teacher and she recently covered a quick intro to trigonometry. She said how she was explaining how they apply to right triangles and how you can find angles with the functions.

My first question was, "Well did you teach them where they come from? Did you mention the unit circle and the relationship between sine and cosine?...(and I rambled for another 10 minutes).

Basically she just said that for what they were doing, they didn't need to know the background, where they came from. It would have been over their head anyways.

I mean, it really depends. I don't use any equation that I cannot derive. However, my line of work often times involves making adjustments to these equations and understanding completely how they work is critical.

 

[bp_psy]

So here is my next question (see what you've done pallidin ) are there ever any classes or homework sets you come along during your education to actually learn and practice this stuff?

Well, if you are a English major then the answer is no. If you are a physics major then more then half the problems in you problem sets will be derivation type problems.

 

[erok81]

I am a physics major, so that's good to know.

I talked to my professor about it as well. He said he won't ever test on it, but I told him to keep explaining them on the board so at least that way I can go home and try to learn them on my own.

Maybe I'll try to find some sites that have practice problems as well.