Proofs, derivations, or both? Feel I've learned math/physics wrong

[DrummingAtom]

I've recently come to the conclusion that I might have made some mistakes along the way. I'm going into my senior year of EE and something just doesn't feel right about my abilities. Over the last couple semesters, I've fallen into the "plug and chug" mode of solving problems. I have some issues finishing tests and exams so the faster I can perform the better in terms of grades but now I feel that I've developed some very bad habits.

I don't have any experiences with proofs. Also, I doubt I can derive basic physics equations such as the uniform circular motion equation. It's embarrassing to admit these things but I want to change and have no idea how now. I'm not sure if I should go back to the basics like trig and calc 1 and relearn those more rigorously or do some higher level physics to expose myself to more derivations. I have taken many math classes up to advanced calculus which was integral transforms/reynold's transport theorem stuff. But my physics classes stopped intro to E&M (physics 2). I have taken the upper level EE version of E&M but it was extremely plug and chug and I gained almost nothing from it.

Ultimately, I want to be able to think like a physicist and be able to model anything I encounter. I'm guessing I should focus on more physics but I just want some guidance. Thanks for any help.

 

[robphy]

What I think helped me a lot was taking the time to compose careful solutions to my homework problems... not just equations, but motivating comments... ideally starting from first principles.

What also helped is learning to distinguish between a "physics equal sign" (equality because of a law of physics or a definition) and a "math equal sign" (equality because of a mathematical operation). In other words, why is this expression equal to that expression?

I also try to visually compose the manipulations into chunks (...this first equation has a physics equal sign because of some law... this group of lines is an algebraic procedure to find x, etc...). This paper "Equation poems" http://dx.doi.org/10.1119/1.18149 was interesting. I also like to create a chain of equalities, clarifying what a particular equal sign means... so that if that equal sign is false for some reason [i.e. a special case doesn't apply], then the left-hand side is generally not equal to the right-hand side.


Along with this, it was helpful for me to be a tutor or TA... this helps to sharpen what you know and tells you what more you really need to know. One of the best ways to learn something is to teach it to somebody else!

 

[verty]

This may or may not help but I offer it only to be helpful. In my experience (not with physics), it helps to refer to the original author or authors. Read the original papers if possible, or contemporary accounts. The reason is that the original author is writing for an audience that does not yet understand and the author is in the best position to describe clearly the motivation of the new theory. It also helps to know what came before so you can contrast the new with the old.

 

[robphy]

This may or may not help but I offer it only to be helpful. In my experience (not with physics), it helps to refer to the original author or authors. Read the original papers if possible, or contemporary accounts. The reason is that the original author is writing for an audience that does not yet understand and the author is in the best position to describe clearly the motivation of the new theory. It also helps to know what came before so you can contrast the new with the old.

Yes, the original authors are helpful...

However, in physics, it is often the case that the authors themselves may be among those who do not understand the implications of the new theory. In addition, sometimes the authors' motivations are heavily influenced by the availability of particular experimental evidence or particular mathematical tools at the time. So, the later contributions of others may lead to a refinement of ideas that the original authors might not have considered [or maybe even agree with]. So, while I think it is enlightening(*) to read the original authors... one should place more weight on the contemporary accounts [which often stand on the shoulders of giants].

Sometimes, it's good to go back to the early authors because there may be other lines of thought that were pursued by them... but not continued on into the contemporary accounts. So, works of the original authors could enlighten(*) an additional line of reasoning that had been forgotten.

(I raise these points because of related discussions in the relativity forum.)

 

[WannabeNewton]

Hi DrummingAtom. So am I correct in assuming that you want a more theoretical/rigorous treatment of introductory mechanics and EM? Keep in mind that derivations in physics are different from proofs in mathematical texts. There's nothing wrong with doing proper math texts but if your main interest is in seeing more rigorous treatments of introductory mechanics and EM then learning proofs from e.g. Spivak's calculus text won't really help you much.

Anyways, as far as first year mechanics goes I would recommend "An Introduction to Mechanics" by Kleppner and Kolenkow: https://www.amazon.com/dp/0521198216/?tag=pfamazon01-20 and
"Introduction to Classical Mechanics: With Problems and Solutions" by Morin: https://www.amazon.com/dp/0521876222/?tag=pfamazon01-20

and for first year EM I would recommend the 3rd edition of Purcell's classic "Electricity and Magnetism": https://www.amazon.com/dp/1107014026/?tag=pfamazon01-20

 

[DrummingAtom]

Hi DrummingAtom. So am I correct in assuming that you want a more theoretical/rigorous treatment of introductory mechanics and EM? Keep in mind that derivations in physics are different from proofs in mathematical texts. There's nothing wrong with doing proper math texts but if your main interest is in seeing more rigorous treatments of introductory mechanics and EM then learning proofs from e.g. Spivak's calculus text won't really help you much.

Yeah, you're correct in that assumption. I feel like I'm missing out on proofs but I know it's not my end goal. I'll check out those books, thanks.

Also, robphy, I'll check out that article when I get back to school so it's free, hopefully. Thanks for the reply.

 

[Simon Bridge]

@wannabeNewton: that was my impression from the first post too: engineering (if I have this right) tends to be results oriented rather than proofs oriented ...

@DrummingAtom: it is not unusual for someone to change emphasis in their education sometime into a particular course. I just want to (re)assure you there.

The trick is to work out if what is needed is supplimental material or a complete change of direction. I'd go along with those references as "supplimental material" (don't change your major).

Formal mathematical proofs are not usually all that important in physics - we tend to trust the mathematicians to get it right - though when something is an important tool the student usually gets a quickie proof or at least a demonstration about why it is reasonable to expect it to work: so there are few "magic wand" answers. OTOH: students are expected to gain the ability to find their own proofs.

There is some caution though ...
I remember a prof using Fermat's last theorem to show something (forget exactly what) without naming it, then leaving the proof of the theorem "as an exercize for the student ... bonus marks!" Fzzl kngl fxbt! TBF he did warn us that it was "quite difficult" >:<

It's been proved now. It's not the sort of proof that would make it into a physics course :)
But there are lots of things in physics that are just true.

 

[jasonRF]

I understand where you are coming from. As an EE myself, I too found that to perform well on exams (which tend to be time limited) the ability to do computations quickly with no errors ends up having a large impact on grades. Like you, for some courses (especially those that I regretted taking half way through the semester) I would revert to plug and chug mode. However, I found that for at least a couple of classes each semester I would be able to make the time to get a better understanding. My suggestion is to work through the derivations that are presented in the book/lecture, and try to understand the approach (why it works) as well as the mechanics of the derivations. The approach is often the most important, but the mechanics will sometimes teach you useful tricks.

For an example from EM: understanding the basic principle used to derive Snell's law and reflection coefficients (boundary conditions!), and then working through the algebra yourself is worthwhile. Then, allow there to be a conductivity (and assume Ohm's law: J = sigma E) and derive it again. Plot what the reflection coefficients look like for various materials, etc. This same approach applies to signals and systems, electronics, etc., not just "physics-like" courses. I find that plotting results that I derive helps gain intuition sometimes, while other times examining the result to see how to make sense of it analytically can help. EDIT: also, take limiting cases to see how things behave in the extreme: dielectric constant -> .infinity, etc. Taking the time to think about results after you answer the homework problem, and examine what they mean and what implications they have is also a very worthwhile habit to get into. Even spending a handful of extra minutes per problem can sometimes help in learning.

cheers,

jason

 

[afreiden]

What I think helped me a lot was taking the time to compose careful solutions to my homework problems... not just equations, but motivating comments... ideally starting from first principles.

What also helped is learning to distinguish between a "physics equal sign" (equality because of a law of physics or a definition) and a "math equal sign" (equality because of a mathematical operation). In other words, why is this expression equal to that expression?

I also try to visually compose the manipulations into chunks (...this first equation has a physics equal sign because of some law... this group of lines is an algebraic procedure to find x, etc...). This paper "Equation poems" http://dx.doi.org/10.1119/1.18149 was interesting. I also like to create a chain of equalities, clarifying what a particular equal sign means... so that if that equal sign is false for some reason [i.e. a special case doesn't apply], then the left-hand side is generally not equal to the right-hand side.


Along with this, it was helpful for me to be a tutor or TA... this helps to sharpen what you know and tells you what more you really need to know. One of the best ways to learn something is to teach it to somebody else!

Excellent advice and very interesting paper, thanks