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Physics students and proof based calculus

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Main Question or Discussion Point

Hey, I have been told to study calculus following Spivak's book.

I was in an Engineering program and I have moved to a Physics one, and I want to retake calculus to really get good at it.

The problem is, Spivak's seems to me like it's very proof based, and I'm having a hard time even with the first problems with elemental numbers and basic proofs. Even when I give up and look them up online, they don't make sense to me, as in why should someone even need to prove that a/b * c/d = (ab)/(cd)

Should I keep going with this book or maybe you know something more appropiate for a physics student?

To be honest I just need a reason to keep going through this book, because I would love to get a deeper understanding of calculus.
 

Answers and Replies

  • #2
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The main reason to prove the obvious is not to prove the obvious, but to sharpen the mind to recognize what is actually needed and used. This might look silly with such an easy example, but can save lives in a structural analysis. One of my favorite quotes is ".. and as the likelihood isn't always on the side of truth, it happened ..." It says that appearance can be deceptive.

You're right that this habit is probably the most distinctive property of mathematics compared with physics and engineering. It is because of the fact, that mathematics always wants to get the most general result, which requires to use only the absolute minimum of assumptions. If you only prove your formulas for flat, Euclidean vector spaces, you might get into trouble the moment your roof looks like an hyperboloid. It's easy to understand, that it will make a difference, whether this roof is build from steel or wood. So whether a function is differentiable or just continuous is merely the mathematical version of it. Don't apply formulas for differentiable functions on continuous functions, only because they look differentiable.

I recently was on the search for good math problems and found a question about vector fields in an engineering exam! One of the last places I had searched for, but if you think about this roof, it makes absolute sense. In this respect it is important to know, when a result can be applied and when not. Your example is only a warm-up. Those proofs help to sharpen your mind to distinguish between necessary and sufficient conditions. No harm will be done, if you fail in a proof, but the practice can once be valuable with a roof.
 
  • #3
PeroK
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Hey, I have been told to study calculus following Spivak's book.

I was in an Engineering program and I have moved to a Physics one, and I want to retake calculus to really get good at it.

The problem is, Spivak's seems to me like it's very proof based, and I'm having a hard time even with the first problems with elemental numbers and basic proofs. Even when I give up and look them up online, they don't make sense to me, as in why should someone even need to prove that a/b * c/d = (ab)/(cd)

Should I keep going with this book or maybe you know something more appropiate for a physics student?

To be honest I just need a reason to keep going through this book, because I would love to get a deeper understanding of calculus.
If ##a,b,c,d## are invertible matrices then it's not true.

One reason to keep going is that the mathematics underlying QM in particular is proof-based. The uncertainty principle, for example, can be proved from the axioms of QM.
 
  • #4
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The main reason to prove the obvious is not to prove the obvious, but to sharpen the mind to recognize what is actually needed and used. This might look silly with such an easy example, but can save lives in a structural analysis. One of my favorite quotes is ".. and as the likelihood isn't always on the side of truth, it happened ..." It says that appearance can be deceptive.

You're right that this habit is probably the most distinctive property of mathematics compared with physics and engineering. It is because of the fact, that mathematics always wants to get the most general result, which requires to use only the absolute minimum of assumptions. If you only prove your formulas for flat, Euclidean vector spaces, you might get into trouble the moment your roof looks like an hyperboloid. It's easy to understand, that it will make a difference, whether this roof is build from steel or wood. So whether a function is differentiable or just continuous is merely the mathematical version of it. Don't apply formulas for differentiable functions on continuous functions, only because they look differentiable.

I recently was on the search for good math problems and found a question about vector fields in an engineering exam! One of the last places I had searched for, but if you think about this roof, it makes absolute sense. In this respect it is important to know, when a result can be applied and when not. Your example is only a warm-up. Those proofs help to sharpen your mind to distinguish between necessary and sufficient conditions. No harm will be done, if you fail in a proof, but the practice can once be valuable with a roof.

This is a very good answer and a very good reason to stay with this book.

I particularly liked "No harm will be done, if you fail in a proof, but the practice can once be valuable with a roof."

Thanks a lot.
 
  • #5
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If ##a,b,c,d## are invertible matrices then it's not true.

One reason to keep going is that the mathematics underlying QM in particular is proof-based. The uncertainty principle, for example, can be proved from the axioms of QM.
It is great to know that I will work with mathematical proofs so that this will actually be helpful.

Oh I didn't even think about matrices. That's also a good point.

Thanks for the answer and for giving me a practical example.
 
  • #6
Stephen Tashi
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why should someone even need to prove that a/b * c/d = (ab)/(cd)
Did you mean ##\frac{a}{b}* \frac{c}{d} = \frac{ac}{bd}##?

Should I keep going with this book or maybe you know something more appropiate for a physics student?
To get good advice, you should explain more.

1) What are the defects in your current understanding of calculus? (Give some examples.) What textbook did you use when you took calculus?

2) Do you envision yourself studying theoretical physics or something more applied and experimental?
 
  • #7
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Did you mean ab∗cd=acbdab∗cd=acbd\frac{a}{b}* \frac{c}{d} = \frac{ac}{bd}?
Yeah that's what I mean, apparently I had to do something like

(a*b^-1) * (c*d^-1) = (a*c) * (b*d)^-1 = (ac)/(bd)

I don't know how to format the way you do it though, sorry about it.

1) What are the defects in your current understanding of calculus? (Give some examples.) What textbook did you use when you took calculus?
I followed my professors notes, which were not very in depth. I'm usually good with the computation side of calculus, so I did pretty well in my engineering exams, where our only "theoretical" questions were about proving things related to infinite series (convergence, divergence...).

So basically I feel like I suck at proofs. I understand them once I see them, but usually I'm not able to work them out by myself. Maybe it's just lack of practice, but it's getting on my nerves with this book.

2) Do you envision yourself studying theoretical physics or something more applied and experimental?
I'm headed towards experimental physics. But I haven't yet gotten my degree so I really can't choose yet, maybe I will find something within theoretical physics that I love.

But for the time being my head is on the experimental side.
 
  • #8
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I don't know how to format the way you do it though, sorry about it.
Have a look: https://www.physicsforums.com/help/latexhelp/
(a*b^-1) * (c*d^-1) = (a*c) * (b*d)^-1 = (ac)/(bd)
Yes, because that is always at the beginning of any problem - math or not: What does it mean?
And then you will find, that ##\frac{}{b}## needs an explanation. It is actually not defined at all. It is merely an abbreviation for ##b^{-1}## since the only defined operation (where the formula makes sense) are multiplicative groups, or rings which also have an addition. But nowhere is anything about division. Division is actually the multiplication by a multiplicative inverse element. So we have to reduce the problem to multiplication in the first place.

That also holds in physics. You can see this e.g. here on PF were the first question of an answer is often: In which reference frame? Unfortunately people tend to think everybody else is part of their local world of thoughts. They are not. So to request proper definitions, frames, additional conditions or whatever is a crucial part of every argumentation - or at least it should be.
 
  • #9
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Have a look: https://www.physicsforums.com/help/latexhelp/

Yes, because that is always at the beginning of any problem - math or not: What does it mean?
And then you will find, that ##\frac{}{b}## needs an explanation. It is actually not defined at all. It is merely an abbreviation for ##b^{-1}## since the only defined operation (where the formula makes sense) are multiplicative groups, or rings which also have an addition. But nowhere is anything about division. Division is actually the multiplication by a multiplicative inverse element. So we have to reduce the problem to multiplication in the first place.

That also holds in physics. You can see this e.g. here on PF were the first question of an answer is often: In which reference frame? Unfortunately people tend to think everybody else is part of their local world of thoughts. They are not. So to request proper definitions, frames, additional conditions or whatever is a crucial part of every argumentation - or at least it should be.

I think part of the problem might be that we only know the basics. We have only worked with one frame.

For example, I don't know what you mean by "rings".

It wasn't until I was introduced to matrices that I worked with something new, with a new basic set of rules to learn.

I will take a look at that link and thanks for the insight.
 
  • #10
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You may need to learn formal logic first.

There is a free book, that I like:

Levin: Discrete Mathematics. Download the 2nd edition free (openstax book). Go over the logic sections and proof technique chapters.
 
  • #11
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Spivak is a garbage calculus book, I think. Just read stewart and move onto elementary real analysis. I mean I would even suggest fitzpatricks book over Spivak. The problem with Spivak is it targets a group which makes no sense. It assumes you don't know calculus but it assumes you understand basic proofs when it's usually the opposite.
 
  • #12
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Spivak is a garbage calculus book, I think. Just read stewart and move onto elementary real analysis. I mean I would even suggest fitzpatricks book over Spivak. The problem with Spivak is it targets a group which makes no sense. It assumes you don't know calculus but it assumes you understand basic proofs when it's usually the opposite.
The book is not garbage. It's target is mathematics majors. Stewart is aimed towards engineering majors, two different groups...
 
  • #13
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The book is not garbage. It's target is mathematics majors. Stewart is aimed towards engineering majors, two different groups...
Tell me under what condition it makes sense to use spivaks book, particularly after having taken calculus already.
 
  • #14
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Tell me under what condition it makes sense to use spivaks book, particularly after having taken calculus already.
If a person is curious and wants to see a rigorous treatment of calculus. It can also be a way for math students to practice proof writing, one of the most important skills of a mathematician, which builds on mathematical maturity. It can ease the transition to higher mathematics, by providing familiarity with something that has been seen before. Making the transition to the abstract, more tangible.

Are you even a mathematics major, mathematics graduate, or interested in Math for the sake of Math? To ask such a question is absurd...
 
  • #15
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i see you are an engineering student....
 
  • #16
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i see you are an engineering student....
Yea, I also covered actual Real Analysis(lebesgue integration). Spivaks book is garbage, if you understand calculus then pick up an actual elementary analysis book. If you don't understand calculus, I doubt you will learn it from Spivaks book.
 
  • #17
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Did you use at least Shilov? Or one of the run of the mill books? Did you google search solutions?
 
  • #18
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Spivak is a garbage calculus book, I think. Just read stewart and move onto elementary real analysis.
Real analysis courses typically require lots of proofs, a skill you're extremely unlikely to get with Stewart's books.

Tell me under what condition it makes sense to use spivaks book, particularly after having taken calculus already.
To be able to actually do proofs. Being able to "find the derivative of ..." and "calculate the area under the curve of ... " require one set of skills, but being able to "prove that ..." requires a completely different skill set. Many students who do well in calculus completely flounder in linear algebra at even very simple problems such as, "Prove that if ##\lambda## is an eigenvalue of a linear transformation with matrix A, then ##\lambda^2## is an eigenvalue of ##A^2##" or the slightly harder "Show that a square matrix A is orthogonal if and only if ##AA^t = I##."

I'm speaking from my experience at teaching college-level math classes for 20+ years.
 
  • #19
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I have progressed with the book.

Slowly but surely, and it's getting easier. Or at least, I'm getting used to make my own proofs much faster.

Questioning every statement I make when trying to come up with a proof is helping a lot, just thinking whether whatever it is that I'm doing is really trivial or not.

Therefore, I'm actually finding Spivak's book enjoyable, and I feel like wether or not it ends up being useful for me, at least I'm getting what I wanted, adeeper understanding of calculus.
 

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