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> 7. stay away from differentials I always had issues with this. My undergraduate math classes were very formal and well thought, every object was defined, alm

> 7. stay away from differentials I always had issues with this. My undergraduate math classes were very formal and well thought, every object was defined, almost all theorems were proved in class. In particular, we were taught differential equations theories, some general theories like systems of differential equations with constant coefficients, Cauchy's theorem, and a bunch of recipes. In a separate class, we were also taught differential forms (e.g. a.dx + b.dy being a linear combination of base elements dx and dy - if I remember well). But then came the physics class, why we could go from dy/dx = f(x) to dy = f(x).dx and integrate both sides remain a mystery. "dx" suddenly became an infinitely small things, when it used to be a notation for a linear form OR a notation for f'(x) === df/dx. Generally speaking, I had a lot of issues in physics because the maths used in that context were not formally defined and seemed to obey to some unwritten set of rules that you were supposed to find out by yourself during your final exam...

notfish 4y ago
Theres definitely a lack of rigor math-wise in physics as a rule. It’s not for everyone. That being said, that lack of rigor is a useful tool sometimes. The dirac delta “function” (functional?) was described in the world of physics first as the only reasonable derivative of a step function/impulse before it was properly defined in the words of mathematics; just because the equational rigor isn’t there yet doesn't mean the argument isn’t solid or the tool is poor.
- ngcc_hk 4y ago
  It is not the lack of rigor. It is more using Math as tool. If it fit the purpose
oplaadpunt 4y ago
This echoes my own experience of differential equations in physics. I once actually asked my teacher directly about the dx schenanigans, and he basically said 'whatever, it's just what it is'. It left me with a rather sour attitude towards the subject as a whole, the idea that you should just manipulate symbols in some arbitrary way
- Hendrikto 4y ago
  > I once actually asked my teacher directly about the dx schenanigans, and he basically said 'whatever, it's just what it is'. I received pretty much the same answer… Did you study in the Netherlands as well? Your name suggests that.
mhh__ 4y ago
One way I explained this the other day, or rather soothed concerns lacking a simpler answer: the notation is designed to (mostly) make you write down things that make sense
sigmoid10 4y ago
Once you reach the concept of differential forms in physics, a lot of these seemingly "ad-hoc" rules suddenly make sense. The issue is that outside of theoretical physics grad students, few people will ever encounter those, since they basically only allow you to re-learn Maxwell's equations on a more fundamental level. But then again a lot of this directly translates into other field theories, including QFT, so there's that. For undergraduates, this seemingly wrong notation is mostly just taught out of convenience and readability. For example, you can do separation of variables more formally in undergrad notation by simply applying the integral as an operator to both sides. In your example that would be Integral(1 dy/dx dx) = Integral(f(x) dx), which works exactly the same, but the mnemonic "separation" of variables is no longer obvious at all.
LudwigNagasena 4y ago
I think what is confusing about differentials is that variables are meta-objects. You can apply function to a number, you can apply function to another function and so on but you cannot really apply function to a variable, you apply it to something that the variable stands for. The introduction of calculus in my university went like this: At first we introduce derivative f’(x) as a limit [f(x+h)-f(x)]/h. It is a linear function. We call d the function that takes f(x) and h and returns f’(x)*h (or it can be defined by currying, whatever). So now we can write df and it makes sense. Ok, so far so good. But what is even dx? We cannot just apply a function to a meta-symbol. Variables are not first-level objects, they are just notational fiction. You may say that it is an identity function, but such explanation breaks down when more free variables are introduced.
- codethief 4y ago
  You're making a good point. However, taking inspiration from differential topology (and differential forms in particular), the notation dx can be made sense of once you interpret x as a coordinate function x: M -> R on a manifold. So in a sense you are replacing the variable x with a function x (whose parameter is usually left unspecified).
  - LudwigNagasena 4y ago
    Yeah, it’s the standard way to do it. But there is a reason the 7th point is called “stay away from differentials.” It’s cryptic and unintuitive for many people. I wonder whether name binding concepts from CS theory a la lambda calculus can be leveraged for an alternative formalization.
    - selimthegrim 4y ago
      I think usually slightly alternate definition of derivative is used - https://en.wikipedia.org/wiki/Total_derivative https://en.wikipedia.org/wiki/Total_derivative
- twos222 4y ago
  If f is a function and x is a variable then dx means that you should express f with respect to an input x, so if A is the area of a a circle then what is dA? since A = pir^2 you have that dA = 2pirdr since A = pi(l/(2pi))^2 where l is the perimeter then you have dA = pi2(l/(2pi))(1/(2pi))dl, so the idea of dA is something related to the diferential of the variable used to define A. The use of the chain rule allow us to operate with differentials. For example, since here r = l/(2pi) we have that dr = dl/(2pi) so that dA = 2pirdr = 2pi(l/(2pi))dl/(2pi) = 1/(2pi)ldl. An finally this implies that dA/dl = 1/(2pi)l
Hendrikto 4y ago
> But then came the physics class, why we could go from dy/dx = f(x) to dy = f(x).dx and integrate both sides remain a mystery. This never made sense to me. When I asked about it, I was just told to go with it.
- rprospero 4y ago
  Perhaps I’m just too much of a physicist, but I’m not sure what there is not to get. Granted, I’m an experimentalist, so that doesn’t help with my rigour. Still, I will encounter students who will have the same concerns, so I’d love to have my blind spot corrected. From my perspective, it’s just two steps: multiply both sides by the same value, then integrate both sides. However, you seem like a smart individual, so it’s probably something more foundational.
  - dan_mctree 4y ago
    I think there's two steps of weirdness. First, many of us see dx/dy not so much as a division of values, but as just some notation for a rate of change strongly linked of f(x). Handling that algebraically seems strange. The explanation of infinitesimals will allow us to sort of begrudgingly accept that such manipulations might be doable, but if we do so we picture dx and dy purely as very small values with a now kind of unclear relationship to f(x). The integration step then becomes a weird mess. First of all most of us are used to only ever taking integrations with respect to a variable. Except now when we integrate, that 'with respect to' part is somehow already pre-filled by the earlier shenanigans. On top of that, we are somehow now integrating with respect to different variables on both sides which intuitively feels like we just did different things on both sides. And what in the world do our new forms now have to do with f(x)? dx and dy were supposed to relate to a change of f(x), but somehow we just took dy out entirely? At this point we've lost all intuition for what we're doing, and feel like we're just staring at some weird notation hack
    - rprospero 4y ago
      Thank you. I’m going to need some time to think about that.
dannymi 4y ago
>But then came the physics class, why we could go from dy/dx = f(x) to dy = f(x).dx and integrate both sides remain a mystery. "dx" suddenly became an infinitely small things, when it used to be a notation for a linear form OR a notation for f'(x) === df/dx. It's the mathematicians who changed it. Infinitesimals used to be how differential equations were defined as in the first place (by Newton and Leibniz). Physicists still think like that because we are end users of mathematics. So the interface has to stay the same. Old text have to keep working etc. Mathematics for us is not so different to software as an end user. In the quest for formalization, infinitesimals have been removed and replaced by limits by mathematicians. And now they have found a way to have a rigorous notation that has infinitesimals again: https://people.math.wisc.edu/~keisler/foundations.pdf https://people.math.wisc.edu/~keisler/foundations.pdf Your example is explained on page 34. On the other hand, a notation of mathematics that is incompatible with infinitesimals will not be a success in physics or engineering, for obvious reasons (compatibility; and also by using infinitesimals one almost always gets to the correct result anyway--even in very complicated cases--, and they are way more intuitive than limits). So there's no problem--it's just that life is short and there's too much to learn. If you want to prove theorems, by all means, use the most rigorous tools--but to use the result it's nice to have a simple interface. Mathematics, for us, is a tool to communicate ideas (about the physical world). Overcomplicating it will obscure the ideas.
- colordrops 4y ago
  So this should be explained to the students instead of letting them feeling around in the dark confused.
  - ehhthing 4y ago
    I just (barely) passed what is known as my university's hardest first year physics course: electricity and magnetism. We didn't use differential equations, but we had the same hand wave "just think about it" relationship with line integrals. The way my professor explained it was "we're engineers, not mathematicians, so we actually know what we're talking about". What he meant by this is that as engineers, we need to have intuition as to what an integral actually is rather than knowing how to prove things with them. To an extent, this is a good way to go about it. Really it's not at all useful to understand things like delta-epsilon as an engineer because those concepts really have no application. It's much more useful to just gain an intuition as to what you're actually doing when you're taking a line integral, and applying that to, for example, calculating an electric field strength as a result of an object. Physics teaches physics, not math.
mkl 4y ago
> But then came the physics class, why we could go from dy/dx = f(x) to dy = f(x).dx and integrate both sides remain a mystery. That is just integrating both sides with respect to x, but you're skipping the step that makes that clear. Start with dy/dx = f(x). Integrate both sides with respect to x: ∫(dy/dx)dx = ∫f(x)dx. Doing the integrals produces: y = F(x) + C, where F(x) is the integral of f(x) and C is an arbitrary constant of integration. Above, the integral on the left doesn't do anything except cancel the derivative. A similar but more substantive method is solving a DE by separation of variables, where the "mystery" is integration by substitution (a.k.a. the chain rule backwards). Start with dy/dx = f(x)g(y). This is a "separable" DE. Rearrange to (1/g(y)) dy/dx = f(x). Integrate both sides with respect to x: ∫(1/g(y))(dy/dx)dx = ∫f(x)dx. We are integrating with respect to x, but where is the x in 1/g(y)? It is hiding inside the y: y = y(x). Define a substitution: Let u = y(x). Then du/dx = dy/dx ⇒ du = (dy/dx)dx, and our equation becomes ∫(1/g(u))du = ∫f(x)dx. Do we really need to introduce another variable? No. We can just use y: ∫(1/g(y))dy = ∫f(x)dx. We often jump straight to this point which makes it pretty confusing, as it looks like we're doing different things to each side of the equation.
captainmuon 4y ago
This is one of the places where I disagree with the article. As an (experimental) physicist, I've always found the dx notation better to understand. When I asked the professor about how it works, he called it a calculus (as in German Kalkül, set of rules to almost mechanically manipulate a mathematical expression, not to be confused with "Calculus"). Of course there exist proofs in terms of limits, or differential forms. But it is something one has to learn, just like div and curl and so on. We had very mathematically rigorous math lectures. I was able to do most of the proofs, but failed to calculate even a simple derivative when thinking in differential forms. (Maybe because I failed to see what is a function and what is a variable in the "mathematical" notation and misapplied the chain rule.) The other thing where I disagree is the part about avoiding word problems. I think the word problems were the only interesing part - how to translate a given physical situation into a differential equation and variables. Solving them is not really interesting - everything interesting for me has been solved before, so I can just ask around the hallway or punch it into a computer algebra system.

notfish 4y ago

Theres definitely a lack of rigor math-wise in physics as a rule. It’s not for everyone. That being said, that lack of rigor is a useful tool sometimes. The dirac delta “function” (functional?) was described in the world of physics first as the only reasonable derivative of a step function/impulse before it was properly defined in the words of mathematics; just because the equational rigor isn’t there yet doesn't mean the argument isn’t solid or the tool is poor.

ngcc_hk 4y ago
It is not the lack of rigor. It is more using Math as tool. If it fit the purpose

oplaadpunt 4y ago

This echoes my own experience of differential equations in physics. I once actually asked my teacher directly about the dx schenanigans, and he basically said 'whatever, it's just what it is'. It left me with a rather sour attitude towards the subject as a whole, the idea that you should just manipulate symbols in some arbitrary way

Hendrikto 4y ago
> I once actually asked my teacher directly about the dx schenanigans, and he basically said 'whatever, it's just what it is'. I received pretty much the same answer… Did you study in the Netherlands as well? Your name suggests that.

mhh__ 4y ago

One way I explained this the other day, or rather soothed concerns lacking a simpler answer: the notation is designed to (mostly) make you write down things that make sense

sigmoid10 4y ago

Once you reach the concept of differential forms in physics, a lot of these seemingly "ad-hoc" rules suddenly make sense. The issue is that outside of theoretical physics grad students, few people will ever encounter those, since they basically only allow you to re-learn Maxwell's equations on a more fundamental level. But then again a lot of this directly translates into other field theories, including QFT, so there's that. For undergraduates, this seemingly wrong notation is mostly just taught out of convenience and readability. For example, you can do separation of variables more formally in undergrad notation by simply applying the integral as an operator to both sides. In your example that would be Integral(1 dy/dx dx) = Integral(f(x) dx), which works exactly the same, but the mnemonic "separation" of variables is no longer obvious at all.

LudwigNagasena 4y ago

I think what is confusing about differentials is that variables are meta-objects. You can apply function to a number, you can apply function to another function and so on but you cannot really apply function to a variable, you apply it to something that the variable stands for. The introduction of calculus in my university went like this: At first we introduce derivative f’(x) as a limit [f(x+h)-f(x)]/h. It is a linear function. We call d the function that takes f(x) and h and returns f’(x)*h (or it can be defined by currying, whatever). So now we can write df and it makes sense. Ok, so far so good. But what is even dx? We cannot just apply a function to a meta-symbol. Variables are not first-level objects, they are just notational fiction. You may say that it is an identity function, but such explanation breaks down when more free variables are introduced.

codethief 4y ago
You're making a good point. However, taking inspiration from differential topology (and differential forms in particular), the notation dx can be made sense of once you interpret x as a coordinate function x: M -> R on a manifold. So in a sense you are replacing the variable x with a function x (whose parameter is usually left unspecified).
- LudwigNagasena 4y ago
  Yeah, it’s the standard way to do it. But there is a reason the 7th point is called “stay away from differentials.” It’s cryptic and unintuitive for many people. I wonder whether name binding concepts from CS theory a la lambda calculus can be leveraged for an alternative formalization.
  - selimthegrim 4y ago
    I think usually slightly alternate definition of derivative is used - https://en.wikipedia.org/wiki/Total_derivative https://en.wikipedia.org/wiki/Total_derivative
twos222 4y ago
If f is a function and x is a variable then dx means that you should express f with respect to an input x, so if A is the area of a a circle then what is dA? since A = pir^2 you have that dA = 2pirdr since A = pi(l/(2pi))^2 where l is the perimeter then you have dA = pi2(l/(2pi))(1/(2pi))dl, so the idea of dA is something related to the diferential of the variable used to define A. The use of the chain rule allow us to operate with differentials. For example, since here r = l/(2pi) we have that dr = dl/(2pi) so that dA = 2pirdr = 2pi(l/(2pi))dl/(2pi) = 1/(2pi)ldl. An finally this implies that dA/dl = 1/(2pi)l

Hendrikto 4y ago

> But then came the physics class, why we could go from dy/dx = f(x) to dy = f(x).dx and integrate both sides remain a mystery. This never made sense to me. When I asked about it, I was just told to go with it.

rprospero 4y ago
Perhaps I’m just too much of a physicist, but I’m not sure what there is not to get. Granted, I’m an experimentalist, so that doesn’t help with my rigour. Still, I will encounter students who will have the same concerns, so I’d love to have my blind spot corrected. From my perspective, it’s just two steps: multiply both sides by the same value, then integrate both sides. However, you seem like a smart individual, so it’s probably something more foundational.
- dan_mctree 4y ago
  I think there's two steps of weirdness. First, many of us see dx/dy not so much as a division of values, but as just some notation for a rate of change strongly linked of f(x). Handling that algebraically seems strange. The explanation of infinitesimals will allow us to sort of begrudgingly accept that such manipulations might be doable, but if we do so we picture dx and dy purely as very small values with a now kind of unclear relationship to f(x). The integration step then becomes a weird mess. First of all most of us are used to only ever taking integrations with respect to a variable. Except now when we integrate, that 'with respect to' part is somehow already pre-filled by the earlier shenanigans. On top of that, we are somehow now integrating with respect to different variables on both sides which intuitively feels like we just did different things on both sides. And what in the world do our new forms now have to do with f(x)? dx and dy were supposed to relate to a change of f(x), but somehow we just took dy out entirely? At this point we've lost all intuition for what we're doing, and feel like we're just staring at some weird notation hack
  - rprospero 4y ago
    Thank you. I’m going to need some time to think about that.

dannymi 4y ago

>But then came the physics class, why we could go from dy/dx = f(x) to dy = f(x).dx and integrate both sides remain a mystery. "dx" suddenly became an infinitely small things, when it used to be a notation for a linear form OR a notation for f'(x) === df/dx. It's the mathematicians who changed it. Infinitesimals used to be how differential equations were defined as in the first place (by Newton and Leibniz). Physicists still think like that because we are end users of mathematics. So the interface has to stay the same. Old text have to keep working etc. Mathematics for us is not so different to software as an end user. In the quest for formalization, infinitesimals have been removed and replaced by limits by mathematicians. And now they have found a way to have a rigorous notation that has infinitesimals again: https://people.math.wisc.edu/~keisler/foundations.pdf https://people.math.wisc.edu/~keisler/foundations.pdf Your example is explained on page 34. On the other hand, a notation of mathematics that is incompatible with infinitesimals will not be a success in physics or engineering, for obvious reasons (compatibility; and also by using infinitesimals one almost always gets to the correct result anyway--even in very complicated cases--, and they are way more intuitive than limits). So there's no problem--it's just that life is short and there's too much to learn. If you want to prove theorems, by all means, use the most rigorous tools--but to use the result it's nice to have a simple interface. Mathematics, for us, is a tool to communicate ideas (about the physical world). Overcomplicating it will obscure the ideas.

colordrops 4y ago
So this should be explained to the students instead of letting them feeling around in the dark confused.
- ehhthing 4y ago
  I just (barely) passed what is known as my university's hardest first year physics course: electricity and magnetism. We didn't use differential equations, but we had the same hand wave "just think about it" relationship with line integrals. The way my professor explained it was "we're engineers, not mathematicians, so we actually know what we're talking about". What he meant by this is that as engineers, we need to have intuition as to what an integral actually is rather than knowing how to prove things with them. To an extent, this is a good way to go about it. Really it's not at all useful to understand things like delta-epsilon as an engineer because those concepts really have no application. It's much more useful to just gain an intuition as to what you're actually doing when you're taking a line integral, and applying that to, for example, calculating an electric field strength as a result of an object. Physics teaches physics, not math.

mkl 4y ago

> But then came the physics class, why we could go from dy/dx = f(x) to dy = f(x).dx and integrate both sides remain a mystery. That is just integrating both sides with respect to x, but you're skipping the step that makes that clear. Start with dy/dx = f(x). Integrate both sides with respect to x: ∫(dy/dx)dx = ∫f(x)dx. Doing the integrals produces: y = F(x) + C, where F(x) is the integral of f(x) and C is an arbitrary constant of integration. Above, the integral on the left doesn't do anything except cancel the derivative. A similar but more substantive method is solving a DE by separation of variables, where the "mystery" is integration by substitution (a.k.a. the chain rule backwards). Start with dy/dx = f(x)g(y). This is a "separable" DE. Rearrange to (1/g(y)) dy/dx = f(x). Integrate both sides with respect to x: ∫(1/g(y))(dy/dx)dx = ∫f(x)dx. We are integrating with respect to x, but where is the x in 1/g(y)? It is hiding inside the y: y = y(x). Define a substitution: Let u = y(x). Then du/dx = dy/dx ⇒ du = (dy/dx)dx, and our equation becomes ∫(1/g(u))du = ∫f(x)dx. Do we really need to introduce another variable? No. We can just use y: ∫(1/g(y))dy = ∫f(x)dx. We often jump straight to this point which makes it pretty confusing, as it looks like we're doing different things to each side of the equation.

captainmuon 4y ago

This is one of the places where I disagree with the article. As an (experimental) physicist, I've always found the dx notation better to understand. When I asked the professor about how it works, he called it a calculus (as in German Kalkül, set of rules to almost mechanically manipulate a mathematical expression, not to be confused with "Calculus"). Of course there exist proofs in terms of limits, or differential forms. But it is something one has to learn, just like div and curl and so on. We had very mathematically rigorous math lectures. I was able to do most of the proofs, but failed to calculate even a simple derivative when thinking in differential forms. (Maybe because I failed to see what is a function and what is a variable in the "mathematical" notation and misapplied the chain rule.) The other thing where I disagree is the part about avoiding word problems. I think the word problems were the only interesing part - how to translate a given physical situation into a differential equation and variables. Solving them is not really interesting - everything interesting for me has been solved before, so I can just ask around the hallway or punch it into a computer algebra system.