Why does treating differential elements like fractions work so well? I studied calculus from a pure math perspective first, and have only basic knowledge of it, so can someone explain with a rigorous proof or a simple explanation why this hack works? Some people told me that treating differentials like fractions is a sin. Can anyone give me an example where treating differentials like fractions fails to work?
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1Have a look at this page https://physics.stackexchange.com/q/92925 – GiorgioP-DoomsdayClockIsAt-90 Dec 24 '20 at 12:17
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Manipulating differentials as fractions is pretty much limited to elementary calculus and simple derivatives. If you move onwards to multivariable calculus, you will find that it does not work for partial derivatives. Moving on further, the notion of a derivative completely changes. Derivatives can now be thought of as vector operators. The $\text{d}$ operator becomes the exterior derivative which brings us to the subject of differential forms and exterior algebra. Therefore, the $\text{d}$ symbol has deeper meaning than just "an infinitesimal change in". – Vincent Thacker Dec 24 '20 at 12:28
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@VincentThacker actually, with a minimum of care it can be extended to multivariable calculus. In z proper treatment, the "d" symbol has never the unclear meaning of "infinitesimal change". – GiorgioP-DoomsdayClockIsAt-90 Dec 24 '20 at 12:38
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@NiharKarve the topic is not only mathematical but of wide (if not highest) interest for the community of Physicists, which keep using these notations and techniques. If this post should be flagged is as a duplicate. – GiorgioP-DoomsdayClockIsAt-90 Dec 24 '20 at 12:41
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@GiorgioP the flagging was prompted by the fact that OP is asking for a "rigorous proof of how treating differentials as fractions works" – Nihar Karve Dec 24 '20 at 12:43
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@hedamnedking17 Your question contains a small inaccuracy. Aren't differential elements that are treated as fractions but derivatives. – GiorgioP-DoomsdayClockIsAt-90 Dec 24 '20 at 12:43
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@NiharKarve, I would also note that the accepted answer in the PSE page I cited is by far the closest and the most convincing for the needs of a physicist. The most upvoted answer to the question you mentioned is still too biased by the nineteenth-century difficulty distinguishing between infinitesimals and differentials. Unfortunately, that bias is still among us. – GiorgioP-DoomsdayClockIsAt-90 Dec 24 '20 at 12:56
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@GiorgioP I did not mean that this question is necessarily a duplicate of the one I linked to. I just felt it would be a better fit for MSE because OP does not explicitly request any physical context. – Nihar Karve Dec 24 '20 at 12:58
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This often works because the notation follows the results that you'd get rigorously from the chain rule or (in reverse under the integral) the result of integration by parts. See here: https://physics.stackexchange.com/questions/572956/the-usage-of-chain-rule-in-physics/573232#573232 – Brick Dec 24 '20 at 13:54
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This issue has been covered at great length in several questions over the years. See here: https://math.stackexchange.com/questions/1784671/when-can-we-not-treat-differentials-as-fractions-and-when-is-it-perfectly-ok – Kevin Carlson Dec 24 '20 at 14:06
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1Does this answer your question? Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? – V.G Dec 24 '20 at 16:58
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Related: https://math.stackexchange.com/questions/3407568/treat-differentials-as-fractions/3407672#3407672 – Ethan Bolker Dec 24 '20 at 17:30
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Physicists do this in more than one context, so a single answer might not be possible that covers all uses. I think you were poorly served by having this migrated to Math SE as some of these uses are related purely to how the physics is modeled and not to the math, although some are mathematical. Some important cases:
- Physics ultimately has to connect to something you can measure in a laboratory or in nature, and you can never measure an infinitesimal $dx$ directly, nor do you typically measure something that you represent by a derivative ("ratio of infinitesimals") directly. What you actually measure is one or more finite values and then interpolate or extrapolate. So an expression like $dx = v\ dt$ is short-hand for $\Delta x \approx v \Delta t$ for finite $\Delta x$ and finite $\Delta t$ and that approximation gets better and better as you make finer and finer resolution measurements. The finite values $\Delta x$ and $\Delta t$ are regular numbers that you can move around using regular arithmetic. It's essentially part of the theory that it doesn't matter a which point you take the limits because the theory asserts something about the extrapolated behavior of finite measurements.
- The physicist accepts that theory may break down as some parameters of the model get big, get small, or take special values, but the physicist keeps writing the equal sign anyway. For example, the physicist doesn't refuse to write $F = dp/dt$ because we know there are higher order terms from relativity or that there are issues with this representation of momentum in quantum mechanics. The physicist separately keeps track of the fact that the theory will break down and where the bounds for that break might be.
- For more purely mathematical operations, a lot of times whatever is done with infinitesimal symbols is ultimately going to be either used as a "ratio" to form the derivative or going to be put under an integral sign. There, although it abuses notation, the chain rule or integration by parts saves many of the manipulations that look like they are treating the infinitesimal symbols like numbers and, e.g., canceling parts of fractions. (See more here: https://physics.stackexchange.com/questions/572956/the-usage-of-chain-rule-in-physics/573232#573232) The chain rule and integration by parts, are, of course rigorously understood.
Brick
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