Derivatives don't always act like fractions (2021)

The key clarification is in one of the comments: if you want to treat partial derivatives like fractions, you need to carry the &quot;constant with respect to foo&quot; modifier along with both nominator and denominator.<p>Once you do that, it&#x27;s clear that you can&#x27;t cancel &quot;dx at constant z&quot; with &quot;dx at constant y&quot; etc. And then the remaining logic works out nicely (see thermodynamics for a perfect application of this).

I’ve never liked to conflation with fractions. Abuse of notation. And it causes so much confusion.<p>Also integrals with “integrate f(x) dx” where people treat “dx” as some number than can be manipulated, when it’s more just part of the notation “integrate_over_x f(x)”<p>Sigh. These are sadly some kind of right-of-passage, or mathematical hazing. Sad.

I still don’t understand what &quot;at constant something&quot; means. I mean formally, mathematically, in a way where I don’t have to kinda guess what the result may be and rely on my poor intuitions and shoot myself continually in the foot in the process.<p>Does someone has a good explanation ?

i&#x27;m looking forward to the day calculus gets rewritten using more intuitive notations.<p>Everytime i manipulate dx i feel like walking on a minefield.

Am I missing something, I don&#x27;t see how the examples are more &quot;intuitive&quot; as they just provide an allied example of using this?<p>My pain was always Hamiltonians and Legendre equations for systems because the lecturer believed in learn by rote rather than explaining something that I&#x27;m sure for him was simply intuitive.

Why would you even tell in the first places derivatives are simply fractions ? They’re not, unless in some very specific physical approximations and in that case don’t try to do anything funky, sticks with the basics stuff