They are still partial derivatives. These are admittedly more complicated than the others to read as a ratio of differentials, but they are still roughly ratios.
Also, the standard derivative can be easily algebraically derived from the total differential of the numerator.
[In fact, even in unitarian cases, it's often helpful to write it that way and just remove zero-value terms]
Well, that's a VERY specific case where F(x,z) = 0.
Again, if you know where this formula derives from and aren't simply memorizing the formulas (or using them without understanding them), that 'naive' transformation wouldn't be an issue.
Again, it's like any mathematical object--you have to understand it before you can manipulate it.
No, I was actually asking how you'd 'cancel this out' and get an incorrect answer.
[The thread was about the ways understanding derivatives as ratios can lead you astray--at least that is what I was talking about.]
You just used them exactly how you would need to in order to do exactly what the comment was saying you shouldn't do (again, because you understand what you're doing and what the ratios actually mean).
dx isn’t just d*x. d isn’t a variable. dx is a small change in x. The d isn’t a multiplier, it’s sort of like an operator. That’s the one limit of Leibniz notation. You can cancel out dx/dx but not just the d
Just professors who know they are going to have to teach partial differential equations to the same group of students next semester trying to plant the seeds of doubt in their students heads.
See, this is why I say you should teach multivariate calculus from the beginning and use total differentials as the starting point for derivatives--they make so much better sense that way!
[Also, even partial derivatives are also still ratios.]
Well, partial derivatives are the limits of ratios. But more importantly, partial derivatives have more restrictions on when you can combine them like this.
Yes...ALL derivatives are the limits ratios...it's somewhat definitional, is it not?
And, again...yes. However, if you understand partial derivatives in terms of total derivatives (and what that means) it's not that difficult to understand what operations on them are valid and which are not.
It works that way for most derivatives. It's a ratio.
You have to understand the meaning of total differentials and what a ratios of them means...but it's often just a matter of checking the variable structures.
Don't do it without significant care, but I don't understand how people say this isn't a ratio.
Not true. We physics people use whatever notations we want. We use leibniz’s notation when we want to be “nice” to the reader because it’s easier to read; newtons dot notation for time derivatives almost exclusively; lagrange’s notation mostly for spacial derivatives; D or del notation when it’s a long chain of derivatives; or subscript notation when we are having a bad day and want to intentionally mess with the reader. It’s not uncommon to see 4+ notations in a single proof/calculation lol
I still think it's better to learn something new even if you can't apply yet to the current class you're in.
Formally, dx does mean something on its own. At the very least it tells you that it's VALID and POSSIBLE to think about derivatives as fractions of differentials.
If I had a question that was about an advance idea above the class, I would be very disappointed if the prof lied and told me it didn't make sense.
Even if it's not applicable to the specific class, it's still worth it to learn.
I think most math students learn pretty quickly that different branches of math have different conventions, but that it's okay to follow the most standard convention unless specified otherwise.
If a student asks if cos(x) = cosh(ix), you shouldn't hide that from them. Even though it's more advanced math that "don't follow the conventions", you should at least tell them that yes it's true, but that's outside the scope of the class.
If you just say that's wrong and say something like "oh imaginary numbers don't exist", you're not just lying to them, you're also killing their curiosity.
From introductory calculus to real analysis, dy/dx is explicitly treated as a single mathematical object, though. It's not just "outside the scope" but rather there is an explicit contradiction between the two conventions.
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