Derivatives: Solving F'(x) = [2f(x)+g(x)]' Problem

[Fredrik]

The first time I can remember being puzzled by this notation conundrum was when I was taking an undergrad course in quantum mechanics. The function $\Psi$ was being introduced and the professor wrote things like this on the board,

$\Psi=\Psi(x)$,

while saying "Psi is a function of x."

My teacher for the introductory courses on classical mechanics did the same thing.

It wasn't until years later that a professor in a graduate math class explained to me the subtleties of that abomination.

I actually tried to use it on a math exam. The professor dismissed it by saying that you can't write down an equality where one side is a function and the other isn't. I tried to explain what I was doing there, but he wasn't even interested in hearing it. I had to figure it out on my own after that. I have always been careful with the notation and the terminology since then.

At the time, though, it was for me just another layer of confusion, piled on top of the other conceptual difficulties associated with quantum mechanics and the meaning of $\Psi$.

Now, as an instructor, I avoid that usage sedulously.
[...]
Things like this can be a source of confusion for students, and even though the confusion can be easily cleared up it instead stays buried in the student's mind because the confusion itself cannot be articulated by the student.

Yes, that's what I'm thinking too. Thanks for posting.

 

[Mark44]

The first time I can remember being puzzled by this notation conundrum was when I was taking an undergrad course in quantum mechanics. The function $\Psi$ was being introduced and the professor wrote things like this on the board,

$\Psi=\Psi(x)$,

while saying "Psi is a function of x."

It wasn't until years later that a professor in a graduate math class explained to me the subtleties of that abomination. At the time, though, it was for me just another layer of confusion, piled on top of the other conceptual difficulties associated with quantum mechanics and the meaning of $\Psi$.

Now, as an instructor, I avoid that usage sedulously. We see it used in introductory physics textbooks, particularly the calculus-based variety, in cases like the treatment of simple harmonic motion.

So it would seem that Fredrik's and your purpose in life is to get all of the authors of physics textbooks to stop using this notation? Good luck with that. I am less concerned about an abuse of notation than I am about being able to communicate an idea. Above, the dependent variable $\Psi$ is some function of x. Perhaps it would be better to write $\Psi = f(x)$, but as long as the idea of connectedness between $\Psi$ and x is communicated, I'm OK with the notation.

I do agree, though, that it would be awful if math texts wrote something like "Let f = f(x) = 2x + 3".

The position is x and to make clear that it's a function of the time t the author will write

$x(t)=A \cos(\omega t + \phi)$.

Is there any evidence that this actually causes confusion?

And then go on to refer to x as both that function and the value of that function as if those two things are the same.

Things like this can be a source of confusion for students, and even though the confusion can be easily cleared up it instead stays buried in the student's mind because the confusion itself cannot be articulated by the student.

 

[Dale]

Closed pending moderation.