For example, if f(x,y)=x2+yx, then (∂f/∂x)y=2x+y, and (∂f/∂y)x=x. We can extend this idea to higher derivatives: ∂2f/∂y2 or ∂^2f/∂y∂x. The latter symbol indicates that we first differentiate f with respect to x, treating y
as a constant, then differentiate the result with respect to y, treating x as a constant. The actual order of differentiation is immaterial: ∂2f/∂x∂y=∂2f/∂y∂x.
Notice: ∂^2f/∂y∂x is clearly written to mean ∂^2f/(∂y∂x).
What an interesting error to point out in support of pemdas.
Clearly the formatting of a paragraph of text in a textbook full of clearly and unambiguously written formulas discussing the very order of operations itself compared to the formatting of an actual formula diagram is going to be less clear. But here you’ve chosen to point to a discussion of why the order is irrelevant in the case under question.
Your example is the conclusion of a review of mathematics.
First we shall review some mathematics.
…
The actual order of differentiation is immaterial:
The fact that the example formula is written sloppy is irrelevant, because at no point is this going to be an actual formula meant to be solved, it’s merely an illustration of why, in this case, the order of a particular operation is “immaterial”.
Even if ∂^2f/∂y∂x is clearly written to mean ∂^2f/(∂y∂x), it doesn’t matter because “∂2f/∂x∂y=∂2f/∂y∂x”. So long as you’re consistently applying pemdas, you’re going to get the same answer whether you derive x first or y.
However, when it’s time to discuss the actual formulas and equations being taught in the example text, clearly and unambiguously written formulas are illustrated as though copied from Ann illustration on a whiteboard instead of inserted into paragraphs that might have simply been transcribed from a lecture. Which, somewhat coincidentally, is exactly what your citation is.
Sure, the definition of grade school doesn’t really matter too much. Because college texts are written in ways that violate pemdas.
Look, for example, at https://www.feynmanlectures.caltech.edu/I_45.html
Notice: ∂^2f/∂y∂x is clearly written to mean ∂^2f/(∂y∂x).
What an interesting error to point out in support of pemdas.
Clearly the formatting of a paragraph of text in a textbook full of clearly and unambiguously written formulas discussing the very order of operations itself compared to the formatting of an actual formula diagram is going to be less clear. But here you’ve chosen to point to a discussion of why the order is irrelevant in the case under question.
Your example is the conclusion of a review of mathematics.
…
The fact that the example formula is written sloppy is irrelevant, because at no point is this going to be an actual formula meant to be solved, it’s merely an illustration of why, in this case, the order of a particular operation is “immaterial”.
Even if ∂^2f/∂y∂x is clearly written to mean ∂^2f/(∂y∂x), it doesn’t matter because “∂2f/∂x∂y=∂2f/∂y∂x”. So long as you’re consistently applying pemdas, you’re going to get the same answer whether you derive x first or y.
However, when it’s time to discuss the actual formulas and equations being taught in the example text, clearly and unambiguously written formulas are illustrated as though copied from Ann illustration on a whiteboard instead of inserted into paragraphs that might have simply been transcribed from a lecture. Which, somewhat coincidentally, is exactly what your citation is.
Under PEMDAS, ∂2f/∂x∂y = (∂2f/∂x) * ∂y = ∂2∂y/∂x