Partition notation

[Sean Fitzpatrick]

Going through the Riemann sums section is turning out to be a bit of a workout... phew.

As I'm going through, I'm wondering about the use of the notation <m>\Delta x</m> to refer to a partition of an interval.

It seems to me that this could be potentially confusing for students, since:

(a) it's not that far off from the notation <m>\Delta x_i</m> used for the length of the ith subinterval

(b) it's used for the common length of each subinterval in a uniform partition.

Is this notation something we (Greg?) are particularly attached to?

If not, what's a good replacement?

I think <m>\mathcal{P}</m> is common in other texts.

[Sean Fitzpatrick]

Related question about partitions:

One of the most common complaints I've had from students/other instructors is that in APEX,

Riemann sums are indexed from 1 to n+1, instead of from 0 to n.

I've always meant to make the change in my version, but never got around to it.

Greg, are you very set on having the index start at 1 instead of 0?

(Or is this a thing where we start at 0 in Canada and everyone starts at 1 in the US?)

[Gregory Hartman]

I don't have other calc books in front of my, but I do remember thinking hard about the \Delta x notation. I didn't like how the same symbol is used to represent two different things, but liked how how it tied a partition to its size via the ||\Delta x|| notation. Plus, the use of \Delta x as a partition is rarely used in the text, so there is minimal times of possible confusion. I guess that could also be argument for replacing it with $P$; it looks like there are only 2 instances of \Delta x representing the partition, so changing it wouldn't cause that much trouble elsewhere.

Open to others' suggestions/preferences here.

As for the numbering of the partition - I'm a pretty hard no on that one. First, it makes more sense to me that the first item be numbered 1. My knowledge of Canadian culture is limited to curling and a particular Molson beer commercial from 2006; maybe numbering from 0 is more common the farther north one goes.

More importantly: changing that notation will change a *ton* in the book (at least, that section), and I'm mindful of the havoc this may wreak on other user's of the text. The class notes of many an instructor will change, and I really try to limit the number of significant changes I make to the text. So - please no to that one. 

[Sean Fitzpatrick]

Ok, that works for me!

I'll leave both as is for now.

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