The loss of LaTeX would just take all the wind out of it, though.

I did note that usually the higher-order derivatives are negligible. I wasn't aware that a factor of 10 would be a good rule of thumb, though, thanks.

That's what I was asking you. Your explanation has a lot to do with, e.g., matrices of operators which seems like a lot of extra work to go through if the result is just as simple as taking the first partials with respect to damage and crit. Partial derivates are incredibly easy to perform and understand; multivariable taylor series seems like a lot of extra theoretical and computational work. Multivariable calculus isn't my specialty so I'm wondering if there is any (or any important) information that is being lost, either in general or in this specific example, by avoiding that whole discussion on Taylor series and simply saying "now we use calculus to see how your damage changes with crit." And, if there is extra information there, does it really matter? Since all we're looking at is how your damage is affected by extra stat points, which on the surface would seem to be the first partials, without anything to do with Taylor series in general.
Well not on purpose, I mean by accident. [Darkmoon Card: Wrath], for example, is written using entirely additive properties but it's affect on crit is a geometric series and its affect on, for example, ISB uptime could very well end up being exponential decay if there isn't a stack limit. As another example, crit on a warrior gives flurry, both of which give rage allowing more specials which have further chance to crit. I don't know if that's a high-order polynomial or an exponential function, but the point is that without intending to their simplistically-designed primary effects can have more complex behavoirs.
On the other hand some of them also have simpler solutions. For example, re: aff warlock nuke measure mentioned above, is not a differential function, it's a recurence relation with a pretty simple closed-form solution. It does happen to be a fraction of various intermediate variables so it might be a bitch to derivate but the stated form isn't as hard.

(Without having tried) I actually don't think that modelling most classes is honestly that hard, except for hidden cooldowns (boooo enh shamans). For example a fire mage 8-1 cycle is just a weighted average of two of your equations above with different numbers, which will probably result in just another equation of the same form with different numbers. The harder ones are piecewise functions, like what a warrior does with more rage as they start butting up against cooldowns. And if all you need are first partials the derivatives are cake.

I was honestly expecting a lot more symbolic computation when I first came to these boards and I was really dissapointed when I found a bunch of brute-force spreadsheets and no calculus =P. Apparently I'm lonelier than I originally anticipated in seeing scaling information easier with raw equations than numerical tables.

Honestly, that was my first thought when reading this thread as well. I mean, it all looks very impressive and is an admirable display of rigor; but I don't see it really allows us to do anything that we weren't already doing. I suppose it does tell us how to get more accurate models by adding higher derivatives, but most of these functions are either trivial and adequately described by the first order approximation, or so insanely messy that taking a 2nd derivative is impractical and not necessarily particularly more accurate anyway.

I'm not trying to be critical, but I guess I'm failing to see the advantage of looking at it in this way.

The fact is, in practice, for a realistic increase of 1 stat (from say, 1 item upgrade or some well-fed buff), has an extremely small effect on the value of other stats that you can just ignore it.

For example if you go to lieuler's spreadsheet, look at the extra dps increase from 10 spell dmg, then the extra dps increase from 10 crit rating, then the dps increase from both 10 dmg and 10 crit rating, you'll see a negligible difference between adding the "benefit from 10 dmg" with the "benefit from 10 crit rating" to what you actually get on the spreadsheet by adding both. That's again, becuase a small change of stats simply doesn't have any even nearly-meaningful effects on the value of stats. You need quite a big change of stats for the value of stats to actually change, in which case you just go to the spreadsheet again and enter your new stats. In reality those kinds of changes happen once in a long time considering the rate at which you get raiding upgrades.

So you haven't looked at enhancement shaman mechanics, I can tell. Each time someone has talked about making closed form expressions for class dps, and how easy it would be, they have been stumped on Enhance shaman mechanics. Tornhoof has a pretty good version at this point, I think. But we have no way to test to see how accurate it is. Blizzard should give us target dummies that act like raid level bosses, imo.

Absolutely true.

Or, we know how the mechanics work, and they're dumb. Expertise, for instance.

Let's say I know the partial derivative of Dd/De (damage with respect to expertise). For enhance shaman, by the way, this is an incredibly hard derivative to find, but pretend it were easy. Since blizzard has made expertise only useful when a multiple of 4 expertise rating is achieved, you now have to express Dd/De as a function of your currently level of expertise.

If I have to choose between using an item with 11 expertise rating and 0 attack power, or one with 28 attack power and 0 expertise rating, I can't do so without knowing my current expertise rating. For these purposes, assume that without this stair step insanity that Blizzard uses, 3AP = 1 Expertise rating, which is about right for Enhance Shaman).

If I currently have 0 expertise rating, then the first item would only give me 8 expertise rating worth (since any expertise rating that doesn't get me to a multiple of 4 is wasted). So, my choices are:

8 e.rating x 3gear-goodness/e.rating = 24 gear goodness
compared to
26 AP x 1 gear-goodness/ap = 26 gear goodness.

Aha! I should pick the second item, as it's a clear winner.

But wait, if I already have 5 expertise rating... well now 5 + 11 = 16, which is evenly divisible by four, so all 11 points of expertise rating on the item are providing benefit. The equations change!

11 e.rating x 3gear-goodness/e.rating = 33 gear goodness
compared to
28 AP x 1 gear-goodness/ap = 28 gear goodness.

Aha! I should pick the first item, as it's a clear winner.

/facepalm

Why Blizzard would do this is a mystery.

Yeah, Mage isn't the greatest example here, since essentially everything is linear. On the other hand, it's actually simple enough to model, and since this is just about mathematical fun anyway, I guess that makes it the best candidate to mess around with.

Int has a slight quadratic term for deep Arcane, I guess.

If the main goal is to get people to properly think more in terms of marginal DPS per stat, maybe a nice post just explaining things at the linear level would be nice.

Also, you might be interested in this:
http://elitistjerks.com/f31/t14250-m...ell_selection/

Isn't a geometric series defined as having a fixed chance for each roll to succeed? So while darkmoon card: wrath with any reasonable crit rate can be more or less approximated as a geometric series (as in, the extra crit doesn't change a lot when it stacks up so the chance for it to fall off is more or less the same, giving you a good general idea of how horrible it is so you don't need to bother with more accurate calculations), combustion changes the crit chance by a LOT (not rare to stack it up to 100% crit), which makes it very different from a geometric series.
I've yet to figure out a way to take combustion into account mathematically (aka no simulator) other than simply calculating the average number of crits out of <insert maximum number of casts with combustion active> casts by calculating the chance to get 3 crits, 4 crits etc up to having all spells crit - and afterwards doing the same calculation without combustion (which would be easy of course) to see the difference.

All of the problematic math that is shaman specific is because of the windfury cooldown, I believe (I can't think of anything else that causes a math problem). But, as you say, it's intrinsic to every aspect of the damage we do.

The other parts that are difficult result from stuff like dumb game mechanics (expertise rating) or unknown true mechanics (how flurry works, proc rates of items, how exactly does the two-roll attack table work for yellow attacks).

I would imagine that you could very easily account for any cooldown - internal or otherwise - could be modeled with the aide of a heavyside function.

I don't really see why you'd expect a bunch of calculus when a brute-force spreadsheet does the job just as well and anyone with a basic understanding of Excel can use one, though.

I did a decent amount of my own multi-variable analysis of the mage damage function with the only variables being crit and damage. With a consistent talent build and base damage values, there were numbers in place of the variables other than hit and crit in your final equation. That stuff just took 3rd semester calculus to calculate, and even adding in haste or other variables into the calculation would not really change things other than the computational complexity. I abandoned work on it when I found that people had developed detailed spreadsheets for calculating damage directly instead of goign through symbolic analysis.

The problem with analyzing other variables is such a rigorous manner is that the gains you would gain from it you would potentially lose because you're basically assuming differentiability to the non-differentiable step-functions that arise in the game. When it comes down to it, all that matters is a proper model of your classes damage and an ability to input all your possible gear and buffs into that model and see what comes out the best. The fact that spreadsheets might list partial derivatives at the current gear state is only to help guide decisions when the spreadsheet doesn't have everything available. The power of calculus in the real world is to approximate solutions where direct calculation is unappealing; if we can use a straightforward "brute force" model that approximates things just as well as an analytical model and can calculate it just as easily, there's no point to the extended analysis.