Symmetry of second partial derivatives

so in the last couple videos I talked about partial derivatives of multivariable functions and here I want to talk about second partial derivatives so I'm going to write some kind of multivariable function let's say it's I don't know sine of X times y squared sine of X multiplied by Y squared and if you take the partial derivative you have two options given that there's two variables you can go one way and say what's the partial derivative partial derivative of F with respect to X and what you do for that X looks like a variable as far as this direction is concerned Y looks like a constant so we differentiate this by saying the derivative of sine of X is cosine X you know you're differentiating with respect to X and then that looks like it's multiplied by a constant so you just continue multiplying that by a constant but you could also go another direction you could also say you know what's the partial derivative with respect to Y and in that case you're considering Y to be the variable so here it looks at Y and says Y squared looks like a variable X looks like a constant sine of X then just looks like sine of a constant which is a constant so that'll be that constant sine of X multiplied by the derivative of Y squared which is going to be 2 times y 2 times y and these these are what you might call first partial derivatives and there's some alternate notation here DF dy you could also say F and then a little subscript Y and over here similarly you'd say F with a little subscript X now each of these two functions these two partial derivatives that you get are also multi variable functions they take in two variables and they output a scalar so we can do something very similar or here you might then apply the partial derivative with respect to X to that partial derivative of your original function with respect to X right it's it's just like a second derivative in ordinary calculus but this time we're doing it partial so when you do it with respect to X cosine X looks like cosine of a variable the derivative of which is negative sine times that variable and Y squared here just looks like a constant so it just stays constant at Y squared and similarly you could go down a different branch of options here and say what if you did your partial derivative with respect to Y of that whole function which itself is a partial derivative with respect to X and if your debt did that then Y y squared now looks like the variable so you're going to take the derivative of that which is 2 y 2y and then what's in front of it just looks like a constant as far as as far as the variable Y is concerned so that stays is cosine of X and the notation here first of all just as in single variable calculus it's common to kind of do a abusive notation with this kind of thing and write partial squared of F divided by partial x squared and this always I don't know when I first learned about these things they always threw me off because here you dislike nents notation you have the great intuition of you know nudging the X and nudging the F but you kind of lose that when you do this but it makes sense if you think of this partial partial X as being an operator and you're just applying it twice and over here the way that that would look it's a little bit funny because you still have that partial squared F on top but and then on the bottom you write partial Y partial X and you know I'm putting them in these order just because it's as if I I wrote it that way right this reflects the fact that first I did the X derivative then I did the Y derivative and you could do this on this side also and it's my field theories but it's actually actually kind of worth doing for a result that we end up seeing here that then I find a little bit surprising actually so here if we go down the path of doing in this case like a partial derivative with respect to X and you know you're thinking this is being applied to your original partial derivative with respect to Y it looks here it says sine of X looks like a variable 2y looks like a constant so what we end up getting is derivative of sine of X cosine X multiplied by that 2y and a pretty cool thing worth pointing out here that maybe you take it for granted maybe you think it's as surprising as I did when I first saw it both of these turn out to be equal right even though it's a very different way that we got that right you first take the partial derivative with respect to X and you get cosine X Y squared which looks very different from sine X to Y and then when you take the derivative with respect to Y you know you get a certain value and when you go down the other path you also get that same value and maybe the way that you'd write this is that you'd say let me just copy this guy over here and what you might say is that the partial derivative of F when you do it the other way around when instead of doing X and then Y you do Y and then X partial X that these guys are equal to each other and that's a pretty pretty cool result and maybe in this case given that the original function just looks like the product of two things you can kind of Reason through why it's the case but what's surprising is that this turns out to be true for I mean not all functions there's actually a certain a certain criterion there's a special theorem it's called Schwarz Schwartz's theorem where if the second partial derivatives of your function are continuous at the relevant point that's the circumstance for this being true but for all intents and purposes the kind of functions you can expect to run into this is the case this order of partial derivatives doesn't matter truth turns out to hold which is actually pretty cool and I've encourage you to play around with some other functions just come up with any multivariable function may be a little bit more complicated than just multiplying two separate things there and see that it's true and maybe try to convince yourself why it's true in certain cases I think that would actually be a really good exercise and just before I go one thing I should probably mention a bit of notation that the people will commonly use with this second partial derivative sometimes instead of saying partial squared F partial x squared they'll just write it as partial and then X X and over here this would be partial let's see first you did it with X then Y so over here you do it first X and then Y kind of the order of these reverses because you're reading less - right but when you do it with this you're kind of reading right to left for how you multiply it in which would mean that this guy let's see this guy over here you know he would be partial first you did the Y and then you did the X so those two those two guys are just different notations for the same thing I mean that can make it a little bit more convenient when you don't want to write out the entire partial squared F divided by partial x squared or things like that and with that I'll call it an end