 Okay, so we've begun talking about differentials, and how if we have some function that depends on some variables, maybe x and y, then we can calculate the differential of that function, which tells how much f changes in response to some changes in x and y, and perhaps if we have more than two variables, additional variables as well. So let's first work a specific example. Let's say we have a simple function like f of x and y is 3x, y squared. So we can go from there to evaluate the differential of f. df is equal to df dx. If I take the partial derivative of this function with respect to x, the x goes away and I just get 3y squared for df dx, for the derivative of that function with respect to x, multiply that by dx, and then likewise for y, the partial derivative of this function with respect to y, y squared gives me a 2y as a derivative, so now I end up with 6. Still have an x and one factor of y now, and that's multiplying dy. So the differential of this expression is this quantity, some amount in response to a change in x, a different amount in response to a change in y. So we've gone in this direction. We can also do this in the reverse direction. If I, as another example, let's say, let's take a fairly similar differential. Suppose I were to give you this quantity, that the differential of some function f is equal to, let's do this. So I've given you the differential, but I've not given you the original function, but of course we can work backwards. We can say what would this function x, y have to be so that its x derivative looks like this term and its y derivative looks like this term. So we need to think about that in the opposite direction. What function has a partial derivative with respect to x that looks like 4y squared? Well that must be 4xy squared because if I take the partial derivative with respect to x, the x disappears and leaves me with just 4y squared. Likewise, if I ask what function would have a partial y derivative of 8xy, the same one, when I go in this direction I get back the same function, 4xy squared. If I take the partial derivative with respect to y, bringing down this exponent of 2 to 4 times 2 gives me 8xy when I drop the power of y. So in this case we've gone in this direction, we've given a df, we could calculate the function f that gives rise to this differential and I suppose to be complete, I could have said my original function might be 4xy squared plus a constant, plus 5 or plus 7 or plus any old integration constant here because when I take the partial derivatives of this function the derivatives of the constant terms will disappear. So I can't tell from just the differential if the original function had a integration constant in it originally or not. So we can go both directions from function to its differential or from a differential back to at least partially a description of the original function but it doesn't always work that smoothly. Suppose we have a differential that looks like, I'll make it look similar 3y squared dx and let's say 12xy dy. So far so good, it sounds like we just want to do this same procedure again if we want to figure out what the function is that gives rise to this differential. If we work backwards from this piece, if this is df dx, what function has a partial x derivative that looks like 3y squared? That must be 3xy squared but when I do this one if I ask what function has a partial y derivative df dy that looks like 12xy, if I integrate 12xy to figure out what that f would be, the function whose y derivative looks like 12xy, that's going to be something different. 6xy squared, if I take its partial y derivative is going to give me 12xy. So I don't get the same function from this term predicting back what f would be as I do from this term predicting back what the f would be. So what that means is there is in fact no function f, neither this one nor this one nor anything else has the complete differential that looks like this. So what we say in a case like this is these differentials are in fact differentials of some original function whether we knew the function originally or whether we could figure it out retroactively. These differentials are differentials of some function. This is actually not the differential of any function at all. I shouldn't be calling it df because there is no f that it is the differential of. What we actually do for quantities like this is we call them inexact differential. So what that is is I've written a d with a cross through it. Some books or some professors may prefer to write a delta in this form. So either one of these is fine if we instead of writing a lowercase d to represent an exact differential, we either put a bar through the d or we use a delta to represent the differential then what we're expressing is this differential is an inexact differential. So that means the thing even though it has the form of the differential looks like some stuff times dx and some stuff times dy. There is no function of x and y that it is the differential of. It's not the exact differential of anything. It's an inexact differential. So inexact differentials turn out to be useful quantities even though they look like they're breaking the rules. So the next thing we'll do is talk about some relationships between inexact differentials and exact differentials.