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Okay. All right, so, today, we have a few more miracles to uncover about distributions,

but soon – and it's all interesting, and it's all useful, but soon we'll have to make our peace

with generalities. There's a lot more detail and a lot more derivations that are given in

detail in the notes, and so I will refer you to that for further reading. I'm not really gonna

say too much more than what I say today, because we really do have to move on.

Again, we should not – you should not feel encumbered to derive everything. You should

be, I think, satisfied, I hope, with the idea that – to look for the derivations, to try to

understand some of them, and just get a general idea how the framework works, because

I think it really is very satisfying.

In the past, when we've done this in class 261, many people who have seen these ideas

before and worked with delta functions in various contexts in different classes have

appreciated the opportunity to at least see what the more general context is and to see

how the arguments work even if not – even if they don't understand all the details and

haven't gone through all the derivations and details.

So really it's mostly – what we've been doing is to give you an idea of how the general

framework works, and some degree of confidence that there is a firm foundation for a lot

of these things, even without all the details. But you should feel free to use – as I said

before, when you were first starting this, it's not that the stuff we've done before was

wrong, it's not that the formulas that we used were incorrect or the applications really not

well founded, so as we go forward, we'll call on those ideas and call on those formulas

really without – say, without fear of recrimination.

But I think it's – I hope you found it satisfying, intellectually, certainly to see how some

of these ideas play out, because it really is quite striking and it's really quite, I think, quite

a remarkable accomplishment to get it all in such a beautiful form.

And then just a few more things that I wanna pick up today, but there's only so much that

I think we're willing to subject each other to, all right?

So the first thing I wanna talk about, or maybe one of the final topics in the general lore

of how distributions work, is the remarkable fact that, as general as they are, one other

operation from calculus carries over to them, and that is the idea of a derivative. So it's

possible to define the derivative of a distribution and, in fact, although I won't do it,

higher order derivates. That is – derivatives turn – different – excuse me, distributions

turn out to be infinitely differentiable in a natural sense.

So the derivative of a distribution. This actually turns out to be a very important operation

on distributions, and one that's of widespread use. So how would we define – so if t is a

given distribution, how to define its derivative, t prime?

All right, when – any time you ask yourself a question like that, if I want a carry over

operation from functions to distributions, the question is how to do it. Remember, I have

to tell you – it's always the case, you give me a test function, I have to tell you how t

prime operates on that test function. That's always the case.

So I have to say – have to define what the pairing is. T prime paired with the test function

phi. And it is always the case, or at least almost always the case, that the way you

but soon – and it's all interesting, and it's all useful, but soon we'll have to make our peace

with generalities. There's a lot more detail and a lot more derivations that are given in

detail in the notes, and so I will refer you to that for further reading. I'm not really gonna

say too much more than what I say today, because we really do have to move on.

Again, we should not – you should not feel encumbered to derive everything. You should

be, I think, satisfied, I hope, with the idea that – to look for the derivations, to try to

understand some of them, and just get a general idea how the framework works, because

I think it really is very satisfying.

In the past, when we've done this in class 261, many people who have seen these ideas

before and worked with delta functions in various contexts in different classes have

appreciated the opportunity to at least see what the more general context is and to see

how the arguments work even if not – even if they don't understand all the details and

haven't gone through all the derivations and details.

So really it's mostly – what we've been doing is to give you an idea of how the general

framework works, and some degree of confidence that there is a firm foundation for a lot

of these things, even without all the details. But you should feel free to use – as I said

before, when you were first starting this, it's not that the stuff we've done before was

wrong, it's not that the formulas that we used were incorrect or the applications really not

well founded, so as we go forward, we'll call on those ideas and call on those formulas

really without – say, without fear of recrimination.

But I think it's – I hope you found it satisfying, intellectually, certainly to see how some

of these ideas play out, because it really is quite striking and it's really quite, I think, quite

a remarkable accomplishment to get it all in such a beautiful form.

And then just a few more things that I wanna pick up today, but there's only so much that

I think we're willing to subject each other to, all right?

So the first thing I wanna talk about, or maybe one of the final topics in the general lore

of how distributions work, is the remarkable fact that, as general as they are, one other

operation from calculus carries over to them, and that is the idea of a derivative. So it's

possible to define the derivative of a distribution and, in fact, although I won't do it,

higher order derivates. That is – derivatives turn – different – excuse me, distributions

turn out to be infinitely differentiable in a natural sense.

So the derivative of a distribution. This actually turns out to be a very important operation

on distributions, and one that's of widespread use. So how would we define – so if t is a

given distribution, how to define its derivative, t prime?

All right, when – any time you ask yourself a question like that, if I want a carry over

operation from functions to distributions, the question is how to do it. Remember, I have

to tell you – it's always the case, you give me a test function, I have to tell you how t

prime operates on that test function. That's always the case.

So I have to say – have to define what the pairing is. T prime paired with the test function

phi. And it is always the case, or at least almost always the case, that the way you

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