• Title: Motivation and Delta Function

  • Series: Distributions

  • YouTube-Title: Distributions 1 | Motivation and Delta Function

  • Bright video: https://youtu.be/gwVEEUg8PBY

  • Dark video: https://youtu.be/iqk9zVXbf_Y

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: dt01_sub_eng.srt

  • Timestamps

    00:00 Intro

    00:16 Analysis

    01:30 Historical Example

    03:55 Delta is not an ordinary function

    06:10 Visualisation

    8:00 Delta Function

    9:20 Outro

  • Subtitle in English

    1 00:00:00,790 –> 00:00:02,720 Hello and welcome to new

    2 00:00:02,730 –> 00:00:04,059 video series where I want

    3 00:00:04,070 –> 00:00:05,489 to show you an introduction

    4 00:00:05,500 –> 00:00:06,769 into the theory of

    5 00:00:06,780 –> 00:00:07,769 distributions

    6 00:00:09,000 –> 00:00:09,239 Here,

    7 00:00:09,250 –> 00:00:10,939 In part one, I want to start

    8 00:00:10,949 –> 00:00:12,340 with the motivation for this

    9 00:00:12,350 –> 00:00:12,979 new topic.

    10 00:00:14,699 –> 00:00:15,920 When you start with real

    11 00:00:15,930 –> 00:00:17,520 analysis, then you learn

    12 00:00:17,530 –> 00:00:19,319 a lot of important concepts.

    13 00:00:20,510 –> 00:00:21,889 For example, you learn what

    14 00:00:21,899 –> 00:00:23,510 a function is, you learn

    15 00:00:23,520 –> 00:00:24,729 what a limit is.

    16 00:00:24,920 –> 00:00:26,559 And most importantly, you

    17 00:00:26,569 –> 00:00:28,399 learn what a derivative is

    18 00:00:29,879 –> 00:00:31,659 with these objects, you can

    19 00:00:31,670 –> 00:00:33,099 already do a lot of

    20 00:00:33,110 –> 00:00:34,060 mathematics.

    21 00:00:34,330 –> 00:00:35,599 But in the end, you will

    22 00:00:35,610 –> 00:00:37,049 reach other fields for

    23 00:00:37,060 –> 00:00:38,740 example, differential

    24 00:00:38,750 –> 00:00:39,639 equations

    25 00:00:40,470 –> 00:00:42,099 for Fourier series

    26 00:00:42,380 –> 00:00:44,090 and even the Fourier

    27 00:00:44,099 –> 00:00:44,930 transform.

    28 00:00:46,889 –> 00:00:48,369 And now here the problems

    29 00:00:48,380 –> 00:00:50,150 begin because

    30 00:00:50,159 –> 00:00:52,110 for differential equations

    31 00:00:52,119 –> 00:00:53,889 also some solutions

    32 00:00:53,900 –> 00:00:55,419 with sharp turns are

    33 00:00:55,430 –> 00:00:56,889 interesting and important.

    34 00:00:58,400 –> 00:01:00,060 And there you already know

    35 00:01:00,139 –> 00:01:01,299 that the notion of a

    36 00:01:01,310 –> 00:01:03,180 derivative is in some

    37 00:01:03,189 –> 00:01:04,940 sense, too strong to

    38 00:01:04,949 –> 00:01:06,779 consider functions with sharp

    39 00:01:06,790 –> 00:01:08,449 turns because at a

    40 00:01:08,459 –> 00:01:10,150 sharp turn, the function

    41 00:01:10,160 –> 00:01:11,580 is not differentiable.

    42 00:01:13,239 –> 00:01:14,459 Therefore, we need something

    43 00:01:14,470 –> 00:01:16,250 new that can also deal with

    44 00:01:16,260 –> 00:01:17,389 such solutions.

    45 00:01:17,400 –> 00:01:19,139 So we would say we need

    46 00:01:19,150 –> 00:01:20,699 something that can deal with

    47 00:01:20,709 –> 00:01:22,169 more general solutions.

    48 00:01:23,489 –> 00:01:23,970 OK.

    49 00:01:23,980 –> 00:01:25,690 We will talk about this later

    50 00:01:25,800 –> 00:01:26,319 Here,

    51 00:01:26,330 –> 00:01:27,790 I want to start with an

    52 00:01:27,800 –> 00:01:29,349 example and indeed, it’s

    53 00:01:29,360 –> 00:01:30,750 a very famous example.

    54 00:01:31,940 –> 00:01:33,360 It goes back to the year

    55 00:01:33,370 –> 00:01:35,279 1927

    56 00:01:35,510 –> 00:01:36,449 and to Paul Dirac,

    57 00:01:38,550 –> 00:01:40,489 what he wanted was a derivative

    58 00:01:40,500 –> 00:01:41,389 of a function.

    59 00:01:42,250 –> 00:01:43,879 And we used letter H

    60 00:01:43,889 –> 00:01:45,610 because this function is

    61 00:01:45,620 –> 00:01:47,129 most of the time known as

    62 00:01:47,139 –> 00:01:48,370 the Heaviside function.

    63 00:01:49,529 –> 00:01:51,330 In spite of having this fancy

    64 00:01:51,339 –> 00:01:53,139 name, the function is indeed

    65 00:01:53,150 –> 00:01:54,099 very simple.

    66 00:01:56,059 –> 00:01:57,699 It is just zero on the

    67 00:01:57,709 –> 00:01:59,669 negative axis and

    68 00:01:59,680 –> 00:02:01,279 one on the positive

    69 00:02:01,290 –> 00:02:01,860 axis.

    70 00:02:03,849 –> 00:02:05,510 And now Dirac wanted

    71 00:02:05,519 –> 00:02:07,029 to have a derivative of this

    72 00:02:07,040 –> 00:02:07,699 function.

    73 00:02:08,000 –> 00:02:09,710 And you see this is no

    74 00:02:09,720 –> 00:02:11,649 problem at all outside

    75 00:02:11,660 –> 00:02:12,389 of zero.

    76 00:02:12,529 –> 00:02:14,240 But in zero, there is a

    77 00:02:14,250 –> 00:02:14,750 jump.

    78 00:02:16,190 –> 00:02:18,169 Hence, in the classical way,

    79 00:02:18,300 –> 00:02:20,009 we have a problem at this

    80 00:02:20,020 –> 00:02:20,770 one point.

    81 00:02:22,509 –> 00:02:24,119 Now Dirac wanted a more

    82 00:02:24,130 –> 00:02:25,889 general derivative and he

    83 00:02:25,899 –> 00:02:27,850 called this one the delta

    84 00:02:27,860 –> 00:02:28,449 function.

    85 00:02:29,800 –> 00:02:31,699 Now more general here means

    86 00:02:31,710 –> 00:02:32,869 of course, it is a

    87 00:02:32,880 –> 00:02:34,339 generalization of the

    88 00:02:34,350 –> 00:02:35,589 classical notion of a

    89 00:02:35,600 –> 00:02:36,380 derivative.

    90 00:02:37,449 –> 00:02:39,330 Hence, we can conclude that

    91 00:02:39,339 –> 00:02:40,940 this new delta function

    92 00:02:41,039 –> 00:02:42,429 has to be zero

    93 00:02:42,600 –> 00:02:44,240 outside of this

    94 00:02:44,250 –> 00:02:46,050 critical point X equals to

    95 00:02:46,059 –> 00:02:46,630 zero.

    96 00:02:51,080 –> 00:02:52,779 However, Dirac also

    97 00:02:52,789 –> 00:02:54,509 wanted that the fundamental

    98 00:02:54,520 –> 00:02:56,440 theorem of calculus still

    99 00:02:56,449 –> 00:02:57,910 holds for this new function

    100 00:02:59,970 –> 00:03:00,949 more concretely.

    101 00:03:00,960 –> 00:03:02,669 This means we can write

    102 00:03:02,679 –> 00:03:04,500 down for an arbitrarily

    103 00:03:04,509 –> 00:03:05,910 small epsilon

    104 00:03:06,600 –> 00:03:07,570 the following integral

    105 00:03:09,440 –> 00:03:10,660 which is just an

    106 00:03:10,669 –> 00:03:12,589 integration around zero

    107 00:03:12,600 –> 00:03:13,759 for a delta function.

    108 00:03:13,970 –> 00:03:15,759 But now we know this is the

    109 00:03:15,770 –> 00:03:17,509 derivative of the Heaviside

    110 00:03:17,520 –> 00:03:18,419 function.

    111 00:03:18,429 –> 00:03:19,949 So we can put in

    112 00:03:20,300 –> 00:03:21,820 the derivative of H here.

    113 00:03:25,250 –> 00:03:27,080 And now if the fundamental

    114 00:03:27,089 –> 00:03:28,940 theorem of calculus holds

    115 00:03:28,949 –> 00:03:30,750 for these functions here

    116 00:03:30,759 –> 00:03:32,690 we have of

    117 00:03:32,720 –> 00:03:33,830 epsilon

    118 00:03:33,889 –> 00:03:34,869 minus

    119 00:03:35,429 –> 00:03:37,229 H of minus epsilon.

    120 00:03:38,729 –> 00:03:40,289 Now, looking at the graph,

    121 00:03:40,300 –> 00:03:42,149 we recognize that this is

    122 00:03:42,160 –> 00:03:43,949 always one on the right hand

    123 00:03:43,960 –> 00:03:45,889 side and always zero

    124 00:03:45,899 –> 00:03:46,990 on the other side.

    125 00:03:47,210 –> 00:03:49,130 So we have one minus zero

    126 00:03:49,179 –> 00:03:51,020 or in other words, always

    127 00:03:51,029 –> 00:03:51,389 one.

    128 00:03:53,979 –> 00:03:55,470 Now this may look

    129 00:03:55,479 –> 00:03:57,240 nice, but we

    130 00:03:57,250 –> 00:03:59,229 already know that if we

    131 00:03:59,240 –> 00:04:01,039 have these two things together,

    132 00:04:01,250 –> 00:04:03,080 this delta can’t be an

    133 00:04:03,089 –> 00:04:04,270 ordinary function.

    134 00:04:05,339 –> 00:04:06,610 Please recall.

    135 00:04:06,619 –> 00:04:08,339 This first part tells

    136 00:04:08,350 –> 00:04:10,009 us that delta

    137 00:04:10,020 –> 00:04:11,320 is zero

    138 00:04:11,330 –> 00:04:12,750 except for one

    139 00:04:12,759 –> 00:04:13,330 point.

    140 00:04:13,339 –> 00:04:14,779 So in our language, this

    141 00:04:14,789 –> 00:04:16,529 would mean delta

    142 00:04:16,540 –> 00:04:18,119 is zero almost

    143 00:04:18,130 –> 00:04:18,809 everywhere.

    144 00:04:20,069 –> 00:04:21,628 To be more concrete, I would

    145 00:04:21,639 –> 00:04:23,598 say this holds with

    146 00:04:23,609 –> 00:04:25,359 respect to the Lebesgue measure.

    147 00:04:27,709 –> 00:04:29,660 And from this, we can conclude

    148 00:04:29,730 –> 00:04:31,589 that every integral for

    149 00:04:31,600 –> 00:04:33,140 this function has to be

    150 00:04:33,149 –> 00:04:33,690 zero.

    151 00:04:36,250 –> 00:04:38,190 In particular, the integral

    152 00:04:38,200 –> 00:04:39,160 from before.

    153 00:04:39,170 –> 00:04:41,010 So minus epsilon to

    154 00:04:41,019 –> 00:04:42,589 epsilon of delta

    155 00:04:42,600 –> 00:04:44,339 X is also

    156 00:04:44,350 –> 00:04:46,209 zero for all epsilon,

    157 00:04:47,929 –> 00:04:49,619 which means it can’t be

    158 00:04:49,630 –> 00:04:51,329 one as we wanted

    159 00:04:51,339 –> 00:04:52,029 before.

    160 00:04:52,290 –> 00:04:54,190 So here’s our contradiction.

    161 00:04:56,910 –> 00:04:58,609 Now it gets even worse

    162 00:04:58,619 –> 00:05:00,390 because Dirac does not stop

    163 00:05:00,399 –> 00:05:00,929 there.

    164 00:05:01,220 –> 00:05:01,359 Dirac

    165 00:05:01,660 –> 00:05:03,589 also wants to calculate with

    166 00:05:03,600 –> 00:05:04,929 the derivatives of this new

    167 00:05:04,940 –> 00:05:05,890 delta function.

    168 00:05:07,269 –> 00:05:09,140 And now we see at this point,

    169 00:05:09,350 –> 00:05:10,750 we have to put a

    170 00:05:10,760 –> 00:05:12,420 meaning into the symbols

    171 00:05:12,429 –> 00:05:12,730 here.

    172 00:05:14,309 –> 00:05:16,209 In order to do this, we

    173 00:05:16,220 –> 00:05:17,589 will define a new

    174 00:05:17,600 –> 00:05:19,589 object and call this a

    175 00:05:19,600 –> 00:05:20,510 distribution.

    176 00:05:20,950 –> 00:05:22,429 And in the end, delta

    177 00:05:22,630 –> 00:05:24,269 will be such a distribution.

    178 00:05:25,209 –> 00:05:26,390 For the reasons you have

    179 00:05:26,399 –> 00:05:28,250 seen above, we want to

    180 00:05:28,260 –> 00:05:29,839 calculate as we have

    181 00:05:29,850 –> 00:05:31,350 ordinary functions, this

    182 00:05:31,359 –> 00:05:33,320 new notion distribution is

    183 00:05:33,329 –> 00:05:35,200 also often called a

    184 00:05:35,209 –> 00:05:36,510 generalized function.

    185 00:05:37,459 –> 00:05:38,899 The overall idea for

    186 00:05:38,910 –> 00:05:40,540 distributions is that we

    187 00:05:40,549 –> 00:05:42,480 see these objects as

    188 00:05:42,489 –> 00:05:43,480 densities.

    189 00:05:43,640 –> 00:05:44,769 So here you should

    190 00:05:44,779 –> 00:05:46,670 visualize a one

    191 00:05:46,690 –> 00:05:48,250 dimensional rode or

    192 00:05:48,260 –> 00:05:50,190 one dimensional bar

    193 00:05:50,549 –> 00:05:52,459 where we have a mass

    194 00:05:52,470 –> 00:05:53,239 density.

    195 00:05:54,679 –> 00:05:56,269 Now, in this picture, the

    196 00:05:56,279 –> 00:05:57,640 delta distribution or the

    197 00:05:57,649 –> 00:05:59,250 delta function would be a

    198 00:05:59,260 –> 00:06:01,140 singular density.

    199 00:06:01,309 –> 00:06:02,899 So a density where

    200 00:06:02,910 –> 00:06:04,730 all the mass is one

    201 00:06:04,739 –> 00:06:05,279 point.

    202 00:06:07,250 –> 00:06:07,750 OK.

    203 00:06:07,760 –> 00:06:09,510 So let’s do a picture for

    204 00:06:09,519 –> 00:06:10,670 this visualization.

    205 00:06:12,570 –> 00:06:13,929 The curve of an ordinary

    206 00:06:13,940 –> 00:06:15,859 function may look like this.

    207 00:06:17,260 –> 00:06:19,000 Now you should see this as

    208 00:06:19,010 –> 00:06:20,760 the mass distribution

    209 00:06:20,899 –> 00:06:22,769 on this one dimensional rode

    210 00:06:22,779 –> 00:06:23,149 here.

    211 00:06:24,019 –> 00:06:25,989 And this means here

    212 00:06:26,000 –> 00:06:27,709 we have a high density

    213 00:06:28,000 –> 00:06:29,489 and at this point, we have

    214 00:06:29,500 –> 00:06:30,559 a low density.

    215 00:06:32,109 –> 00:06:33,630 Now for finding the

    216 00:06:33,640 –> 00:06:35,600 density in the real world,

    217 00:06:35,609 –> 00:06:36,850 you would use a measurement

    218 00:06:36,859 –> 00:06:38,850 device in

    219 00:06:38,859 –> 00:06:40,549 mathematics, we then use

    220 00:06:40,559 –> 00:06:42,440 so called test functions.

    221 00:06:43,989 –> 00:06:45,750 So functions phi that

    222 00:06:45,760 –> 00:06:47,109 should be nice enough.

    223 00:06:47,119 –> 00:06:48,929 So at least continuous and

    224 00:06:48,940 –> 00:06:50,529 localized in some sense

    225 00:06:51,890 –> 00:06:53,230 for me, that means they are

    226 00:06:53,239 –> 00:06:55,149 zero except for one

    227 00:06:55,160 –> 00:06:56,190 small bump.

    228 00:06:56,510 –> 00:06:58,200 And that’s the region you

    229 00:06:58,209 –> 00:06:58,779 measure in.

    230 00:07:00,779 –> 00:07:02,040 Now, in order to find the

    231 00:07:02,049 –> 00:07:03,779 mass in that region, we have

    232 00:07:03,790 –> 00:07:05,200 to calculate an integral.

    233 00:07:05,209 –> 00:07:06,649 So an integral over

    234 00:07:06,660 –> 00:07:08,239 R function

    235 00:07:08,250 –> 00:07:10,059 f and function

    236 00:07:10,200 –> 00:07:10,480 Phi

    237 00:07:14,149 –> 00:07:16,119 here we get out a real number

    238 00:07:16,440 –> 00:07:17,839 which describes

    239 00:07:17,850 –> 00:07:19,839 our density

    240 00:07:20,119 –> 00:07:22,089 here in that region.

    241 00:07:22,429 –> 00:07:24,269 Yeah, roughly in that way.

    242 00:07:26,829 –> 00:07:28,570 And now the idea is that

    243 00:07:28,579 –> 00:07:30,089 we use a lot of these

    244 00:07:30,100 –> 00:07:31,260 small bumps.

    245 00:07:31,290 –> 00:07:32,760 So we get out a lot of real

    246 00:07:32,769 –> 00:07:33,679 numbers.

    247 00:07:33,690 –> 00:07:34,950 In other words, we have a

    248 00:07:34,959 –> 00:07:36,619 map that maps

    249 00:07:36,920 –> 00:07:38,720 all the phis to

    250 00:07:38,730 –> 00:07:40,359 these real numbers given

    251 00:07:40,369 –> 00:07:41,149 by the integral.

    252 00:07:42,480 –> 00:07:44,149 Or in other words, instead

    253 00:07:44,160 –> 00:07:45,640 of looking at the function

    254 00:07:45,649 –> 00:07:47,410 F itself, we can

    255 00:07:47,420 –> 00:07:49,359 just look at the collection

    256 00:07:49,380 –> 00:07:50,970 of all these real

    257 00:07:50,980 –> 00:07:51,839 numbers here.

    258 00:07:53,279 –> 00:07:54,679 Indeed, that is what we will

    259 00:07:54,690 –> 00:07:56,100 do in the next videos, we

    260 00:07:56,109 –> 00:07:57,410 will substitute the function

    261 00:07:57,420 –> 00:07:59,329 F with this linear

    262 00:07:59,339 –> 00:07:59,890 map.

    263 00:08:01,380 –> 00:08:02,920 Now, the big advantage of

    264 00:08:02,929 –> 00:08:04,720 this is of course, that

    265 00:08:04,730 –> 00:08:06,440 now we can also deal with

    266 00:08:06,450 –> 00:08:08,160 this strange delta function.

    267 00:08:09,260 –> 00:08:10,750 Obviously, we can’t draw

    268 00:08:10,760 –> 00:08:12,239 a graph for this delta

    269 00:08:12,250 –> 00:08:13,809 function because it’s not

    270 00:08:13,820 –> 00:08:14,799 an ordinary function.

    271 00:08:16,369 –> 00:08:17,709 However, we have our

    272 00:08:17,720 –> 00:08:19,260 visualization that it is

    273 00:08:19,269 –> 00:08:20,649 a point mass.

    274 00:08:20,790 –> 00:08:22,630 So this means it is zero

    275 00:08:22,640 –> 00:08:24,299 except for one point

    276 00:08:24,940 –> 00:08:26,519 and we have put that point

    277 00:08:26,529 –> 00:08:27,440 in zero.

    278 00:08:27,779 –> 00:08:29,589 So here we have a jump

    279 00:08:29,959 –> 00:08:31,320 and this jump should be

    280 00:08:31,329 –> 00:08:32,510 infinitely high.

    281 00:08:34,669 –> 00:08:36,328 So what does this mean?

    282 00:08:36,619 –> 00:08:38,179 Yeah, it means if you want

    283 00:08:38,188 –> 00:08:40,049 to measure with our test

    284 00:08:40,058 –> 00:08:41,688 functions, you get

    285 00:08:41,698 –> 00:08:43,568 out the value of the test

    286 00:08:43,578 –> 00:08:45,198 function at this one

    287 00:08:45,208 –> 00:08:45,828 point.

    288 00:08:49,059 –> 00:08:51,039 So this makes sense for our

    289 00:08:51,049 –> 00:08:52,960 measurement devices here.

    290 00:08:53,429 –> 00:08:55,390 And also for the integral

    291 00:08:55,400 –> 00:08:57,080 we have seen before for the

    292 00:08:57,090 –> 00:08:57,919 delta function.

    293 00:08:59,500 –> 00:09:01,450 Indeed, we generalize this

    294 00:09:01,460 –> 00:09:02,559 integral here.

    295 00:09:02,919 –> 00:09:04,440 So now we want to put in

    296 00:09:04,450 –> 00:09:05,929 a new function, delta

    297 00:09:06,210 –> 00:09:08,179 function instead of f

    298 00:09:08,190 –> 00:09:10,179 but we learn we can’t see

    299 00:09:10,190 –> 00:09:11,960 that as a real integral.

    300 00:09:11,969 –> 00:09:13,830 What we have indeed is a

    301 00:09:13,840 –> 00:09:15,000 generalization of this

    302 00:09:15,010 –> 00:09:16,989 concept to this linear

    303 00:09:17,000 –> 00:09:17,359 map.

    304 00:09:19,750 –> 00:09:20,890 Well, I think that’s good

    305 00:09:20,900 –> 00:09:21,890 enough for today.

    306 00:09:21,900 –> 00:09:23,130 So now you see the

    307 00:09:23,140 –> 00:09:24,590 motivation for

    308 00:09:24,599 –> 00:09:25,659 distributions.

    309 00:09:25,849 –> 00:09:27,270 And in the next video, I

    310 00:09:27,280 –> 00:09:28,739 will give the exact

    311 00:09:28,750 –> 00:09:30,210 definition of such a

    312 00:09:30,219 –> 00:09:30,969 distribution.

    313 00:09:32,299 –> 00:09:33,840 This means that we will learn

    314 00:09:33,849 –> 00:09:35,440 what the test functions are

    315 00:09:35,450 –> 00:09:37,169 actually and also

    316 00:09:37,179 –> 00:09:38,750 what this linear map then

    317 00:09:38,760 –> 00:09:39,260 is

    318 00:09:41,219 –> 00:09:42,479 then thank you very much

    319 00:09:42,489 –> 00:09:43,760 for listening and see you

    320 00:09:43,770 –> 00:09:44,440 next time.

    321 00:09:44,849 –> 00:09:45,510 Bye.

  • Quiz Content

    Q1: What is the definition of the Heaviside function?

    A1: $$ H(x) = \begin{cases} 1,, ~~ x \geq 1 \ 0,, ~~ x < 1 \end{cases} $$

    A2: $$ H(x) = \begin{cases} 1,, ~~ x > 0 \ 0,, ~~ x \leq 0 \end{cases} $$

    A3: $$ H(x) = \begin{cases} -1,, ~~ x > 0 \ 0,, ~~ x \leq 0 \end{cases} $$

    Q2: Is the Heaviside function $H$ differentiable?

    A1: Yes!

    A2: No!

    Q3: What is a conclusion of the fundamental theorem of calculus for a continuously differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$?

    A1: $$ \int_a^b f^\prime(x) , dx = f(b) - f(a)$$

    A2: $$ \int_a^b f^\prime(x) , dx = f^\prime(b) - f^\prime(a)$$

    A3: $$ \int_a^b f(x), dx = f(b) - f(a)$$

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