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Title: Motivation and Delta Function
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Series: Distributions
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YouTube-Title: Distributions 1 | Motivation and Delta Function
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Bright video: https://youtu.be/gwVEEUg8PBY
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Dark video: https://youtu.be/iqk9zVXbf_Y
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: dt01_sub_eng.srt
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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
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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.
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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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Last update: 2024-11
