1 00:00:00,000 --> 00:00:00,380 2 00:00:00,380 --> 00:00:02,439 In the last video, I introduced you to what is 3 00:00:02,439 --> 00:00:04,849 probably the most bizarro function that you've 4 00:00:04,849 --> 00:00:05,949 encountered so far. 5 00:00:05,950 --> 00:00:09,234 And that was the Dirac delta function. 6 00:00:09,234 --> 00:00:13,529 7 00:00:13,529 --> 00:00:16,189 And I defined it to be-- and I'll do the 8 00:00:16,190 --> 00:00:17,400 shifted version of it. 9 00:00:17,399 --> 00:00:21,969 You're already hopefully reasonably familiar with it. 10 00:00:21,969 --> 00:00:25,379 So Dirac delta of t minus c. 11 00:00:25,379 --> 00:00:31,369 We can say that it equals 0, when t does not equal c, so it 12 00:00:31,370 --> 00:00:33,774 equals 0 everywhere, but it 13 00:00:33,774 --> 00:00:35,449 essentially pops up to infinity. 14 00:00:35,450 --> 00:00:37,470 And we have to be careful with this infinity. 15 00:00:37,469 --> 00:00:38,759 I'm going to write it in quotes. 16 00:00:38,759 --> 00:00:40,969 It pops up to infinity. 17 00:00:40,969 --> 00:00:42,699 And we even saw in the previous video, it's kind of 18 00:00:42,700 --> 00:00:45,100 different degrees of infinity, because you can still multiply 19 00:00:45,100 --> 00:00:48,510 this by other numbers to get larger Dirac delta functions 20 00:00:48,509 --> 00:00:50,559 when t is equal to c. 21 00:00:50,560 --> 00:00:53,170 But more important than this, and this is kind of a 22 00:00:53,170 --> 00:00:56,750 pseudodefinition here, is the idea that when we take the 23 00:00:56,750 --> 00:01:00,859 integral, when we take the area under the curve over the 24 00:01:00,859 --> 00:01:04,838 entire x- or the entire t-axis, I guess we could say, 25 00:01:04,838 --> 00:01:08,730 when we take the area under this curve, and obviously, it 26 00:01:08,730 --> 00:01:13,890 equals zero everywhere except at t is equal to c, when we 27 00:01:13,890 --> 00:01:17,930 take this area, this is the important point, that the area 28 00:01:17,930 --> 00:01:19,180 is equal to 1. 29 00:01:19,180 --> 00:01:22,630 30 00:01:22,629 --> 00:01:25,829 And so this is what I meant by pseudoinfinity, because if I 31 00:01:25,829 --> 00:01:32,730 have 2 times the Dirac delta function, and if I'm taking 32 00:01:32,730 --> 00:01:37,140 the area under the curve of that, of 2 times the Dirac 33 00:01:37,140 --> 00:01:41,370 delta function t minus c dt, this should be equal to 2 34 00:01:41,370 --> 00:01:46,410 times-- the area of just under the Dirac delta function 2 35 00:01:46,409 --> 00:01:50,569 times from minus infinity to infinity of the delta function 36 00:01:50,569 --> 00:01:55,119 shifted by c dt, which is just 2 times-- we already showed 37 00:01:55,120 --> 00:01:58,090 you, I just said, by definition, this is 1, so this 38 00:01:58,090 --> 00:02:00,060 will be equal to 2. 39 00:02:00,060 --> 00:02:04,620 So if I put a 2 out here, this infinity will have to be twice 40 00:02:04,620 --> 00:02:07,370 as high, so that the area is now 2. 41 00:02:07,370 --> 00:02:09,870 That's why I put that infinity in parentheses. 42 00:02:09,870 --> 00:02:11,180 But it's an interesting function. 43 00:02:11,180 --> 00:02:14,099 I talked about it at the end of the last video that it can 44 00:02:14,099 --> 00:02:17,719 help model things that kind of jar things all of a sudden, 45 00:02:17,719 --> 00:02:21,969 but they impart a fixed amount of impulse on something and a 46 00:02:21,969 --> 00:02:24,009 fixed amount of change in momentum. 47 00:02:24,009 --> 00:02:26,659 And we'll understand that a little bit more in the future. 48 00:02:26,659 --> 00:02:30,299 But let's kind of get the mathematical tools completely 49 00:02:30,300 --> 00:02:30,719 understood. 50 00:02:30,719 --> 00:02:33,590 And let's try to figure out what the Dirac delta function 51 00:02:33,590 --> 00:02:36,200 does when we multiply it, what it does to the Laplace 52 00:02:36,199 --> 00:02:38,549 transform when we multiply it times some function. 53 00:02:38,550 --> 00:02:42,105 So let's say I have my Dirac delta function and I'm going 54 00:02:42,104 --> 00:02:43,169 to shift it. 55 00:02:43,169 --> 00:02:44,559 That's a little bit more interesting. 56 00:02:44,560 --> 00:02:46,479 And if you want to unshift it, you just say, OK, 57 00:02:46,479 --> 00:02:47,439 well, c equals 0. 58 00:02:47,439 --> 00:02:49,030 What happens when c equals 0? 59 00:02:49,030 --> 00:02:52,520 And I'm going to shift it and multiply it times some 60 00:02:52,520 --> 00:02:54,719 arbitrary function f of t. 61 00:02:54,719 --> 00:02:57,590 If I wanted to figure out the Laplace transform of just the 62 00:02:57,590 --> 00:02:59,800 delta function by itself, I could say f of 63 00:02:59,800 --> 00:03:01,390 t is equal to 1. 64 00:03:01,389 --> 00:03:08,019 So let's take our Laplace transform of this. 65 00:03:08,020 --> 00:03:10,219 And we can just use the definition of the Laplace 66 00:03:10,219 --> 00:03:16,740 transform, so this is equal to the area from 0 to infinity, 67 00:03:16,740 --> 00:03:21,510 or we could call it the integral from 0 to infinity of 68 00:03:21,509 --> 00:03:24,340 e to the minus -- that's just part of the Laplace transform 69 00:03:24,340 --> 00:03:27,094 definition-- times this thing-- and I'll just write it 70 00:03:27,094 --> 00:03:31,080 in this order-- times f of t times 71 00:03:31,080 --> 00:03:33,960 our Dirac delta function. 72 00:03:33,960 --> 00:03:38,599 Delta t minus c and times dt. 73 00:03:38,599 --> 00:03:41,359 74 00:03:41,360 --> 00:03:43,090 Now, here I'm going to make a little bit of 75 00:03:43,090 --> 00:03:44,670 an intuitive argument. 76 00:03:44,669 --> 00:03:47,759 A lot of the math we do is kind of-- especially if you 77 00:03:47,759 --> 00:03:50,409 want to be very rigorous and formal, the Dirac delta 78 00:03:50,409 --> 00:03:53,659 function starts to break down a lot of tools that you might 79 00:03:53,659 --> 00:03:55,169 have not realized it would break down, but I think 80 00:03:55,169 --> 00:03:56,919 intuitively, we can still work with it. 81 00:03:56,919 --> 00:04:00,424 So I'm going to solve this integral for you intuitively, 82 00:04:00,425 --> 00:04:02,990 and I think it'll make some sense. 83 00:04:02,990 --> 00:04:05,379 So let's draw this. 84 00:04:05,379 --> 00:04:08,789 Let me draw this, what we're trying to do. 85 00:04:08,789 --> 00:04:10,310 So let me draw what we're trying to take 86 00:04:10,310 --> 00:04:11,375 the integral of. 87 00:04:11,375 --> 00:04:13,599 And we only care from zero to infinity, so I'll only do it 88 00:04:13,599 --> 00:04:14,609 from zero to infinity. 89 00:04:14,610 --> 00:04:17,819 And I'll assume that c is greater than zero, that the 90 00:04:17,819 --> 00:04:19,560 delta function pops up someplace in 91 00:04:19,560 --> 00:04:21,870 the positive t-axis. 92 00:04:21,870 --> 00:04:25,620 So what is this first part going to look like? 93 00:04:25,620 --> 00:04:27,610 What is that going to look like? e to the minus 94 00:04:27,610 --> 00:04:28,780 st times f of t? 95 00:04:28,779 --> 00:04:29,219 I don't know. 96 00:04:29,220 --> 00:04:31,460 It's going to be some function. e to the minus st 97 00:04:31,459 --> 00:04:34,319 starts at 1 and drops down, but we're multiplying it times 98 00:04:34,319 --> 00:04:38,269 some arbitrary function, so I'll just draw it like this. 99 00:04:38,269 --> 00:04:40,509 Maybe it looks something like this. 100 00:04:40,509 --> 00:04:45,569 This right here is e to the minus st times f of t. 101 00:04:45,569 --> 00:04:48,860 And the f of t is what kind of gives it its arbitrary shape. 102 00:04:48,860 --> 00:04:50,199 Fair enough. 103 00:04:50,199 --> 00:04:53,110 Now, let's graph our Dirac delta function. 104 00:04:53,110 --> 00:04:57,980 With zero everywhere except right at c, right at c right 105 00:04:57,980 --> 00:05:01,840 there, it pops up infinitely high, but we only draw an 106 00:05:01,839 --> 00:05:05,539 arrow that is of height 1 to show that its area is 1. 107 00:05:05,540 --> 00:05:07,260 I mean, normally when you graph things you don't draw 108 00:05:07,259 --> 00:05:10,110 arrows, but this arrow shows that the area under this 109 00:05:10,110 --> 00:05:12,290 infinitely high thing is 1. 110 00:05:12,290 --> 00:05:14,260 So we do a 1 there. 111 00:05:14,259 --> 00:05:16,629 So if we multiply this, we care about the area under this 112 00:05:16,629 --> 00:05:17,409 whole thing. 113 00:05:17,410 --> 00:05:21,120 When we multiply these two functions, when we multiply 114 00:05:21,120 --> 00:05:24,050 this times this times the delta function, this is-- let 115 00:05:24,050 --> 00:05:24,569 me write this. 116 00:05:24,569 --> 00:05:27,800 This is the delta function shifted to c. 117 00:05:27,800 --> 00:05:30,949 If I multiply that times that, what do I get? 118 00:05:30,949 --> 00:05:34,060 This is kind of the key intuition here. 119 00:05:34,060 --> 00:05:37,410 Let me redraw my axes. 120 00:05:37,410 --> 00:05:41,500 Let me see if I can do it a little bit straighter. 121 00:05:41,500 --> 00:05:44,930 Don't judge me by the straightness of my axes. 122 00:05:44,930 --> 00:05:46,259 So that's t. 123 00:05:46,259 --> 00:05:48,480 So what happens when I multiply these two? 124 00:05:48,480 --> 00:05:52,480 Everywhere, when t equals anything other than c, the 125 00:05:52,480 --> 00:05:55,569 Dirac delta function is zero. 126 00:05:55,569 --> 00:05:57,290 So it's zero times anything. 127 00:05:57,290 --> 00:05:58,560 I don't care what this function is going to do, it's 128 00:05:58,560 --> 00:05:59,920 going to be zero. 129 00:05:59,920 --> 00:06:02,759 So it's going to be zero everywhere, except something 130 00:06:02,759 --> 00:06:07,050 interesting happens at t is equal to c. 131 00:06:07,050 --> 00:06:10,110 At t equals c, what's the value of the function? 132 00:06:10,110 --> 00:06:12,000 Well, it's going to be the value of 133 00:06:12,000 --> 00:06:14,069 the Dirac delta function. 134 00:06:14,069 --> 00:06:17,759 It's going to be the Dirac delta function times whatever 135 00:06:17,759 --> 00:06:20,029 height this is. 136 00:06:20,029 --> 00:06:23,409 This is going to be this point right here or this right 137 00:06:23,410 --> 00:06:26,530 there, that point. 138 00:06:26,529 --> 00:06:30,029 This is going to be this function evaluated at c. 139 00:06:30,029 --> 00:06:33,829 I'll mark it right here on the y-axis, or on the f of t, 140 00:06:33,829 --> 00:06:34,949 whatever you want to call it. 141 00:06:34,949 --> 00:06:39,764 This is going to be e to the minus sc times f of c. 142 00:06:39,764 --> 00:06:42,969 All I'm doing is I'm just evaluating this function at c, 143 00:06:42,970 --> 00:06:45,150 so that's the point right there. 144 00:06:45,149 --> 00:06:48,179 So if you take this point, which is just some number, it 145 00:06:48,180 --> 00:06:51,840 could be 5, 5 times this, you're just getting 5 times 146 00:06:51,839 --> 00:06:53,139 the Dirac delta function. 147 00:06:53,139 --> 00:06:54,539 Or in this case, it's not 5. 148 00:06:54,540 --> 00:06:55,930 It's this little more abstract thing. 149 00:06:55,930 --> 00:06:58,660 I could just draw it like this. 150 00:06:58,660 --> 00:07:02,080 When I multiply this thing times my little delta function 151 00:07:02,079 --> 00:07:04,259 there, I get this. 152 00:07:04,259 --> 00:07:08,560 The height, it's a delta function, but it's scaled now. 153 00:07:08,560 --> 00:07:11,420 It's scaled, so now my new thing is going 154 00:07:11,420 --> 00:07:12,949 to look like this. 155 00:07:12,949 --> 00:07:17,079 If I just multiply that times that, I essentially get e to 156 00:07:17,079 --> 00:07:20,819 the minus sc times f of c. 157 00:07:20,819 --> 00:07:23,240 This might look like some fancy function, but it's just 158 00:07:23,240 --> 00:07:25,600 a number when we consider it in terms of t. 159 00:07:25,600 --> 00:07:29,240 s, it becomes something when we go into the Laplace world, 160 00:07:29,240 --> 00:07:32,250 but from t's point of view, it's just a constant. 161 00:07:32,250 --> 00:07:34,100 All of these are just constants, so this might as 162 00:07:34,100 --> 00:07:35,430 well just be 5. 163 00:07:35,430 --> 00:07:41,060 So it's this constant times my Dirac delta function, times 164 00:07:41,060 --> 00:07:43,360 delta of t minus c. 165 00:07:43,360 --> 00:07:46,600 When I multiply that thing times that thing, all I'm left 166 00:07:46,600 --> 00:07:48,680 with is this thing. 167 00:07:48,680 --> 00:07:51,314 And this height is still going to be infinitely high, but 168 00:07:51,314 --> 00:07:57,069 it's infinitely high scaled in such a way that its area is 169 00:07:57,069 --> 00:07:59,129 going to be not 1. 170 00:07:59,129 --> 00:08:00,500 And I'll show it to you. 171 00:08:00,500 --> 00:08:02,850 So what's the integral of this thing? 172 00:08:02,850 --> 00:08:04,980 Taking the integral of this thing from minus infinity to 173 00:08:04,980 --> 00:08:07,870 infinity, since this thing is this thing, it should be the 174 00:08:07,870 --> 00:08:09,379 same thing as taking the integral of this thing from 175 00:08:09,379 --> 00:08:10,709 minus infinity to infinity. 176 00:08:10,709 --> 00:08:12,219 So let's do that. 177 00:08:12,220 --> 00:08:13,500 Actually, we don't have to do it from minus infinity. 178 00:08:13,500 --> 00:08:15,220 I said from zero to infinity. 179 00:08:15,220 --> 00:08:19,850 So if we take from zero to infinity, what I'm saying is 180 00:08:19,850 --> 00:08:21,390 taking this integral is equivalent 181 00:08:21,389 --> 00:08:23,159 to taking this integral. 182 00:08:23,160 --> 00:08:31,260 So e to the minus sc f of c times my delta function t 183 00:08:31,259 --> 00:08:34,269 minus c dt. 184 00:08:34,269 --> 00:08:37,178 Now, this thing right here, let me make this very clear, 185 00:08:37,178 --> 00:08:41,279 I'm claiming that this is equivalent to this. 186 00:08:41,279 --> 00:08:44,529 Because everywhere else, the delta function zeroes out this 187 00:08:44,529 --> 00:08:47,299 function, so we only care about this function, or e to 188 00:08:47,299 --> 00:08:50,429 the minus st f of t when t is equal to c. 189 00:08:50,429 --> 00:08:52,809 And so that's why we were able to turn it into a constant. 190 00:08:52,809 --> 00:08:54,919 But since this is a constant, we can bring it out of the 191 00:08:54,919 --> 00:08:59,379 integral, and so this is equal to-- I'm going to go backwards 192 00:08:59,379 --> 00:09:01,860 here just to kind of save space and still give you these 193 00:09:01,860 --> 00:09:03,120 things to look at. 194 00:09:03,120 --> 00:09:07,690 If we take out the constants from inside of the integral, 195 00:09:07,690 --> 00:09:14,540 we get e to the minus sc times f of c times the integral from 196 00:09:14,539 --> 00:09:21,649 0 to infinity of f of t minus c dt. 197 00:09:21,649 --> 00:09:23,829 Oh sorry, not f of t minus c. 198 00:09:23,830 --> 00:09:27,520 199 00:09:27,519 --> 00:09:28,370 This is not an f. 200 00:09:28,370 --> 00:09:29,379 I have to be very careful. 201 00:09:29,379 --> 00:09:31,000 This is a delta. 202 00:09:31,000 --> 00:09:33,159 Let me do that in a different color. 203 00:09:33,159 --> 00:09:35,959 I took out the constant terms there, and it's going to be a 204 00:09:35,960 --> 00:09:39,710 delta of t minus c dt. 205 00:09:39,710 --> 00:09:42,410 206 00:09:42,409 --> 00:09:43,759 Let me get the right color. 207 00:09:43,759 --> 00:09:46,350 208 00:09:46,350 --> 00:09:49,300 Now, what is this thing by definition? 209 00:09:49,299 --> 00:09:50,370 This thing is 1. 210 00:09:50,370 --> 00:09:52,419 I mean, we could put it from minus infinity to infinity, it 211 00:09:52,419 --> 00:09:53,149 doesn't matter. 212 00:09:53,149 --> 00:09:55,169 The only time where it has any area is right under c. 213 00:09:55,169 --> 00:09:57,110 So this thing is equal to 1. 214 00:09:57,110 --> 00:10:01,389 So this whole integral right there has been reduced to this 215 00:10:01,389 --> 00:10:04,039 right there, because this is just equal to 1. 216 00:10:04,039 --> 00:10:07,459 So the Laplace transform of our shifted delta function 217 00:10:07,460 --> 00:10:13,060 times some other function is equal to e to the minus sc 218 00:10:13,059 --> 00:10:15,199 times f of c. 219 00:10:15,200 --> 00:10:18,160 Let me write that again down here. 220 00:10:18,159 --> 00:10:20,649 Let me write it all at once. 221 00:10:20,649 --> 00:10:25,590 So the Laplace transform of our shifted delta function t 222 00:10:25,590 --> 00:10:32,899 minus c times some function f of t, it equals e 223 00:10:32,899 --> 00:10:35,569 to the minus c. 224 00:10:35,570 --> 00:10:38,420 Essentially, we're just evaluating e to the minus st 225 00:10:38,419 --> 00:10:39,240 evaluated at c. 226 00:10:39,240 --> 00:10:44,110 So e to the minus cs times f of c. 227 00:10:44,110 --> 00:10:47,419 We're essentially just evaluating these things at c. 228 00:10:47,419 --> 00:10:48,679 This is what it equals. 229 00:10:48,679 --> 00:10:51,159 So from this we can get a lot of interesting things. 230 00:10:51,159 --> 00:10:55,079 What is the Laplace transform-- actually, what is 231 00:10:55,080 --> 00:10:57,490 the Laplace transform of just the plain 232 00:10:57,490 --> 00:11:00,250 vanilla delta function? 233 00:11:00,250 --> 00:11:05,879 Well, in this case, we have c is equal to 0, and f of t is 234 00:11:05,879 --> 00:11:06,879 equal to 1. 235 00:11:06,879 --> 00:11:08,340 It's just a constant term. 236 00:11:08,340 --> 00:11:11,090 So if we do that, then the Laplace transform of this 237 00:11:11,090 --> 00:11:16,095 thing is just going to be e to the minus 0 times s times 1, 238 00:11:16,095 --> 00:11:18,040 which is just equal to 1. 239 00:11:18,039 --> 00:11:21,740 So the Laplace transform of our delta function is 1, which 240 00:11:21,740 --> 00:11:23,850 is a nice clean thing to find out. 241 00:11:23,850 --> 00:11:26,269 And then if we wanted to just figure out the Laplace 242 00:11:26,269 --> 00:11:28,529 transform of our shifted function, the Laplace 243 00:11:28,529 --> 00:11:35,870 transform of our shifted delta function, this is just a 244 00:11:35,870 --> 00:11:37,812 special case where f of t is equal to 1. 245 00:11:37,812 --> 00:11:40,439 246 00:11:40,440 --> 00:11:42,870 We could write it times 1, where f of t is equal to 1. 247 00:11:42,870 --> 00:11:47,810 So this is going to be equal to e to the minus cs times f 248 00:11:47,809 --> 00:11:51,839 of c, but f is just a constant, f is just 1 here. 249 00:11:51,840 --> 00:11:55,670 So it's times 1, or it's just e to the mine cs. 250 00:11:55,669 --> 00:12:01,549 So just like that, using a kind of visual evaluation of 251 00:12:01,549 --> 00:12:03,819 the integral, we were able to figure out the Laplace 252 00:12:03,820 --> 00:12:07,850 transforms for a bunch of different situations involving 253 00:12:07,850 --> 00:12:09,509 the Dirac delta function. 254 00:12:09,509 --> 00:12:12,399 Anyway, hopefully, you found that reasonably useful.