WEBVTT 1 00:00:00.080 --> 00:00:03.839 You know, when you drive over a massive suspension bridge, 2 00:00:03.879 --> 00:00:05.839 or maybe when you stand on the fiftieth floor of 3 00:00:05.839 --> 00:00:10.359 a skyscraper, there is this comforting expectation of absolute precisions. 4 00:00:10.400 --> 00:00:12.000 Oh yeah, one hundred percent. 5 00:00:12.119 --> 00:00:14.960 We look at those massive concrete pillars and the braided 6 00:00:15.000 --> 00:00:18.000 steel cables and we just assume it's well, it's binary. 7 00:00:18.039 --> 00:00:19.120 It holds or it doesn't. 8 00:00:19.280 --> 00:00:19.879 Right, right. 9 00:00:20.000 --> 00:00:23.879 We like to think it's perfectly safe, backed by absolute rigid, 10 00:00:23.879 --> 00:00:27.320 mathematical certainty. Okay, let's unpack this. 11 00:00:27.440 --> 00:00:30.640 Because it is incredibly comforting to think of the physical 12 00:00:30.679 --> 00:00:33.439 world as perfectly rigid. I mean, as an engineer, we 13 00:00:33.479 --> 00:00:36.520 want to believe that the materials beneath our feet behave 14 00:00:36.640 --> 00:00:39.399 with one hundred percent certainty every single time. 15 00:00:39.719 --> 00:00:42.920 But then you step into the actual world of civil 16 00:00:42.960 --> 00:00:47.719 engineering and suddenly that illusion of absolute certainty completely shatters. Oh, 17 00:00:47.759 --> 00:00:50.920 it really does, because reality is messy. We are looking 18 00:00:50.960 --> 00:00:55.600 at a mathematical landscape that is honestly a lot more dynamic, fluid, 19 00:00:55.679 --> 00:00:58.679 and chaotic than just, you know, a solid block of concrete. 20 00:00:58.799 --> 00:01:03.000 Yeah, nature doesn't like st eight lines are perfect numbers exactly. 21 00:01:02.640 --> 00:01:05.439 So welcome to today's deep dive. We are taking a 22 00:01:05.480 --> 00:01:08.400 fascinating look at the hidden mathematics that actually keeps our 23 00:01:08.439 --> 00:01:12.040 physical world standing. And to do this we are leaning 24 00:01:12.079 --> 00:01:15.560 on a brilliant foundational manual Numerical and Statistical Methods for 25 00:01:15.640 --> 00:01:19.120 Civil Engineering by ravish Ur Singh and mocal Bot. 26 00:01:19.280 --> 00:01:21.680 And our mission for you today is to explore how 27 00:01:21.760 --> 00:01:26.120 abstract mathematics acts as the ultimate toolkit for engineers. We 28 00:01:26.159 --> 00:01:29.959 are going to decode how engineers take the unpredictable, chaotic 29 00:01:30.079 --> 00:01:32.920 nature of the real world and turn it into safe, 30 00:01:33.040 --> 00:01:37.200 reliable structures. And we'll do it using clear scientific language 31 00:01:37.239 --> 00:01:39.640 without getting buried in impenetrable jargon. 32 00:01:39.760 --> 00:01:42.159 Because before an engineer can pour a single ounce of 33 00:01:42.159 --> 00:01:46.200 concrete or even analyzed collected field data, they have to 34 00:01:46.239 --> 00:01:49.079 confront a fundamental truth. The real world does not operate 35 00:01:49.120 --> 00:01:50.159 in absolute certainties. 36 00:01:50.280 --> 00:01:51.760 No, it operates in probabilities. 37 00:01:51.879 --> 00:01:54.319 Right like if I am buying steel beams, I cannot 38 00:01:54.359 --> 00:01:57.879 know with absolute certainty that every single microscopic iron bond 39 00:01:57.959 --> 00:01:59.599 in that beam is flawless. 40 00:02:00.000 --> 00:02:02.640 You literally can't. I mean, when you are sourcing materials, 41 00:02:02.920 --> 00:02:06.799 predicting weather impacts, estimating wear and tear, you are dealing 42 00:02:06.840 --> 00:02:11.960 with randomness. To manage this, engineers lean heavily on probability 43 00:02:12.360 --> 00:02:13.479 and Bayes theorem. 44 00:02:13.599 --> 00:02:14.280 Oh base theorem. 45 00:02:14.639 --> 00:02:17.400 Let's frame this with a practical scenario from the text. 46 00:02:17.960 --> 00:02:21.639 Imagine you are managing a massive construction site and you 47 00:02:21.719 --> 00:02:25.960 have two different manufacturing plants supplying your hydraulic machines. Okay, 48 00:02:26.360 --> 00:02:29.840 so Plant A manufactures seventy percent of your total machines 49 00:02:30.319 --> 00:02:34.479 and Plant B supplies the remaining thirty percent. Now, historical 50 00:02:34.560 --> 00:02:37.120 data tells you that at Plant A, eighty percent of 51 00:02:37.159 --> 00:02:39.439 their machines are rated as standard quality. 52 00:02:39.639 --> 00:02:41.039 Right, But Plant B is different. 53 00:02:41.240 --> 00:02:43.439 Yeah, A, Plant B they have a tighter process and 54 00:02:43.520 --> 00:02:46.199 ninety percent of theirs meet the standard quality mark. 55 00:02:46.560 --> 00:02:48.800 So I have a massive warehouse full of these machines 56 00:02:48.840 --> 00:02:51.159 mixed together. If I just grab one machine completely at 57 00:02:51.240 --> 00:02:53.319 random and I test it and it turns out to 58 00:02:53.319 --> 00:02:57.159 be standard quality, base theorem allows me to mathematically reverse 59 00:02:57.199 --> 00:03:01.319 engineer the probability of which specific plant that machine came from. 60 00:03:01.400 --> 00:03:04.479 It's essentially a mathematical formula that lets you update your 61 00:03:04.520 --> 00:03:06.560 beliefs based on new evidence. 62 00:03:06.639 --> 00:03:08.719 That is the perfect way to phrase it. It updates 63 00:03:08.759 --> 00:03:13.719 your baseline assumptions. You aren't just guessing, you are quantifying 64 00:03:13.759 --> 00:03:16.039 the likelihood of an origin based on the result you 65 00:03:16.080 --> 00:03:16.879 hold in your hand. 66 00:03:17.759 --> 00:03:21.159 But applying probability to engineering feels a bit like, I 67 00:03:21.159 --> 00:03:25.400 don't know, being a highly scientific weather forecaster. You can't 68 00:03:25.400 --> 00:03:28.560 know precisely if one individual brick is going to fail 69 00:03:28.599 --> 00:03:32.080 in a hurricane, right, but you can confidently predict the 70 00:03:32.120 --> 00:03:36.479 structural behavior of the entire brick wall. The pushback I have, though, 71 00:03:36.759 --> 00:03:40.639 is how an engineer practically designs for randomness without just 72 00:03:40.719 --> 00:03:41.879 over engineering everything. 73 00:03:42.000 --> 00:03:44.080 Oh that's the classic trap, because. 74 00:03:43.800 --> 00:03:46.000 I mean, if I don't know when a part will fail, 75 00:03:46.400 --> 00:03:48.879 my instinct is just make it ten times thicker, which 76 00:03:48.919 --> 00:03:51.680 obviously completely blows the construction budget. 77 00:03:51.719 --> 00:03:55.280 What's fascinating here is that this is where probability distributions 78 00:03:55.319 --> 00:03:58.039 step in, and they provide the answer to that exact 79 00:03:58.120 --> 00:04:02.759 problem of over engineering. Distributions give engineers a mathematical boundary 80 00:04:02.759 --> 00:04:03.400 for safety. 81 00:04:03.599 --> 00:04:04.479 Okay, a boundary. 82 00:04:04.800 --> 00:04:08.680 Yeah. They are essentially models of how randomness behaves over 83 00:04:08.759 --> 00:04:11.319 time or across large numbers. 84 00:04:11.000 --> 00:04:15.520 Of items, like the binomial, Pusson and normal distributions. Right. 85 00:04:16.279 --> 00:04:19.959 The text outlines those and the practical applications are pretty wild. 86 00:04:20.160 --> 00:04:23.839 Take the binomial distribution. This deals with scenarios that have 87 00:04:23.959 --> 00:04:28.600 exactly two outcomes like pass or fail, works or. 88 00:04:28.560 --> 00:04:30.600 Breaks yep, the classic binary. 89 00:04:30.959 --> 00:04:33.560 The authors use this great example of electric bulbs on 90 00:04:33.600 --> 00:04:36.800 a street pole. If you know that the probability of 91 00:04:36.839 --> 00:04:39.519 a single bulb surviving for one hundred hours is roughly 92 00:04:39.560 --> 00:04:42.279 thirty percent, you don't just guess how many bulbs to. 93 00:04:42.240 --> 00:04:44.600 Install, No, you'd be replacing them constantly. 94 00:04:44.720 --> 00:04:48.000 Right. You use the binomial distribution to calculate exactly how 95 00:04:48.000 --> 00:04:50.160 many bulbs need to be on that pole to ensure 96 00:04:50.519 --> 00:04:52.600 with ninety nine percent safety that at least one of 97 00:04:52.639 --> 00:04:55.279 them will still be shining after one hundred hours. You 98 00:04:55.319 --> 00:04:57.600 are mathematically guaranteeing a safety threshold. 99 00:04:57.680 --> 00:05:00.600 And then you have rare random events that happen over 100 00:05:00.639 --> 00:05:04.160 a specific period of time or space. For those engineers 101 00:05:04.279 --> 00:05:07.920 use the Poisson distribution. Yeah. And the text uses a 102 00:05:07.959 --> 00:05:11.800 sobering but highly critical example for civil and mining engineers, 103 00:05:12.040 --> 00:05:15.079 which is estimating the probability of fatal accidents in a 104 00:05:15.079 --> 00:05:15.720 coal mine. 105 00:05:15.759 --> 00:05:16.480 Wow. Ok. 106 00:05:16.639 --> 00:05:19.639 If historical data tells us the average number of accidents 107 00:05:19.720 --> 00:05:23.959 per year, the Poisson distribution allows an engineer to calculate 108 00:05:23.959 --> 00:05:27.639 the precise probability of seeing zero, two, or even a 109 00:05:27.680 --> 00:05:31.839 maximum of three accidents in the coming year, So it models. 110 00:05:31.480 --> 00:05:34.879 The predictable nature of entirely unpredictable. 111 00:05:34.240 --> 00:05:37.639 Events exactly, so safety protocols and emergency budgets can be 112 00:05:37.680 --> 00:05:39.000 accurately scaled. 113 00:05:38.959 --> 00:05:41.680 Which brings us to the famous Bell curve for the 114 00:05:41.759 --> 00:05:46.600 normal distribution. The manual applies this to agricultural engineering. If 115 00:05:46.639 --> 00:05:49.199 you know the mean yield of a particular crop is, 116 00:05:49.279 --> 00:05:52.639 say six hundred and eighty kilograms, with a certain standard variants, 117 00:05:53.120 --> 00:05:55.360 you can calculate the exact number of blots out of 118 00:05:55.399 --> 00:05:58.519 a thousand that will yield over seven hundred kilograms or 119 00:05:58.720 --> 00:06:01.519 fall below six hundred and fife to kilograms. You are 120 00:06:01.560 --> 00:06:03.360 literally quantifying the chaos of. 121 00:06:03.399 --> 00:06:06.920 Nature, you are, But probability gives us the theoretical risk. 122 00:06:07.240 --> 00:06:10.560 The next step is reality. The mud on the boots, Yeah, exactly. 123 00:06:10.920 --> 00:06:13.000 When you're standing in the mud on a job site 124 00:06:13.000 --> 00:06:16.600 holding a clipboard full of messy real world measurements, how 125 00:06:16.639 --> 00:06:20.199 do you make sense of that. Once engineers establish the probabilities, 126 00:06:20.279 --> 00:06:22.240 they have to go out into the field collect actual 127 00:06:22.360 --> 00:06:24.720 data and find the patterns and all that noise. 128 00:06:25.079 --> 00:06:29.480 So we move from theoretical probability into statistics. In regression, 129 00:06:30.199 --> 00:06:33.360 you gather this massive chaotic data set from the field 130 00:06:33.839 --> 00:06:38.319 thousands of soil density readings or you know, windspeed. 131 00:06:37.920 --> 00:06:39.680 Logs, and it's just a wall of numbers. 132 00:06:39.759 --> 00:06:43.439 Right. To make that usable, you need descriptive statistics, your mean, 133 00:06:44.120 --> 00:06:47.560 median mode, standard deviation, and skewness. 134 00:06:47.720 --> 00:06:51.399 Because the human brain cannot comprehend a spreadsheet with ten 135 00:06:51.519 --> 00:06:53.199 thousand individual stress tests. 136 00:06:53.319 --> 00:06:53.680 No way. 137 00:06:53.879 --> 00:06:57.680 Descriptive statistics summarize that massive chaos into a few digestible 138 00:06:57.720 --> 00:07:00.560 numbers that actually tell a story about the central tendencies 139 00:07:00.639 --> 00:07:03.959 and the spread of your materials. But engineering rarely looks 140 00:07:03.959 --> 00:07:08.680 at just one thing. In isolation. Engineers constantly deal with bivariate. 141 00:07:08.120 --> 00:07:11.360 Distributions, analyzing two variables simultaneously to see if they. 142 00:07:11.319 --> 00:07:13.480 Move together exactly correlation and regression. 143 00:07:13.759 --> 00:07:18.720 So by calculating the Carl Pearson coefficient of correlation, engineers 144 00:07:18.759 --> 00:07:22.560 can mathematically prove the strength of the link between two variables. 145 00:07:22.720 --> 00:07:26.279 Yes, think of the correlation coefficient as measuring the strength 146 00:07:26.319 --> 00:07:30.240 of the invisible anchor tying two phenomena together, and regression 147 00:07:30.279 --> 00:07:32.879 lines allow them to take that relationship and use it 148 00:07:32.920 --> 00:07:34.199 to make future predictions. 149 00:07:35.000 --> 00:07:38.040 The book gives a really grounded example of this, predicting 150 00:07:38.079 --> 00:07:40.720 agricultural output based on rainfall data. 151 00:07:40.839 --> 00:07:42.279 Oh, that's a great one, right. 152 00:07:42.560 --> 00:07:45.079 If you have historical data showing how different amounts of 153 00:07:45.160 --> 00:07:49.839 rain affect crop yield, you can construct a regression line. Then, 154 00:07:49.959 --> 00:07:53.399 if the seasonal forecast calls for exactly twenty nine centimeters 155 00:07:53.399 --> 00:07:56.120 of rain, you can plug that twenty nine into your 156 00:07:56.120 --> 00:07:59.920 regression equation and get a highly accurate mathematical estimate of 157 00:08:00.040 --> 00:08:00.839 you're expected yield. 158 00:08:01.040 --> 00:08:02.560 It gives you a specific tart it. 159 00:08:02.680 --> 00:08:05.199 Yeah, there's another example, dealing with the maintenance cost of 160 00:08:05.199 --> 00:08:07.519 a car based on its age at two, four, six, 161 00:08:07.519 --> 00:08:10.480 and eight years. You can struct the line and certainly 162 00:08:10.519 --> 00:08:12.439 you can predict what a ten year old car will 163 00:08:12.439 --> 00:08:14.519 cost you in repairs before you ever get there. 164 00:08:14.759 --> 00:08:15.800 It's predictive power. 165 00:08:16.040 --> 00:08:19.000 But I have to push back a little here. Isn't 166 00:08:19.040 --> 00:08:22.360 correlation just a fancy way of stating the obvious. I mean, 167 00:08:22.399 --> 00:08:25.160 if it rains, more crops grow, If a car gets older, 168 00:08:25.240 --> 00:08:28.959 it breaks down more. Do we really need a complex 169 00:08:29.160 --> 00:08:31.959 Carl Pearson coefficient matrix to tell us that? 170 00:08:32.440 --> 00:08:34.720 It is a fair question. But if we connect this 171 00:08:34.840 --> 00:08:38.799 to the bigger picture, here's the critical distinction for civil engineering. 172 00:08:39.159 --> 00:08:44.080 Engineering requires exact, quantifiable parameters to make safe decisions. 173 00:08:44.240 --> 00:08:45.639 Right, because lives are on the line. 174 00:08:45.720 --> 00:08:48.720 Simple observation tells you that more rain generally means more 175 00:08:48.720 --> 00:08:51.960 crops or more runoff. But what if you were designing 176 00:08:51.960 --> 00:08:56.360 a municipal drainage system for that agricultural field. Ye, you 177 00:08:56.440 --> 00:08:59.759 cannot design a concrete pipe based on the phrase generally 178 00:08:59.799 --> 00:09:03.039 more runoff. You need to know the mathematical certainty of 179 00:09:03.080 --> 00:09:05.039 the volume of water to expect per hour. 180 00:09:05.480 --> 00:09:07.639 So a regression line gives you the precise rate. 181 00:09:07.519 --> 00:09:11.559 Of change exactly. It elevates a casual observation into a predictive, 182 00:09:11.639 --> 00:09:15.360 quantifiable tool that can be tested, verified, and most importantly 183 00:09:15.399 --> 00:09:15.919 built upon. 184 00:09:16.200 --> 00:09:19.320 That puts it in perspective. You need the exact numbers 185 00:09:19.360 --> 00:09:21.840 to know what diameter pipe to buy so the town 186 00:09:21.960 --> 00:09:26.039 doesn't flood. But knowing two variables are correlated is great 187 00:09:26.039 --> 00:09:30.399 for broad trends. How do engineers predict precise values that 188 00:09:30.440 --> 00:09:32.320 they haven't directly measured in the field? 189 00:09:32.639 --> 00:09:34.799 Ah reading between the lines, right. 190 00:09:35.000 --> 00:09:37.159 If I take a measurement at ten feet and another 191 00:09:37.159 --> 00:09:39.879 at twenty feet, what is happening at fifteen feet? 192 00:09:40.039 --> 00:09:43.759 That is the essence of curve fitting and interpolation. Let's 193 00:09:43.759 --> 00:09:46.840 start with curve fitting using the least square method. Okay, 194 00:09:47.080 --> 00:09:50.799 sometimes your scattered data doesn't form a perfect straight line. 195 00:09:51.080 --> 00:09:54.440 It might form a quadratic curve or an exponential curve. 196 00:09:55.120 --> 00:09:57.519 The least square method is a mathematical way to find 197 00:09:57.559 --> 00:10:00.200 the best fit curve through a cloud of data points. 198 00:10:00.159 --> 00:10:03.279 And the way it works is fascinating. You're essentially trying 199 00:10:03.320 --> 00:10:05.440 to draw a line through the data where you minimize 200 00:10:05.480 --> 00:10:08.000 the square of the errors. It's like snapping a rubber 201 00:10:08.000 --> 00:10:10.200 band through the center of the points the good visual 202 00:10:10.399 --> 00:10:13.399 You want the overall tension or distance from the points 203 00:10:13.399 --> 00:10:16.200 to the line to be as small as possible. The 204 00:10:16.200 --> 00:10:18.879 booth applies this to finding the mechanical pull required to 205 00:10:18.919 --> 00:10:21.120 lift a load using a pulley block. 206 00:10:21.159 --> 00:10:22.600 The PNW example. 207 00:10:22.720 --> 00:10:26.840 Yeah, calculating the maximum deflection of a simply supported structural 208 00:10:26.840 --> 00:10:30.519 beam under various loads. You plot your physical test loads 209 00:10:30.759 --> 00:10:33.320 fit the curve using the least square method, and suddenly 210 00:10:33.360 --> 00:10:36.879 you have a continuous algebraic equation that models the physical 211 00:10:36.919 --> 00:10:37.720 beam perfectly. 212 00:10:38.080 --> 00:10:41.080 It translates a physical object into a mathematical function. 213 00:10:41.600 --> 00:10:44.960 But here's where it gets really interesting for me. Interpolation. 214 00:10:45.639 --> 00:10:48.480 I like to think of interpolation like having the first 215 00:10:48.519 --> 00:10:51.679 and last frame of a movie scene and using math 216 00:10:51.919 --> 00:10:55.039 to perfectly draw all the individual frames in between. 217 00:10:55.159 --> 00:10:58.320 That is a brilliant way to visualize it. Interpolation is 218 00:10:58.360 --> 00:11:01.480 the process of computing inner mediate values of a function 219 00:11:01.679 --> 00:11:05.000 from a given set of tabular values. You literally read 220 00:11:05.039 --> 00:11:06.519 between the lines of your data table. 221 00:11:06.879 --> 00:11:09.840 And the text introduces several methods for this, right, starting 222 00:11:09.840 --> 00:11:12.919 with Newton's forward and backward interpolation formulas. 223 00:11:12.559 --> 00:11:15.159 Right, and those are used when your data points are 224 00:11:15.200 --> 00:11:17.799 spaced at strictly equal intervals. 225 00:11:17.600 --> 00:11:19.799 Like estimating the population of a town in the year 226 00:11:19.840 --> 00:11:23.039 eighteen ninety five based on dicable census data from eighteen 227 00:11:23.120 --> 00:11:25.639 ninety one, nineteen oh one, nineteen eleven, and so on. Yeah, 228 00:11:25.679 --> 00:11:28.720 the intervals are exactly ten years apart. Yeah, Or finding 229 00:11:28.759 --> 00:11:31.600 the exact area of a circle with a diameter of 230 00:11:31.600 --> 00:11:33.919 one hundred and five units based on a table of 231 00:11:34.000 --> 00:11:37.799 known areas for diameters s based exactly ten units apart. 232 00:11:37.960 --> 00:11:41.320 But reality, again, isn't always neat. Sometimes your data is 233 00:11:41.360 --> 00:11:44.559 taken at unequal intervals. Maybe a sensor failed during a reading, 234 00:11:44.840 --> 00:11:48.600 or a rock formation meant you couldn't access a certain 235 00:11:48.639 --> 00:11:49.919 depth for a soil. 236 00:11:49.679 --> 00:11:51.519 Sample, so the intervals get messed up. 237 00:11:51.559 --> 00:11:55.440 Exactly for unequal intervals, engineers have to adapt the text 238 00:11:55.480 --> 00:12:00.440 teaches Lagrange's interpolation formula and Newton's divided difference formula. By 239 00:12:00.440 --> 00:12:04.240 dividing the differences between the irregular intervals, the math adjusts 240 00:12:04.279 --> 00:12:05.480 for the uneven spacing. 241 00:12:05.840 --> 00:12:08.120 There's a great civil engineering example of this in the 242 00:12:08.120 --> 00:12:11.679 book Computing the Sheer Stress in a clay stratum. The 243 00:12:11.720 --> 00:12:15.679 engineers have soil samples taken at highly irregular depths because 244 00:12:15.720 --> 00:12:18.519 of the drilling process, like one point nine meters, three 245 00:12:18.519 --> 00:12:19.960 point one meters four point. 246 00:12:19.720 --> 00:12:21.440 Two meters, super random depths. 247 00:12:21.559 --> 00:12:23.840 Right, But the architect needs to know the exact shear 248 00:12:23.919 --> 00:12:27.039 stress at exactly four point five meters to design the 249 00:12:27.080 --> 00:12:30.519 foundation footing. Lagrong's formula solves that. Oh, and there's also 250 00:12:30.519 --> 00:12:31.960 Stirling's formula mentioned right. 251 00:12:32.080 --> 00:12:36.320 Yes, Sterling's formula uses central differences. It is incredibly useful 252 00:12:36.320 --> 00:12:39.120 when the value you are trying to estimate lies right 253 00:12:39.240 --> 00:12:42.480 near the middle of your tabulated data set, rather than 254 00:12:42.559 --> 00:12:43.759 at the very beginning or. 255 00:12:43.679 --> 00:12:45.559 The end, so it zeros in on the middle. 256 00:12:45.720 --> 00:12:48.279 Yeah, it takes an average of the forward and backward 257 00:12:48.279 --> 00:12:51.440 differences to pinpoint that central value with high accuracy. 258 00:12:51.840 --> 00:12:55.639 Hold on, though we are talking about interpolation. But I 259 00:12:55.720 --> 00:12:59.840 hear the word extrapolation thrown around a lot in data analysis. 260 00:13:00.159 --> 00:13:01.360 Are they basically the same thing? 261 00:13:01.519 --> 00:13:05.120 Oh? No, that is a vital distinction, especially regarding safety. 262 00:13:05.879 --> 00:13:08.919 Interpolation is estimating a value within the range of your 263 00:13:08.960 --> 00:13:12.639 known data points. Okay, Extrapolation is estimating a value outside 264 00:13:12.679 --> 00:13:16.759 your known range. Interpolation is generally highly reliable because it 265 00:13:16.799 --> 00:13:21.360 is bounded by actual physical measurements you took. Extrapolation is much. 266 00:13:21.240 --> 00:13:23.360 Riskier because you're guessing what happens next. 267 00:13:23.559 --> 00:13:27.559 You are assuming the mathematical trend continues indefinitely into the unknown. 268 00:13:28.159 --> 00:13:32.600 Physical materials rarely do that. Eventually, steel bends or concrete fractures. 269 00:13:32.879 --> 00:13:37.039 That makes total sense. So we've used probability to understand 270 00:13:37.039 --> 00:13:41.080 our risks, statistics define patterns in the mud, and curve 271 00:13:41.080 --> 00:13:44.720 fitting to generate smooth equations from our choppy data. But 272 00:13:44.799 --> 00:13:47.519 what happens when the mathematical functions generated by all this 273 00:13:47.639 --> 00:13:51.960 curve fitting are simply too complex to solve using standard 274 00:13:51.960 --> 00:13:53.039 textbook calculus. 275 00:13:53.519 --> 00:13:56.919 That is where we enter the art of approximation, numerical integration, 276 00:13:57.320 --> 00:14:01.240 and the numerical solutions of differential equations. This is what 277 00:14:01.279 --> 00:14:03.799 you do when analytical math hits a brick wall. 278 00:14:03.879 --> 00:14:06.879 Let's talk about numerical integration first. We're looking at things 279 00:14:06.960 --> 00:14:10.039 like the trapezoidal rule and Simpson's one third and three 280 00:14:10.039 --> 00:14:14.000 eighths rules, and the examples here are incredibly tactile, very 281 00:14:14.039 --> 00:14:17.360 hands on. Imagine you are a civil engineer and you 282 00:14:17.399 --> 00:14:20.320 need to calculate the exact cross sectional area of a 283 00:14:20.399 --> 00:14:23.240 river that is eighty meters wide so you can build 284 00:14:23.240 --> 00:14:25.879 a bridge support. You obviously can't drain the river to 285 00:14:25.919 --> 00:14:26.320 measure it. 286 00:14:26.360 --> 00:14:27.799 Now, you'd get fired pretty quickly for. 287 00:14:27.759 --> 00:14:30.519 That, right, So you take depth measurements every ten meters, 288 00:14:30.679 --> 00:14:31.120 and if. 289 00:14:30.960 --> 00:14:33.720 You just connected those depth readings with straight lines, you'd 290 00:14:33.759 --> 00:14:37.480 get blocky trapezoids, which doesn't reflect the smooth natural curve 291 00:14:37.519 --> 00:14:38.720 of a river bed at all. 292 00:14:38.840 --> 00:14:41.120 It would look like a retro video game exactly. 293 00:14:41.320 --> 00:14:44.559 But by applying Simpson's one third rule to those discrete 294 00:14:44.639 --> 00:14:49.559 depth readings, the math draws smooth, curved parabolas over small 295 00:14:49.600 --> 00:14:52.559 slices of the river. It hugs the natural shape of 296 00:14:52.600 --> 00:14:56.000 the river bed much more accurately, allowing you to approximate 297 00:14:56.039 --> 00:14:59.279 the total area of the cross section with incredible precision. 298 00:14:59.600 --> 00:15:03.080 Another brilliant application of numerical integration in the text is 299 00:15:03.200 --> 00:15:06.559 estimating the total distance of train covers in eighteen minutes 300 00:15:07.039 --> 00:15:10.120 just by using discrete velocity readings taken every three minutes. 301 00:15:10.200 --> 00:15:12.960 That's a classic one. We're calculating the isothermal work done 302 00:15:13.039 --> 00:15:15.559 when compressing a gas from twenty two liters down to 303 00:15:15.600 --> 00:15:18.720 two leaders using pressure readings taken at intervals. You are 304 00:15:18.720 --> 00:15:22.879 calculating the exact area under a curve without actually possessing 305 00:15:22.960 --> 00:15:26.120 the continuous algebraic equation of the curve itself. 306 00:15:26.240 --> 00:15:29.279 It really is mathematical magic. The text also covers solving 307 00:15:29.399 --> 00:15:33.399 massive systems of linear equations. When you are analyzing the 308 00:15:33.440 --> 00:15:36.000 forces on a steel trust bridge, you don't just have 309 00:15:36.080 --> 00:15:38.840 one equation. You might have dozens of equations with dozens 310 00:15:38.879 --> 00:15:42.519 of variables representing the intersecting forces at every single joint. 311 00:15:42.559 --> 00:15:47.399 The book walks through direct methods like Gass elimination, but 312 00:15:47.440 --> 00:15:51.919 also iterative methods like the Gau's Cidal method. I love 313 00:15:51.960 --> 00:15:54.000 the Gauss ciital method because it feels like playing a 314 00:15:54.039 --> 00:15:55.279 massive game of Sudoku. 315 00:15:55.360 --> 00:15:56.919 That's a really good analogy. 316 00:15:56.759 --> 00:15:59.679 Right, You make a highly educated initial guess for your 317 00:15:59.799 --> 00:16:02.000 very you plug it in and you see how it 318 00:16:02.039 --> 00:16:05.360 impacts the rest of the board. It will be slightly wrong. 319 00:16:05.519 --> 00:16:07.480 So you adjust your guess and you run it through 320 00:16:07.480 --> 00:16:10.240 the mathematical loop again and again until all the numbers 321 00:16:10.279 --> 00:16:11.480 lock into place perfectly. 322 00:16:11.840 --> 00:16:15.519 That iterative loop is exactly right. They also tackle finding 323 00:16:15.559 --> 00:16:18.799 the roots of equations, essentially finding where a complex function 324 00:16:18.919 --> 00:16:23.039 equals zero. That compare different iterative methods here, highlighting that 325 00:16:23.080 --> 00:16:26.240 the Newton Raftsin method converges on the true answer. 326 00:16:26.080 --> 00:16:27.320 Very quickly, super fast. 327 00:16:27.399 --> 00:16:30.120 Yeah. It has a quadratic convergence rate or an order 328 00:16:30.120 --> 00:16:30.720 of two. 329 00:16:30.720 --> 00:16:33.120 Which means with every single guess you make in the loop, 330 00:16:33.399 --> 00:16:36.919 the number of correct decimal places in your answer roughly doubles. 331 00:16:37.360 --> 00:16:40.480 It zeros in on the truth incredibly fast, whereas the 332 00:16:40.480 --> 00:16:43.360 second method has a slower convergence rate of one point 333 00:16:43.399 --> 00:16:44.919 six one eight exactly. 334 00:16:45.360 --> 00:16:49.480 And finally, the text addresses ordinary differential equations or odes, 335 00:16:49.759 --> 00:16:54.440 using methods like run Jakuda or Milne's predictor corrector. Odes 336 00:16:54.480 --> 00:16:56.480 are crucial because they model change. 337 00:16:56.240 --> 00:16:58.080 Over time like a dynamic system. 338 00:16:58.320 --> 00:17:01.080 Yeah, if you are analyzing a pen ingelum swinging or 339 00:17:01.120 --> 00:17:04.599 a water tank draining, the rate of change is constantly 340 00:17:04.680 --> 00:17:08.319 shifting based on the current state of the system. Runjikoud 341 00:17:08.480 --> 00:17:10.920 is a step by step method that calculates the slope 342 00:17:10.920 --> 00:17:14.440 at various points within a small time step, averages them out, 343 00:17:14.680 --> 00:17:16.880 and predicts exactly where the system will be in the 344 00:17:16.880 --> 00:17:17.480 next moment. 345 00:17:17.640 --> 00:17:20.119 Okay, I have a major question here. We keep using 346 00:17:20.160 --> 00:17:26.200 the word approximation, numerical approximation, Yeah, guessing and iterating. So 347 00:17:26.279 --> 00:17:29.480 what does this all mean? Why rely on approximation when 348 00:17:29.519 --> 00:17:31.799 buildings and bridges need to be completely exact? 349 00:17:32.039 --> 00:17:32.960 I hear this a lot. 350 00:17:33.240 --> 00:17:36.119 Doesn't the phrase numerical approximation sound a bit risky for 351 00:17:36.160 --> 00:17:38.359 an engineer building a suspension bridge I have to drive 352 00:17:38.359 --> 00:17:39.000 my family over. 353 00:17:39.160 --> 00:17:41.680 This raises an important question, and it's a very common 354 00:17:41.720 --> 00:17:46.000 misconception about the word approximation in applied mathematics. In engineering, 355 00:17:46.039 --> 00:17:49.279 a numerical approximation is not a blind guess. It is 356 00:17:49.319 --> 00:17:54.000 a highly rigorous calculation computed to a known safe tolerance level. 357 00:17:54.799 --> 00:17:57.559 When we use these numerical methods, we also use error 358 00:17:57.599 --> 00:18:01.480 formulas to calculate the maximum possible inaccuracy of our results, 359 00:18:01.480 --> 00:18:01.799 so you. 360 00:18:01.799 --> 00:18:04.119 Know exactly how wrong you could possibly be. 361 00:18:04.400 --> 00:18:08.279 Yes, we iterate the calculation, running that Sudoku loop until 362 00:18:08.319 --> 00:18:11.759 that error margin becomes physically negligible. We are talking less 363 00:18:11.799 --> 00:18:15.200 than a fraction of a millimeter. We are balancing computational 364 00:18:15.240 --> 00:18:18.400 effort with the precise level of physical precision required to 365 00:18:18.519 --> 00:18:19.880 ensure absolute safety. 366 00:18:20.400 --> 00:18:24.039 So basically, we are trading infinite computational time for a 367 00:18:24.079 --> 00:18:27.799 physical reality that is perfectly structurally safe. An error of 368 00:18:27.880 --> 00:18:30.200 one thousandth of an inch just doesn't matter when you 369 00:18:30.200 --> 00:18:31.920 are pouring massive concrete. 370 00:18:31.559 --> 00:18:33.880 Pillars, exactly. It's about practical perfection. 371 00:18:34.319 --> 00:18:38.000 So, to summarize the massive toolkit we've explored today, probability 372 00:18:38.000 --> 00:18:41.400 gives us the theoretical boundaries of random real world behavior. 373 00:18:41.599 --> 00:18:44.440 Statistics allow us to find the meaningful signals in messi 374 00:18:44.480 --> 00:18:47.559 field data. Curve Fitting and interpolation let us bridge the 375 00:18:47.559 --> 00:18:49.599 gaps in our physical measurements. 376 00:18:49.359 --> 00:18:53.720 And numerical integration and differential equations give us the tools 377 00:18:53.759 --> 00:18:57.240 to solve the unsolvable math, computing everything to a safe, 378 00:18:57.400 --> 00:19:02.440 verifiable tolerance. Abstract math truly is the bridge to physical reality. 379 00:19:02.680 --> 00:19:07.759 It is, and while modern engineering software handles these calculations instantly, Today, 380 00:19:08.400 --> 00:19:12.480 understanding the underlying manual iterative processes like tracing the step 381 00:19:12.519 --> 00:19:15.240 by step gass c sital method or the run Shakoda steps 382 00:19:15.599 --> 00:19:18.279 is what separates a software user from a true engineer. 383 00:19:18.440 --> 00:19:21.359 A true engineer can look at an automated software output, 384 00:19:21.680 --> 00:19:25.000 verify it against their deep understanding of the mathematical mechanics, 385 00:19:25.200 --> 00:19:26.720 and actually trust the results. 386 00:19:26.839 --> 00:19:28.319 I want to leave you with a thought that builds 387 00:19:28.319 --> 00:19:32.319 on all this. We talked about how foundational probability distributions 388 00:19:32.359 --> 00:19:35.720 like the Poisson or normal distributions, rely on historical data 389 00:19:35.720 --> 00:19:38.559 to predict things like one hundred year floods or extreme 390 00:19:38.599 --> 00:19:41.720 wind shears right the baseline data, but with climate change 391 00:19:41.799 --> 00:19:45.559 rapidly altering weather patterns globally, we have to ask what 392 00:19:45.680 --> 00:19:49.559 happens when the historical data itself becomes obsolete. The pure 393 00:19:49.599 --> 00:19:53.519 mathematics of the press On distribution remain perfectly sound, but 394 00:19:53.559 --> 00:19:57.039 if the baseline historical data is shifting, engineers are going 395 00:19:57.079 --> 00:20:00.839 to have to entirely recalculate the probabilistic founds of our 396 00:20:00.880 --> 00:20:05.480 modern infrastructure. It is a massive, ongoing mathematical challenge. 397 00:20:05.079 --> 00:20:07.839 That is a phenomenal point. The math is only as 398 00:20:07.839 --> 00:20:09.559 good as the reality it represents. 399 00:20:10.359 --> 00:20:13.319 Before we go as promised, here's a quick review exercise 400 00:20:13.359 --> 00:20:15.200 for you to test what we've covered in this deep dive. 401 00:20:15.960 --> 00:20:17.680