1 00:00:00,000 --> 00:00:00,590 2 00:00:00,590 --> 00:00:03,139 Where we left off in the last video, we were finding the 3 00:00:03,140 --> 00:00:06,580 surface area of a torus, or a doughnut shape. 4 00:00:06,580 --> 00:00:09,210 And we were doing it by taking a surface integral. 5 00:00:09,210 --> 00:00:11,920 And in order to take a surface integral, we had to find the 6 00:00:11,919 --> 00:00:15,310 partial of our parameterization with respect to s, and the 7 00:00:15,310 --> 00:00:18,920 partial with respect to t, and now we're ready to take 8 00:00:18,920 --> 00:00:19,910 the cross product. 9 00:00:19,910 --> 00:00:21,940 And then we can take the magnitude of the cross product. 10 00:00:21,940 --> 00:00:24,810 And then we can actually take this double integral and 11 00:00:24,809 --> 00:00:26,820 figure out the surface area. 12 00:00:26,820 --> 00:00:28,670 So let's just do it step by step. 13 00:00:28,670 --> 00:00:31,360 Here we could take the cross product, which is not 14 00:00:31,359 --> 00:00:33,329 a non-hairy operation. 15 00:00:33,329 --> 00:00:36,149 This is why you don't see many surface integrals actually get 16 00:00:36,149 --> 00:00:38,399 done, or many examples done. 17 00:00:38,399 --> 00:00:42,219 Let's take the cross product of these two fellows. 18 00:00:42,219 --> 00:00:46,560 So the partial of r with respect to s, crossed with-- 19 00:00:46,560 --> 00:00:50,719 in magenta-- the partial of r with respect to t. 20 00:00:50,719 --> 00:00:52,609 This will be a little bit of review of cross 21 00:00:52,609 --> 00:00:53,390 products for you. 22 00:00:53,390 --> 00:00:55,070 You might remember this is going to be equal 23 00:00:55,070 --> 00:00:57,659 to the determinant. 24 00:00:57,659 --> 00:01:00,359 I'm going to write the unit vectors up here. 25 00:01:00,359 --> 00:01:05,239 The first row is i, j, and k. 26 00:01:05,239 --> 00:01:07,629 And then the next 2 rows are going to be-- let me do that in 27 00:01:07,629 --> 00:01:13,759 that yellow color-- the next 2 rows are going to be the 28 00:01:13,760 --> 00:01:15,460 components of these guys. 29 00:01:15,459 --> 00:01:18,669 So let me copy and paste them. 30 00:01:18,670 --> 00:01:20,739 You have that right there. 31 00:01:20,739 --> 00:01:23,619 Copy and paste. 32 00:01:23,620 --> 00:01:26,050 Put that guy right there. 33 00:01:26,049 --> 00:01:28,929 Then you have this fellow right there. 34 00:01:28,930 --> 00:01:30,140 Copy and paste. 35 00:01:30,140 --> 00:01:31,510 Put him right there. 36 00:01:31,510 --> 00:01:34,130 And then you got this guy right here. 37 00:01:34,129 --> 00:01:35,349 This'll save us some time. 38 00:01:35,349 --> 00:01:37,159 Copy and paste. 39 00:01:37,159 --> 00:01:38,899 Put him right there. 40 00:01:38,900 --> 00:01:42,440 Then the last row is going to be this guy's components. 41 00:01:42,439 --> 00:01:44,259 Copy and paste. 42 00:01:44,260 --> 00:01:46,719 Put him right here. 43 00:01:46,719 --> 00:01:48,679 Almost done. 44 00:01:48,680 --> 00:01:51,740 This guy-- copy and paste. 45 00:01:51,739 --> 00:01:52,869 Put him right there. 46 00:01:52,870 --> 00:01:54,890 Make sure we know that these are separate terms. 47 00:01:54,890 --> 00:01:57,280 And finally, we don't have to copy and paste it, but just 48 00:01:57,280 --> 00:01:59,430 since we did for all of the other terms, I'll do it 49 00:01:59,430 --> 00:02:01,490 for that 0, as well. 50 00:02:01,489 --> 00:02:04,009 So the cross product of these is literally the determinant 51 00:02:04,010 --> 00:02:06,275 of this matrix right here. 52 00:02:06,275 --> 00:02:11,349 53 00:02:11,349 --> 00:02:13,599 And so, just as a bit of a refresher of taking 54 00:02:13,599 --> 00:02:17,489 determinants, this is going to be i times the subdeterminant 55 00:02:17,490 --> 00:02:20,700 right here, if you cross out this column and that row. 56 00:02:20,699 --> 00:02:26,349 So it's going to be equal to i-- you're not used to seeing 57 00:02:26,349 --> 00:02:28,769 the unit vector written first, but we can switch the order 58 00:02:28,770 --> 00:02:32,330 later-- times i times the submatrix right here. 59 00:02:32,330 --> 00:02:34,290 If you cross out this column and that row. 60 00:02:34,289 --> 00:02:38,370 So it's going to be this term times 0-- which is just 61 00:02:38,370 --> 00:02:42,490 0-- minus this term times that term. 62 00:02:42,490 --> 00:02:45,930 So minus this term times this term- the negative signs are 63 00:02:45,930 --> 00:02:47,520 going to cancel out, so this'll be positive. 64 00:02:47,520 --> 00:02:50,510 So it's just going to be i times this term times this 65 00:02:50,509 --> 00:02:52,239 term, without a negative sign right there. 66 00:02:52,240 --> 00:02:58,850 So i times this term, which is a cosine of s. 67 00:02:58,849 --> 00:03:01,629 It's really that term times that term, minus that term 68 00:03:01,629 --> 00:03:03,229 times that term, but the negatives cancel out. 69 00:03:03,229 --> 00:03:04,369 That times that is 0. 70 00:03:04,370 --> 00:03:05,099 So that's how we can do this. 71 00:03:05,099 --> 00:03:14,229 It's a cosine of s times b plus a cosine of s-- I'll just all 72 00:03:14,229 --> 00:03:18,139 switch to the same color-- sine of t. 73 00:03:18,139 --> 00:03:20,699 So we've got our i term for the cross product. 74 00:03:20,699 --> 00:03:24,189 Now it's going to be minus j-- remember when you take the 75 00:03:24,189 --> 00:03:27,590 determinant, you actually have this, kind of, you have to 76 00:03:27,590 --> 00:03:29,060 checker board of switching sines. 77 00:03:29,060 --> 00:03:34,370 So now it's going to be minus j times-- and you cross out that 78 00:03:34,370 --> 00:03:38,390 row and that column-- and it's going to be this term times 79 00:03:38,389 --> 00:03:43,000 this term-- which is just 0-- minus this term 80 00:03:43,000 --> 00:03:44,159 times this term. 81 00:03:44,159 --> 00:03:46,770 And once again, when you have-- oh, sorry. 82 00:03:46,770 --> 00:03:49,020 When you cross out this column and that row. 83 00:03:49,020 --> 00:03:54,750 So it's going to be that guy times that guy, minus 84 00:03:54,750 --> 00:03:58,229 this guy times this guy. 85 00:03:58,229 --> 00:04:00,689 So it's going to be minus this guy times this guy-- so it's 86 00:04:00,689 --> 00:04:03,439 going to be-- let me do it in yellow. 87 00:04:03,439 --> 00:04:11,979 So the negative times negative that guy, b plus a cosine of s 88 00:04:11,979 --> 00:04:16,680 cosine of t times this guy, a cosine of s. 89 00:04:16,680 --> 00:04:19,209 90 00:04:19,209 --> 00:04:21,573 We'll clean it up in a little bit. 91 00:04:21,574 --> 00:04:24,420 Well, we'll clean this up, and you see this negative and that 92 00:04:24,420 --> 00:04:25,500 negative will cancel out. 93 00:04:25,500 --> 00:04:27,519 We're just multiplying everything. 94 00:04:27,519 --> 00:04:29,799 And then finally, the k term. 95 00:04:29,800 --> 00:04:35,470 So plus-- I'll go to the next line-- plus k times-- cross out 96 00:04:35,470 --> 00:04:38,800 that row, that column-- it's going to be that times that, 97 00:04:38,800 --> 00:04:42,340 minus that times that. 98 00:04:42,339 --> 00:04:44,989 So that looks like a kind of a beastly thing. 99 00:04:44,990 --> 00:04:46,519 But I think if we take it step by step, it 100 00:04:46,519 --> 00:04:47,389 shouldn't be too bad. 101 00:04:47,389 --> 00:04:48,300 So that times that. 102 00:04:48,300 --> 00:04:51,254 The negatives are going to cancel out. 103 00:04:51,254 --> 00:04:56,269 So this term right here is going to be a sine 104 00:04:56,269 --> 00:04:59,769 of t, sine of s. 105 00:04:59,769 --> 00:05:05,959 And then this term right here is b plus a 106 00:05:05,959 --> 00:05:09,375 cosine of s sine of t. 107 00:05:09,375 --> 00:05:13,209 108 00:05:13,209 --> 00:05:15,489 So that's that times that-- and the negatives canceled out, 109 00:05:15,490 --> 00:05:17,889 that's why I didn't put any negatives here-- minus 110 00:05:17,889 --> 00:05:19,370 this times this. 111 00:05:19,370 --> 00:05:22,000 So this times this is going to be a negative number. 112 00:05:22,000 --> 00:05:23,660 But if you take the negative of it, it's going to 113 00:05:23,660 --> 00:05:24,480 be a positive value. 114 00:05:24,480 --> 00:05:31,470 So it's going to be plus that a cosine of t 115 00:05:31,470 --> 00:05:34,780 sine of s times that. 116 00:05:34,779 --> 00:05:40,709 Times b plus a cosine of s cosine of t. 117 00:05:40,709 --> 00:05:42,849 Now you see why you don't see many examples of surface 118 00:05:42,850 --> 00:05:44,420 integrals being done. 119 00:05:44,420 --> 00:05:49,069 Let's see if we can clean this up a little bit, especially if 120 00:05:49,069 --> 00:05:51,930 we can clean up this last term a bit. 121 00:05:51,930 --> 00:05:54,660 So let's see what we can do to simplify it. 122 00:05:54,660 --> 00:05:55,985 So our first term. 123 00:05:55,985 --> 00:05:58,040 So let's just multiply it out, I guess is the 124 00:05:58,040 --> 00:05:59,510 easiest way to do it. 125 00:05:59,509 --> 00:06:02,769 Actually, the easiest first step would just be factor out 126 00:06:02,769 --> 00:06:04,959 the b plus a cosine of s. 127 00:06:04,959 --> 00:06:07,680 Because that's in every term. b plus a cosine of s. 128 00:06:07,680 --> 00:06:09,530 b plus a cosine of s. 129 00:06:09,529 --> 00:06:12,479 b plus a cosine of s. b plus a cosine of s. 130 00:06:12,480 --> 00:06:13,710 So let's just factor that out. 131 00:06:13,709 --> 00:06:20,310 So this whole crazy thing can be written as b plus a cosine 132 00:06:20,310 --> 00:06:24,370 of s-- so we factored it out-- times--. 133 00:06:24,370 --> 00:06:27,139 I'll put in some brackets here, so you don't multiply 134 00:06:27,139 --> 00:06:28,789 times every component. 135 00:06:28,790 --> 00:06:31,125 So the i component, when you factor this guy out, is going 136 00:06:31,125 --> 00:06:34,310 to be a cosine of s sine of t. 137 00:06:34,310 --> 00:06:36,480 Let me write it in green. 138 00:06:36,480 --> 00:06:46,770 So it's going to be a cosine of s sine of t times i-- you're 139 00:06:46,769 --> 00:06:48,699 not used to seeing the i before, so I'm going to write 140 00:06:48,699 --> 00:06:53,289 the i here-- and then plus--. 141 00:06:53,290 --> 00:06:56,170 We're factoring this guy out, so you're just going to be 142 00:06:56,170 --> 00:06:58,750 left with cosine of t, a cosine of s. 143 00:06:58,750 --> 00:07:06,350 Or we can write it as a cosine of s cosine of t-- that's that 144 00:07:06,350 --> 00:07:09,050 right there, just putting it in the same order as that-- 145 00:07:09,050 --> 00:07:12,110 times the unit vector j. 146 00:07:12,110 --> 00:07:15,590 And then when we factored this guy out-- so we're not going 147 00:07:15,589 --> 00:07:21,289 to see that or that anymore. 148 00:07:21,290 --> 00:07:24,360 When you factor that out, we can multiply this 149 00:07:24,360 --> 00:07:25,990 out, and what do we get? 150 00:07:25,990 --> 00:07:29,139 So in green, I'll write again. 151 00:07:29,139 --> 00:07:32,219 So if you multiply sine of t times this thing over here-- 152 00:07:32,220 --> 00:07:35,500 because that's all that we have left after we factor out this 153 00:07:35,500 --> 00:07:44,550 thing-- we get a sine of s, sine squared of t, right? 154 00:07:44,550 --> 00:07:46,360 We have sine of t times sine of t. 155 00:07:46,360 --> 00:07:50,250 156 00:07:50,250 --> 00:07:51,910 So that's that over there. 157 00:07:51,910 --> 00:07:54,460 Plus-- what do we have over here? 158 00:07:54,459 --> 00:07:57,519 We have a sine of s times cosine squared of t. 159 00:07:57,519 --> 00:08:05,000 160 00:08:05,000 --> 00:08:08,449 And all of that times the k unit vector. 161 00:08:08,449 --> 00:08:12,509 162 00:08:12,509 --> 00:08:15,310 And so things are looking a little bit more simplified, 163 00:08:15,310 --> 00:08:17,579 but you might see something jump out at you. 164 00:08:17,579 --> 00:08:19,699 You have a sine squared and a cosine squared. 165 00:08:19,699 --> 00:08:22,159 So somehow, if I can just make that just sine squared plus 166 00:08:22,160 --> 00:08:24,240 cosine squared of t, those will simplify to 1. 167 00:08:24,240 --> 00:08:25,579 And we can. 168 00:08:25,579 --> 00:08:29,279 And this term right here, we can-- if we just focus on that 169 00:08:29,279 --> 00:08:32,220 term-- and this is all kind of algebraic manipulation. 170 00:08:32,220 --> 00:08:36,120 If we just focus on that term, this term right here can be 171 00:08:36,120 --> 00:08:42,169 rewritten as a sine of s-- if we factor that out-- times sine 172 00:08:42,169 --> 00:08:47,399 squared of t plus cosine squared of t times 173 00:08:47,399 --> 00:08:50,360 our unit vector, k. 174 00:08:50,360 --> 00:08:50,590 Right? 175 00:08:50,590 --> 00:08:53,480 I just factored out an a sine of s from both of these terms. 176 00:08:53,480 --> 00:08:56,360 And this is our most fundamental trig identity 177 00:08:56,360 --> 00:08:57,330 from the unit circle. 178 00:08:57,330 --> 00:08:59,950 This is equal to 1. 179 00:08:59,950 --> 00:09:05,110 So this last term simplifies to a sine of s times k. 180 00:09:05,110 --> 00:09:07,919 So, so far we've gotten pretty far. 181 00:09:07,919 --> 00:09:11,839 We were able to figure out the cross product of these 2, I 182 00:09:11,840 --> 00:09:14,445 guess, partial derivatives of the vector valued, 183 00:09:14,445 --> 00:09:16,710 or our original parameterization there. 184 00:09:16,710 --> 00:09:20,820 We were able to figure out what this thing right here, before 185 00:09:20,820 --> 00:09:24,070 we take the magnitude of it, it translates to this 186 00:09:24,070 --> 00:09:25,370 thing right here. 187 00:09:25,370 --> 00:09:27,269 Let me rewrite it-- well, I don't need to rewrite it. 188 00:09:27,269 --> 00:09:27,779 You know it. 189 00:09:27,779 --> 00:09:28,819 Well, I'll rewrite it. 190 00:09:28,820 --> 00:09:31,393 So that's equal to-- I'll rewrite it neatly and we'll use 191 00:09:31,393 --> 00:09:38,000 this in the next video-- b plus a cosine of s times open 192 00:09:38,000 --> 00:09:46,809 bracket a cosine of s sine of t times i plus-- switch back to 193 00:09:46,809 --> 00:09:56,839 the blue-- plus a cosine of s cosine of t times j plus-- 194 00:09:56,840 --> 00:09:59,899 switch back to the blue-- this thing-- plus-- this simplified 195 00:09:59,899 --> 00:10:04,870 nicely-- a sine of s times k. 196 00:10:04,870 --> 00:10:07,315 Times the unit vector k. 197 00:10:07,315 --> 00:10:10,000 198 00:10:10,000 --> 00:10:15,019 This right here is this expression right there. 199 00:10:15,019 --> 00:10:16,740 And I'll finish this video, since I'm already 200 00:10:16,740 --> 00:10:17,970 over 10 minutes. 201 00:10:17,970 --> 00:10:18,899 And in the next video, we're going to take 202 00:10:18,899 --> 00:10:20,179 the magnitude of it. 203 00:10:20,179 --> 00:10:22,539 And then, if we have time, actually take 204 00:10:22,539 --> 00:10:23,469 this double integral. 205 00:10:23,470 --> 00:10:24,470 And we'll all be done. 206 00:10:24,470 --> 00:10:28,389 We'll figure out the surface area of this torus. 207 00:10:28,389 --> 00:10:28,799