1 00:00:00,000 --> 00:00:00,550 2 00:00:00,550 --> 00:00:07,160 Let's have the vector valued function r of s and t is equal 3 00:00:07,160 --> 00:00:09,980 to-- well, x is going to be a function of s and t. 4 00:00:09,980 --> 00:00:14,089 So we'll just write it as x of s and t times the x unit 5 00:00:14,089 --> 00:00:22,179 vector, or i, plus y of s and t times the y unit factor, or j, 6 00:00:22,179 --> 00:00:28,239 plus x of s and t times the z unit vector, k. 7 00:00:28,239 --> 00:00:32,280 So given that we have this vector valued function, let's 8 00:00:32,280 --> 00:00:34,770 define or let's think about what it means to take the 9 00:00:34,770 --> 00:00:38,800 partial derivative of this vector valued function with 10 00:00:38,799 --> 00:00:41,619 respect to one of the parameters, s or t. 11 00:00:41,619 --> 00:00:44,039 I think it's going to be pretty natural, nothing 12 00:00:44,039 --> 00:00:45,420 completely bizarre here. 13 00:00:45,420 --> 00:00:49,030 We've taken partial derivatives of non-vector valued functions 14 00:00:49,030 --> 00:00:52,850 before, where we only vary one of the variables. 15 00:00:52,850 --> 00:00:54,280 We only take it with respect to one variable. 16 00:00:54,280 --> 00:00:55,770 You hold the other one constant. 17 00:00:55,770 --> 00:00:57,750 We're going to do the exact same thing here. 18 00:00:57,750 --> 00:01:00,920 And we've taken regular derivatives of vector 19 00:01:00,920 --> 00:01:01,719 valued functions. 20 00:01:01,719 --> 00:01:04,120 The path in those just ended up being the regular derivative 21 00:01:04,120 --> 00:01:05,490 of each of the terms. 22 00:01:05,489 --> 00:01:07,179 And we're going to see, it's going to be the same thing here 23 00:01:07,180 --> 00:01:08,950 with the partial derivative. 24 00:01:08,950 --> 00:01:14,670 So let's define the partial derivative of 25 00:01:14,670 --> 00:01:16,969 r with respect to s. 26 00:01:16,969 --> 00:01:19,719 And everything I do with respect to s, you can just swap 27 00:01:19,719 --> 00:01:21,929 it with t, and you're going to get the same exact result. 28 00:01:21,930 --> 00:01:28,800 I'm going to define it as being equal to the limit as delta 29 00:01:28,799 --> 00:01:36,239 s approaches 0 of r of s plus delta s. 30 00:01:36,239 --> 00:01:38,250 Only finding the limit with respect to a 31 00:01:38,250 --> 00:01:41,569 change in s comma t. 32 00:01:41,569 --> 00:01:44,339 We're holding t, as you can imagine, constant for given 33 00:01:44,340 --> 00:01:50,189 t, minus r of s and t. 34 00:01:50,189 --> 00:01:56,629 All of that over delta s. 35 00:01:56,629 --> 00:01:59,269 Now, if you do a little bit of algebra here, you literally, 36 00:01:59,269 --> 00:02:01,979 you know-- r of s plus delta s comma t, that's the same thing 37 00:02:01,980 --> 00:02:06,310 as x of s plus delta s t i, plus y of s plus 38 00:02:06,310 --> 00:02:07,900 delta s t j, plus z. 39 00:02:07,900 --> 00:02:09,740 All that minus this thing. 40 00:02:09,740 --> 00:02:11,370 If you do a little bit of algebra with that, and if you 41 00:02:11,370 --> 00:02:13,069 don't believe me, try it out. 42 00:02:13,069 --> 00:02:22,949 This is going to be equal to the limit of delta s 43 00:02:22,949 --> 00:02:25,619 approaching 0-- and I'm going to write it small because it'd 44 00:02:25,620 --> 00:02:33,120 take up a lot of space-- of x of s plus delta s comma t minus 45 00:02:33,120 --> 00:02:38,340 x of s and t, I think you know where I'm going. 46 00:02:38,340 --> 00:02:40,280 This is all a little bit monotonous to write it all 47 00:02:40,280 --> 00:02:43,439 out, but never hurts. 48 00:02:43,439 --> 00:02:47,750 Times s or divided by delta s times i-- and then I'll do it 49 00:02:47,750 --> 00:02:51,099 in different colors, so it's less monotonous-- plus y. 50 00:02:51,099 --> 00:02:53,289 Where every-- those limited delta s [? approaches ?] 51 00:02:53,289 --> 00:02:55,719 0 applies to every term I'm writing out here. 52 00:02:55,719 --> 00:03:04,590 y of s plus delta s comma t minus y of s comma t, all of 53 00:03:04,590 --> 00:03:09,560 that over delta s times j. 54 00:03:09,560 --> 00:03:19,240 And then finally, plus z of s plus delta s comma t minus z of 55 00:03:19,240 --> 00:03:27,909 s and t, all of that over delta s times the z unit vector, k. 56 00:03:27,909 --> 00:03:29,889 And this all comes out of this definition. 57 00:03:29,889 --> 00:03:32,989 If you literally just put s plus delta s in place for s-- 58 00:03:32,990 --> 00:03:34,600 you evaluate all this, do a little algebra-- you're going 59 00:03:34,599 --> 00:03:36,500 to get the exact same thing. 60 00:03:36,500 --> 00:03:40,110 And this, hopefully, pops out at you as, gee, we're just 61 00:03:40,110 --> 00:03:42,590 taking the partial derivative of each of these functions 62 00:03:42,590 --> 00:03:44,479 with respect to s. 63 00:03:44,479 --> 00:03:49,039 And these functions right here, this x of s and t, this is a 64 00:03:49,039 --> 00:03:50,669 non-vector valued function. 65 00:03:50,669 --> 00:03:53,449 This y, this is also a non-vector valued function. 66 00:03:53,449 --> 00:03:56,239 z is also a non-vector valued function. 67 00:03:56,240 --> 00:03:59,540 When you put them all together, it becomes a vector valued 68 00:03:59,539 --> 00:04:01,409 function, because we're multiplying the first 69 00:04:01,409 --> 00:04:02,270 one times a vector. 70 00:04:02,270 --> 00:04:03,620 The second one times another vector. 71 00:04:03,620 --> 00:04:05,210 The third one times another vector. 72 00:04:05,210 --> 00:04:06,570 But independently, these functions are 73 00:04:06,569 --> 00:04:07,569 non-vector valued. 74 00:04:07,569 --> 00:04:10,139 So this is just the definition of the regular 75 00:04:10,139 --> 00:04:11,109 partial derivatives. 76 00:04:11,110 --> 00:04:13,170 Where we're taking the limit as delta s approaches 0 77 00:04:13,169 --> 00:04:15,019 in each of these cases. 78 00:04:15,020 --> 00:04:17,379 So this is the exact same thing. 79 00:04:17,379 --> 00:04:21,529 This is equal to-- this is the exact same thing as the partial 80 00:04:21,529 --> 00:04:30,059 derivative of x with respect to s times i plus the partial 81 00:04:30,060 --> 00:04:38,000 derivative y with respect to s times j plus the partial 82 00:04:38,000 --> 00:04:43,699 derivative of z with respect to s times k. 83 00:04:43,699 --> 00:04:47,159 I'm going to do one more thing here and this is pseudo mathy, 84 00:04:47,160 --> 00:04:49,620 but it's going to come out-- the whole reason I'm even doing 85 00:04:49,620 --> 00:04:52,649 this video, is it's going to give us some good tools in our 86 00:04:52,649 --> 00:04:55,469 tool kit for the videos that I'm about to do on 87 00:04:55,470 --> 00:04:57,190 surface integrals. 88 00:04:57,189 --> 00:04:59,509 So I'm going to do one thing here that's a little pseudo 89 00:04:59,509 --> 00:05:01,990 mathy, and that's really because differentials are these 90 00:05:01,990 --> 00:05:04,600 things that are very hard to define rigorously, but I think 91 00:05:04,600 --> 00:05:07,510 it'll give you the intuition of what's going on. 92 00:05:07,509 --> 00:05:10,740 So this thing right here, I'm going to say this is also equal 93 00:05:10,740 --> 00:05:13,819 to-- and you're not going to see this in any math textbook, 94 00:05:13,819 --> 00:05:16,519 and hard core mathematicians are going to kind of cringe 95 00:05:16,519 --> 00:05:17,949 when they see me do this. 96 00:05:17,949 --> 00:05:20,159 But I like to do it because I think it'll give you the 97 00:05:20,160 --> 00:05:23,090 intuition on what's going on when we take our 98 00:05:23,089 --> 00:05:23,989 surface integrals. 99 00:05:23,990 --> 00:05:28,340 So I'm going to say that this whole thing right here, that 100 00:05:28,339 --> 00:05:36,909 that is equal to r of s plus the differential of s-- a super 101 00:05:36,910 --> 00:05:44,130 small change in s-- t minus r of s and t, all of that 102 00:05:44,129 --> 00:05:48,899 over that same super small change in s. 103 00:05:48,899 --> 00:05:51,439 So hopefully you understand at least why I view 104 00:05:51,439 --> 00:05:52,420 things this way. 105 00:05:52,420 --> 00:05:56,930 When I take the limit as delta s approaches 0, these delta s's 106 00:05:56,930 --> 00:05:58,810 are going to get super duper duper small. 107 00:05:58,810 --> 00:06:03,370 And in my head, that's how I imagine differentials. 108 00:06:03,370 --> 00:06:07,365 When someone writes the derivative of y with respect to 109 00:06:07,365 --> 00:06:12,870 x-- and let's say that they say that that is 2-- and we've done 110 00:06:12,870 --> 00:06:15,290 a little bit of math with differentials before. 111 00:06:15,290 --> 00:06:19,400 You can imagine multiplying both sides by dx, and you 112 00:06:19,399 --> 00:06:22,209 could get dy is equal to 2dx. 113 00:06:22,209 --> 00:06:24,669 We've done this throughout calculus. 114 00:06:24,670 --> 00:06:27,870 The way I imagine it is super small change in y-- infinitely 115 00:06:27,870 --> 00:06:31,829 small change in y-- is equal to 2 times-- though, you can 116 00:06:31,829 --> 00:06:35,419 imagine an equally small change in x. 117 00:06:35,420 --> 00:06:39,879 So it's a-- well, if you have a super small change in x, your 118 00:06:39,879 --> 00:06:42,290 change in y is going to be still super small, but it's 119 00:06:42,290 --> 00:06:43,510 going to be 2 times that. 120 00:06:43,509 --> 00:06:44,709 I guess that's the best way to view it. 121 00:06:44,709 --> 00:06:48,310 But in general, I view differentials as super small 122 00:06:48,310 --> 00:06:50,420 changes in a variable. 123 00:06:50,420 --> 00:06:53,600 So with that out of the way, and me explaining to you that 124 00:06:53,600 --> 00:06:56,290 many mathematicians would cringe at what I just wrote, 125 00:06:56,290 --> 00:06:59,290 hopefully this gives you a little-- this isn't like 126 00:06:59,290 --> 00:07:01,080 some crazy thing I did. 127 00:07:01,079 --> 00:07:04,240 I'm just saying, oh, delta s as delta approaches 0, I 128 00:07:04,240 --> 00:07:06,639 kind of imagine that as ds. 129 00:07:06,639 --> 00:07:09,829 And the whole reason I did that, is if you take this side 130 00:07:09,829 --> 00:07:12,609 and that side, and multiply both sides times this 131 00:07:12,610 --> 00:07:16,689 differential ds, then what happens? 132 00:07:16,689 --> 00:07:23,250 The left hand side, you get the partial of r with respect to 133 00:07:23,250 --> 00:07:28,120 s is equal to this times ds. 134 00:07:28,120 --> 00:07:30,189 I'll do ds in maybe pink. 135 00:07:30,189 --> 00:07:32,769 Times ds-- this is just a regular differential, 136 00:07:32,769 --> 00:07:33,889 super small change in s. 137 00:07:33,889 --> 00:07:37,219 This is a kind of a partial, with respect to s. 138 00:07:37,220 --> 00:07:42,740 That's going to be equal to-- well, if you multiply this side 139 00:07:42,740 --> 00:07:45,230 of the equation times ds, this guy's going to disappear. 140 00:07:45,230 --> 00:07:49,700 So it's going to be r of s, plus our super small change 141 00:07:49,699 --> 00:07:55,079 in s, t minus r of s and t. 142 00:07:55,079 --> 00:07:57,019 Now let me put a little square around this. 143 00:07:57,019 --> 00:07:59,649 This is going to be valuable for us in the next video. 144 00:07:59,649 --> 00:08:01,979 We're going to actually think about what this means and how 145 00:08:01,980 --> 00:08:05,210 to visualize this on a surface. 146 00:08:05,209 --> 00:08:07,379 As you can imagine, this is a vector right here. 147 00:08:07,379 --> 00:08:09,259 You have 2 vector valued functions and you're 148 00:08:09,259 --> 00:08:09,990 taking the difference. 149 00:08:09,990 --> 00:08:11,439 And we're going to visualize it in the next video. 150 00:08:11,439 --> 00:08:13,759 It's going to really help us with surface integrals. 151 00:08:13,759 --> 00:08:17,269 By the same exact logic, we can do everything we did 152 00:08:17,269 --> 00:08:20,139 here with s, we can do it with t, as well. 153 00:08:20,139 --> 00:08:24,959 So we can define the partial-- I'll draw a little-- I can 154 00:08:24,959 --> 00:08:28,649 define the partial of r with respect-- let me do it in a 155 00:08:28,649 --> 00:08:31,079 different color, completely different color. 156 00:08:31,079 --> 00:08:32,269 It's orange. 157 00:08:32,269 --> 00:08:36,579 The partial of r with respect to t-- the definition 158 00:08:36,580 --> 00:08:37,370 is just right here. 159 00:08:37,370 --> 00:08:46,750 The limit as delta t approaches 0 of r of s t plus delta 160 00:08:46,750 --> 00:08:51,100 t minus r of s and t. 161 00:08:51,100 --> 00:08:52,769 In this situation we're holding the s, you can 162 00:08:52,769 --> 00:08:53,879 imagine, in constant. 163 00:08:53,879 --> 00:08:59,220 We're finding its change in t, all of that over delta t. 164 00:08:59,220 --> 00:09:00,379 And the same thing falls out. 165 00:09:00,379 --> 00:09:04,269 This is equal to the partial of x with respect to ti plus y 166 00:09:04,269 --> 00:09:07,579 with respect to tj, plus z with respect to tk. 167 00:09:07,580 --> 00:09:11,190 Same exact thing, you just kind of swap the s's and the t's. 168 00:09:11,190 --> 00:09:13,740 And by that same logic, you'd have the same 169 00:09:13,740 --> 00:09:15,990 result but in terms of t. 170 00:09:15,990 --> 00:09:18,970 If you do this pseudo mathy thing that I did up here, then 171 00:09:18,970 --> 00:09:25,680 you would get the partial of r with respect to t times a super 172 00:09:25,679 --> 00:09:28,799 small change in t. dt, our t differential, you could 173 00:09:28,799 --> 00:09:40,359 imagine, is equal to r of st plus dt minus r of s and t. 174 00:09:40,360 --> 00:09:43,009 So let's box these two guys away. 175 00:09:43,009 --> 00:09:44,340 And in the next video, we're going to actually 176 00:09:44,340 --> 00:09:46,790 visualize what these mean. 177 00:09:46,789 --> 00:09:49,639 And sometimes, when you kind of do a bunch of like, silly math 178 00:09:49,639 --> 00:09:50,669 like this, you're always like, all right, what is 179 00:09:50,669 --> 00:09:51,240 this all about? 180 00:09:51,240 --> 00:09:53,110 Remember, all I did is I said, what does it mean to take the 181 00:09:53,110 --> 00:09:55,539 derivative of this with respect to s or t? 182 00:09:55,539 --> 00:09:58,969 Played around with it a little bit, I got this result. 183 00:09:58,970 --> 00:10:01,220 These 2 are going to be very valuable for us, I think, in 184 00:10:01,220 --> 00:10:03,870 getting the intuition for why surface integrals 185 00:10:03,870 --> 00:10:06,450 look the way they do.