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00:00:00I will write down the 4th and 12th
00:00:04I will write down the 4th and 12th
00:00:12This is about the directional derivative
00:00:17We will use the scalar function f
00:00:22We will use the scalar function f
00:00:26Find the factor of the direction of the f
00:00:33Find the fraction of the direction
00:00:36Find the factor of the direction of the f
00:00:41from the point P to the point Q.
00:00:45The point is that we have to make a unit vector PQ
00:00:49in the direction of the unit vector PQ.
00:00:55Therefore, when we do it, we have to do it step by step.
00:01:00Step 1. This is the first step.
00:01:05We have to think about the unit vector PQ first.
00:01:11Step 12.
00:01:14The unit vector PQ is the point Q.
00:01:17The point Q is P.
00:01:19The point Q is Q.
00:01:211 plus 1 is 2.
00:01:250 plus 1 is 1.
00:01:304 plus 2 is 2.
00:01:332 plus 1 is 2.
00:01:36These are the vector PQ, vector IJK.
00:01:39Let's think about the pq value, is it equal to 1?
00:01:44If it's equal to 1, then it's done. But if it's not equal to 1, then we need to find the value.
00:01:55It's the square root of 9, so we get 3.
00:01:59See? It's not equal to 1.
00:02:01Therefore, the unit vector pq that we need to use is equal to pq times pq.
00:02:14We get negative 2 over 3, 1 over 3, 2 over 3.
00:02:19We need to use this one instead.
00:02:21We need to check if it's equal to 1.
00:02:31If it's not equal to 1, then we need to find the value.
00:02:33We need to make it equal to 1.
00:02:35Now it's equal to 1.
00:02:37We need to find the unit vector value.
00:02:39We need to find the pq value.
00:02:41Step 2.
00:02:49Think about the gradient.
00:02:51Think about the gradient.
00:02:53Yesterday I just learned about the gradient.
00:02:55The gradient of f.
00:02:57The gradient of f.
00:03:07What was the gradient yesterday?
00:03:09Did you forget?
00:03:11It's a vector.
00:03:13It's x of f.
00:03:17It's y of f.
00:03:21It's z of f.
00:03:23It's z of f.
00:03:25It's a vector.
00:03:27We use the f that we get.
00:03:29We use the f that we get.
00:03:31We use the f that we get.
00:03:33We use the f that we get.
00:03:35We use the f that we get.
00:03:37We use the f that we get.
00:03:39We use the f that we get.
00:03:41We use the f that we get.
00:03:43We use the f that we get.
00:03:45We use the f that we get.
00:03:47We use the f that we get.
00:03:49We use the f that we get.
00:03:51We use the f that we get.
00:03:53We use the f that we get.
00:03:55We use the f that we get.
00:03:59We use the f that we get.
00:04:05Again!
00:05:071 minus 1 is equal to 2, and then multiply by 2 to get the gradient at point 1 minus 1 is equal to 2.
00:05:13Let's think about the gradient first.
00:05:19What is the value of z?
00:05:21z is equal to 2.
00:05:23Minus 8 is equal to i.
00:05:2812 is equal to j.
00:05:31Minus 2 is equal to k.
00:05:33Minus 2 is equal to k.
00:05:59Step 2.
00:06:01Step 3.
00:06:11Step 3.
00:06:19Step 3.
00:06:31Step 4.
00:06:37Step 4.
00:06:43Step 4.
00:06:57Step 4.
00:07:01Step 5.
00:07:11Step 5.
00:07:15Step 5.
00:07:21Step 5.
00:07:25Step 5.
00:07:27อันออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออ�
00:07:57อออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออ�
00:08:27อออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออ�
00:08:57อออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออ�
00:09:27อออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออ�
00:09:57อออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออ�
00:10:27อออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออ�
00:10:57อออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออออ�
00:11:27It's a directional sentence.
00:11:33He said that f's high directional deliverance at point 11 is 3 root 5.
00:11:41f's high directional deliverance in the V area is one u and one v.
00:11:49This is u, this is v.
00:11:52See? You can differentiate between u and v.
00:11:55There's a unit vector u and a unit vector v. This is a unit anyway because it's already divided.
00:12:01If we think about u and add a square root to it, it will be equal to 1.
00:12:12Is it 1?
00:12:14It's equal to 1.
00:12:16So we don't need to divide this one. It's equal to 1.
00:12:20This is a unit.
00:12:22This is a unit of u and a unit of v.
00:12:27It's already a unit vector.
00:12:30Find the gradient at point 1.
00:12:34Find the gradient at point 1.
00:12:36Suppose that the gradient at point 1 is a and b.
00:12:39It's a vector a and b.
00:12:41So we need to find how much a and b is.
00:12:43First, we're going to think about it one by one.
00:12:46Because there are both u and v.
00:12:48I'll take u first.
00:12:50The directional derivative of f at point 1 in u is 3 over root 5.
00:12:58Let's take this example first.
00:13:00Let's take one example at a time.
00:13:02I'll take the first one.
00:13:04According to the formula, this will be a gradient, right?
00:13:11Dot with what?
00:13:13u.
00:13:14Since u is already a unit.
00:13:16Dot u.
00:13:183 over root 5.
00:13:22We don't know the gradient at point 1.
00:13:25Let's take an example.
00:13:27Let's take an example of a and b.
00:13:31Right?
00:13:32We already know u.
00:13:34What is u?
00:13:381 over root 5 plus 2 over root 5 is equal to 3 over root 5.
00:13:49This is u.
00:13:51Dot it.
00:13:53It will be i times i times j times j when dot.
00:13:56a over root 5.
00:13:582b over root 5 is equal to 3 over root 5.
00:14:04We always multiply root 5.
00:14:08So, we get a plus 2b is equal to 3.
00:14:14It's the first equation.
00:14:16Let's continue.
00:14:46Let's keep equation 1.
00:14:50What is the value of v at point 1?
00:14:585 over root 5.
00:15:005 over root 5.
00:15:02Let's continue.
00:15:08Let's set the gradient at point 1.
00:15:12Dot with v is equal to 5 over root 5.
00:15:26The gradient at point 1 is ab.
00:15:28The gradient at point 1 is ab.
00:15:30What is v?
00:15:32Look at v.
00:15:36The vector v is above.
00:15:382 over root 5 plus 1 over root 5 is equal to 5 over root 5.
00:15:46Dot it.
00:15:48Dot this and dot that.
00:15:502a plus b over root 5.
00:15:565 over root 5.
00:15:585 over root 5.
00:16:00We always multiply root 5.
00:16:022a plus b is equal to 5.
00:16:04The second equation.
00:16:06Can you solve the equation 2?
00:16:08ab.
00:16:10Solve it.
00:16:12The first equation is 2.
00:16:14There are 2 equations.
00:16:16Solve it.
00:16:18What is the value of ab?
00:16:22Let's multiply equation 1 by 2.
00:16:262a plus 4b is equal to 6.
00:16:32The third equation.
00:16:34Solve the third equation minus the second equation.
00:16:383b is equal to 1.
00:16:40b is 1 over 3.
00:16:42The first equation.
00:16:44Solve the first equation.
00:16:56b is 1 over 3.
00:17:002a plus 1 over 3 is equal to 5.
00:17:04Solve it.
00:17:062a is 14 over 3.
00:17:1014 over 3.
00:17:12Is it 14?
00:17:1414 over 3.
00:17:162a is 7 over 3.
00:17:18This is the answer.
00:17:20But don't solve the value of ab.
00:17:22It has to be the vector gradient.
00:17:24So the gradient at point 11
00:17:26will be 1 over 3i plus 7 over 3j.
00:17:34It has to be the gradient.
00:17:36This is the answer.
00:17:40Let's switch.
00:17:42a is 7 over 3.
00:17:44b is 1 over 3.
00:17:46a is above b.
00:18:10Are you sleepy?
00:18:20Are you sleepy?
00:18:24Yes, I'm sleepy.
00:18:32I'm lonely. I miss you so much.
00:18:34You have a bad temper.
00:18:36Do you know why?
00:18:38I went to a cafe
00:18:40and sent you a cup of coffee.
00:18:42Did you see it?
00:18:44I told you
00:18:46next week
00:18:48or tomorrow
00:18:50I'll send you a cup of coffee
00:18:52at the cafe.
00:19:08I'll send you a cup of coffee
00:19:10at the cafe.
00:19:12I'll send you a cup of coffee
00:19:14at the cafe.
00:19:16I'll send you a cup of coffee
00:19:18at the cafe.
00:19:20I'll send you a cup of coffee
00:19:22at the cafe.
00:19:24I'll send you a cup of coffee
00:19:26at the cafe.
00:19:28I'll send you a cup of coffee
00:19:30at the cafe.
00:19:32I'll send you a cup of coffee
00:19:34at the cafe.
00:19:36I'll send you a cup of coffee
00:19:38at the cafe.
00:19:40I'll send you a cup of coffee
00:19:42at the cafe.
00:19:44I'll send you a cup of coffee
00:19:46at the cafe.
00:20:02function f
00:20:04f of x, y, z is equal to 5 square root of x, y, z
00:20:07must be an equation like this
00:20:10therefore, f of x, y, z
00:20:12you can move 13 to the left or right
00:20:14or equal to 0, it's up to you
00:20:15it's already bifurcated, so it doesn't matter
00:20:18so it's just this
00:20:19x is equal to 2
00:20:20plus 5y is equal to 2
00:20:22plus 3z is equal to 2
00:20:24and c is equal to 13
00:20:28it's just this because it's bifurcated
00:20:3013 is irrelevant
00:20:32it's not used
00:20:34with the stage
00:20:50oh
00:20:51here we go
00:20:53the key character here is the Gai στην
00:20:56I'm supposed to have the Gai
00:20:57must have the Gai
00:20:58it will be a vector
00:21:00if it is a status bar
00:21:01it's because of our 2 forces
00:21:03If we want to think, there must be a gradient that is a vector that sets the scene first, right?
00:21:10There must be a gradient that is the hero.
00:21:14So we have to think about the gradient first.
00:21:18At this point, there is a point, see?
00:21:20There is a point P0, this is point P0.
00:21:23At the point negative 1, 0, 2, we have to find the gradient at negative 1, 0, 2 first.
00:21:31It's a vector that sets the scene and the hero.
00:21:34That's step 1.
00:21:41Gradient x, y, z first.
00:21:45Can I just skip it? I'm too lazy to write it.
00:21:47This one is long.
00:21:49Okay, I'll just write this one.
00:21:51It's f of i, j, y, and j.
00:21:57I'll just write the formula. I'm too lazy to write it.
00:21:59It's too long.
00:22:01I'll just skip it.
00:22:03I know that I have to skip this one.
00:22:21How many steps can I skip?
00:22:23Step x, dx, i.
00:22:28The next one is 10, 10, y, j.
00:22:33The next one is 6, z.
00:22:37Because the others can skip to 0.
00:22:396, z, k.
00:22:41And then substitute the value at the point.
00:22:43What is the point?
00:22:45At the point negative 1, 0, 2.
00:22:47Substitute the value negative 1, 0, 2.
00:22:54It becomes negative 4, i, 0, j, 12, k.
00:23:02Negative 4, 0, 12.
00:23:06And this is the vector ABC.
00:23:10It's the value of ABC that we're going to use in the formula.
00:23:17Step 2, go to the tangent plane.
00:23:22Tangent plane.
00:23:27At the point negative 1, 0, 2.
00:23:34Negative 1, 0, 2 is x, 0, y, 0, z, 0.
00:23:37It's in the formula.
00:23:39ax minus x, 0 plus by minus y, 0 plus cz minus z, 0 is equal to 0.
00:23:51Tangent plane formula.
00:23:54ABC, x, 0, y, 0.
00:23:56If you know the formula, put it in and answer.
00:24:07x is 12, z is negative 2, equal to 0.
00:24:16This is the tangent plane.
00:24:25This is the tangent plane.
00:24:29Spread it out.
00:24:35Negative 4, x, negative 4, 0, 12, z, negative 24, equal to 0.
00:24:46It becomes negative 4, x, plus 12, z, negative 28, equal to 0.
00:24:53Negative 4, x, minus 3z, plus 7, equal to 0.
00:25:10It becomes x, minus 3z, plus 7, equal to 0.
00:25:14This is the tangent plane.
00:25:17You can put x and z in the same term.
00:25:34Step 3.
00:25:36Normal line.
00:25:38Normal line.
00:25:45Normal line.
00:25:47Normal line.
00:25:49x, y, z.
00:26:14Remember the formula?
00:26:16y, 0, z, 0, plus ABC, t, t, t.
00:26:19x, 0, y, 0, z, 0, minus 1, 0, 2, minus 4, 0, 12, minus 4, t, 0, t, 12, t.
00:26:38x, minus 1, minus 4, t, y, 0, z, 2, plus 12, t.
00:26:48This is the tangent plane.
00:26:56Remember the formula?
00:26:58x, 0, y, 0, z, 0, plus ABC, t, t, t.
00:27:07x, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1,
00:27:37minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1
00:28:07Minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus
00:28:371, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1,
00:29:07minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1, minus 1
00:29:37Do you have it?
00:29:39Can you show me one more page?
00:29:4110 and 12.
00:29:43I made a mistake.
00:29:45This is number 11.
00:29:47Okay.
00:29:51I'll do number 10 first.
00:29:53I'll give you 5 minutes for each question.
00:29:57Number 9 is homework.
00:29:59Try to do it.
00:30:03So you can get a new role.
00:30:05You can get a new role.
00:30:07Directional is a must.
00:30:094 points, 5 points.
00:30:11Tangent plane, normal line, 4 points, 5 points.
00:30:13These two questions are 10 points.
00:30:15It's not hard.
00:30:17It's all the same steps.
00:30:25Remember the steps.
00:30:27Create a PQ first.
00:30:29Find a gradient instead of a curve.
00:30:31Then dot.
00:30:33Do it.
00:30:35Do it.
00:30:37Do it.
00:30:39Do it.
00:30:41Do it.
00:30:43Do it.
00:30:45Do it.
00:30:47Do it.
00:30:49Do it.
00:30:51Do it.
00:30:53Do it.
00:30:55Do it.
00:30:57Do it.
00:30:59Do it.
00:31:01Do it.
00:31:03Do it.
00:31:05Do it.
00:31:07Do it.
00:31:09Do it.
00:31:11Do it.
00:31:13Do it.
00:31:15Do it.
00:31:17Do it.
00:31:19Do it.
00:31:21Do it.
00:31:23Do it.
00:31:25Do it.
00:31:27Do it.
00:31:29Root 14.
00:31:33Unit PQ.
00:31:35Do the same pattern.
00:31:59Repeat.
00:32:09Gradient.
00:32:17Gradient.
00:32:23Gradient.
00:32:25Gradient.
00:32:29Gradient.
00:32:33Gradient.
00:32:37Gradient.
00:32:41Pseudo.
00:32:47Pseudo.
00:32:49Pseudo.
00:32:53Pseudo.
00:32:59Pseudo.
00:33:03Pseudo.
00:33:07Pseudo.
00:33:11Pseudo.
00:33:17Gradient.
00:33:19Gradient.
00:33:27Gradient.
00:33:33Gradient.
00:33:37Gradient.
00:33:43Gradient.
00:35:4512, no, 6, 3, no, 2, 7, 10, 72, and 7, 14.
00:36:06Now, I'm going to add a question.
00:36:08The question is the same, right?
00:36:12I want to know how much is the maximum value.
00:36:16Add a little bit.
00:36:22Add a little bit of how much is the maximum value.
00:36:25It's the same, right?
00:36:27If it's the same, you have to think of the maximum value.
00:36:30It's the norm of the gradient.
00:36:32The norm of the gradient is the answer.
00:36:34If they ask how much is the maximum value,
00:36:36write the reason first and then think of the gradient.
00:36:412.
00:36:43Think of the gradient to be the same value as u.
00:36:523.
00:36:54Think of the gradient to be the same value as u.
00:37:21The norm of the gradient is the answer.
00:37:524.
00:37:54The norm of the gradient is the answer.
00:37:56But if the opposite equation is negative,
00:37:58the slope will decrease.
00:38:00This will be the lowest slope.
00:38:02.
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