[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.33,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.33,0:00:03.11,Default,,0000,0000,0000,,One of the most fundamental\Nideas in all of physics
Dialogue: 0,0:00:03.11,0:00:05.38,Default,,0000,0000,0000,,is the idea of work.
Dialogue: 0,0:00:05.38,0:00:08.45,Default,,0000,0000,0000,,Now when you first learn work,\Nyou just say, oh, that's
Dialogue: 0,0:00:08.45,0:00:10.12,Default,,0000,0000,0000,,just force times distance.
Dialogue: 0,0:00:10.12,0:00:12.20,Default,,0000,0000,0000,,But then later on, when you\Nlearn a little bit about
Dialogue: 0,0:00:12.20,0:00:14.77,Default,,0000,0000,0000,,vectors, you realize that the\Nforce isn't always going in
Dialogue: 0,0:00:14.77,0:00:17.61,Default,,0000,0000,0000,,the same direction as\Nyour displacement.
Dialogue: 0,0:00:17.61,0:00:21.45,Default,,0000,0000,0000,,So you learn that work is\Nreally the magnitude, let me
Dialogue: 0,0:00:21.45,0:00:33.07,Default,,0000,0000,0000,,write this down, the magnitude\Nof the force, in the direction,
Dialogue: 0,0:00:33.07,0:00:39.46,Default,,0000,0000,0000,,or the component of the force\Nin the direction
Dialogue: 0,0:00:39.46,0:00:41.74,Default,,0000,0000,0000,,of displacement.
Dialogue: 0,0:00:41.74,0:00:44.21,Default,,0000,0000,0000,,Displacement is just distance\Nwith some direction.
Dialogue: 0,0:00:44.21,0:00:49.97,Default,,0000,0000,0000,,
Dialogue: 0,0:00:49.97,0:00:55.29,Default,,0000,0000,0000,,Times the magnitude of the\Ndisplacement, or you could say,
Dialogue: 0,0:00:55.29,0:00:56.70,Default,,0000,0000,0000,,times the distance displaced.
Dialogue: 0,0:00:56.70,0:01:00.81,Default,,0000,0000,0000,,
Dialogue: 0,0:01:00.81,0:01:02.33,Default,,0000,0000,0000,,And the classic example.
Dialogue: 0,0:01:02.33,0:01:06.25,Default,,0000,0000,0000,,Maybe you have an ice cube,\Nor some type of block.
Dialogue: 0,0:01:06.25,0:01:08.74,Default,,0000,0000,0000,,I just ice so that there's\Nnot a lot of friction.
Dialogue: 0,0:01:08.74,0:01:12.51,Default,,0000,0000,0000,,Maybe it's standing on a bigger\Nlake or ice or something.
Dialogue: 0,0:01:12.51,0:01:15.03,Default,,0000,0000,0000,,And maybe you're pulling on\Nthat ice cube at an angle.
Dialogue: 0,0:01:15.03,0:01:17.61,Default,,0000,0000,0000,,Let's say, you're pulling\Nat an angle like that.
Dialogue: 0,0:01:17.61,0:01:20.82,Default,,0000,0000,0000,,That is my force, right there.
Dialogue: 0,0:01:20.82,0:01:24.08,Default,,0000,0000,0000,,Let's say my force is\Nequal to-- well, that's
Dialogue: 0,0:01:24.08,0:01:25.16,Default,,0000,0000,0000,,my force vector.
Dialogue: 0,0:01:25.16,0:01:33.87,Default,,0000,0000,0000,,Let's say the magnitude of\Nmy force vector, let's
Dialogue: 0,0:01:33.87,0:01:35.31,Default,,0000,0000,0000,,say it's 10 newtons.
Dialogue: 0,0:01:35.31,0:01:37.65,Default,,0000,0000,0000,,And let's say the direction of\Nmy force vector, right, any
Dialogue: 0,0:01:37.65,0:01:41.08,Default,,0000,0000,0000,,vector has to have a magnitude\Nand a direction, and the
Dialogue: 0,0:01:41.08,0:01:44.92,Default,,0000,0000,0000,,direction, let's say it has a\N30 degree angle, let's say a 60
Dialogue: 0,0:01:44.92,0:01:47.77,Default,,0000,0000,0000,,degree angle, above horizontal.
Dialogue: 0,0:01:47.77,0:01:49.56,Default,,0000,0000,0000,,So that's the direction\NI'm pulling in.
Dialogue: 0,0:01:49.56,0:01:52.60,Default,,0000,0000,0000,,And let's say I displace it.
Dialogue: 0,0:01:52.60,0:01:55.93,Default,,0000,0000,0000,,This is all review, hopefully.
Dialogue: 0,0:01:55.93,0:01:59.22,Default,,0000,0000,0000,,If you're displacing it, let's\Nsay you displace it 5 newtons.
Dialogue: 0,0:01:59.22,0:02:02.57,Default,,0000,0000,0000,,So let's say the displacement,\Nthat's the displacement vector
Dialogue: 0,0:02:02.57,0:02:10.29,Default,,0000,0000,0000,,right there, and the magnitude\Nof it is equal to 5 meters.
Dialogue: 0,0:02:10.29,0:02:13.46,Default,,0000,0000,0000,,So you've learned from the\Ndefinition of work, you can't
Dialogue: 0,0:02:13.46,0:02:16.94,Default,,0000,0000,0000,,just say, oh, I'm pulling with\N10 newtons of force and
Dialogue: 0,0:02:16.94,0:02:18.36,Default,,0000,0000,0000,,I'm moving it 5 meters.
Dialogue: 0,0:02:18.36,0:02:22.56,Default,,0000,0000,0000,,You can't just multiply the 10\Nnewtons times the 5 meters.
Dialogue: 0,0:02:22.56,0:02:25.66,Default,,0000,0000,0000,,You have to find the magnitude\Nof the component going in the
Dialogue: 0,0:02:25.66,0:02:29.05,Default,,0000,0000,0000,,same direction as\Nmy displacement.
Dialogue: 0,0:02:29.05,0:02:31.86,Default,,0000,0000,0000,,So what I essentially need to\Ndo is, the length, if you
Dialogue: 0,0:02:31.86,0:02:34.93,Default,,0000,0000,0000,,imagine the length of this\Nvector being 10, that's the
Dialogue: 0,0:02:34.93,0:02:37.75,Default,,0000,0000,0000,,total force, but you need to\Nfigure out the length of the
Dialogue: 0,0:02:37.75,0:02:40.77,Default,,0000,0000,0000,,vector, that's the component of\Nthe force, going in the same
Dialogue: 0,0:02:40.77,0:02:43.46,Default,,0000,0000,0000,,direction as my displacement.
Dialogue: 0,0:02:43.46,0:02:45.57,Default,,0000,0000,0000,,And a little simple\Ntrigonometry, you know that
Dialogue: 0,0:02:45.57,0:02:53.12,Default,,0000,0000,0000,,this is 10 times the cosine of\N60 degrees, or that's equal to,
Dialogue: 0,0:02:53.12,0:02:58.01,Default,,0000,0000,0000,,cosine of 60 degrees is 1/2, so\Nthat's just equal to 5.
Dialogue: 0,0:02:58.01,0:03:00.38,Default,,0000,0000,0000,,So this magnitude, the\Nmagnitude of the force going
Dialogue: 0,0:03:00.38,0:03:02.41,Default,,0000,0000,0000,,in the same direction of\Nthe displacement in this
Dialogue: 0,0:03:02.41,0:03:04.81,Default,,0000,0000,0000,,case, is 5 newtons.
Dialogue: 0,0:03:04.81,0:03:07.50,Default,,0000,0000,0000,,
Dialogue: 0,0:03:07.50,0:03:09.85,Default,,0000,0000,0000,,And then you can\Nfigure out the work.
Dialogue: 0,0:03:09.85,0:03:19.56,Default,,0000,0000,0000,,You could say that the work is\Nequal to 5 newtons times, I'll
Dialogue: 0,0:03:19.56,0:03:20.63,Default,,0000,0000,0000,,just write a dot for times.
Dialogue: 0,0:03:20.63,0:03:22.29,Default,,0000,0000,0000,,I don't want you to think\Nit's cross product.
Dialogue: 0,0:03:22.29,0:03:26.68,Default,,0000,0000,0000,,Times 5 meters, which is 25\Nnewton meters, or you could
Dialogue: 0,0:03:26.68,0:03:31.25,Default,,0000,0000,0000,,even say 25 Joules of\Nwork have been done.
Dialogue: 0,0:03:31.25,0:03:35.28,Default,,0000,0000,0000,,And this is all review of\Nsomewhat basic physics.
Dialogue: 0,0:03:35.28,0:03:36.72,Default,,0000,0000,0000,,But just think about\Nwhat happened, here.
Dialogue: 0,0:03:36.72,0:03:37.43,Default,,0000,0000,0000,,What was the work?
Dialogue: 0,0:03:37.43,0:03:39.19,Default,,0000,0000,0000,,If I write in the abstract.
Dialogue: 0,0:03:39.19,0:03:42.55,Default,,0000,0000,0000,,The work is equal\Nto the 5 newtons.
Dialogue: 0,0:03:42.55,0:03:46.70,Default,,0000,0000,0000,,That was the magnitude of my\Nforce vector, so it's the
Dialogue: 0,0:03:46.70,0:03:52.63,Default,,0000,0000,0000,,magnitude of my force vector,\Ntimes the cosine of this angle.
Dialogue: 0,0:03:52.63,0:03:53.86,Default,,0000,0000,0000,,So you know, let's\Ncall that theta.
Dialogue: 0,0:03:53.86,0:03:55.01,Default,,0000,0000,0000,,Let's say it a\Nlittle generally.
Dialogue: 0,0:03:55.01,0:03:58.15,Default,,0000,0000,0000,,So times the cosine\Nof the angle.
Dialogue: 0,0:03:58.15,0:04:01.74,Default,,0000,0000,0000,,This is the amount of my force\Nin the direction of the
Dialogue: 0,0:04:01.74,0:04:04.96,Default,,0000,0000,0000,,displacement, the cosine of the\Nangle between them, times the
Dialogue: 0,0:04:04.96,0:04:06.80,Default,,0000,0000,0000,,magnitude of the displacement.
Dialogue: 0,0:04:06.80,0:04:12.26,Default,,0000,0000,0000,,So times the magnitude\Nof the displacement.
Dialogue: 0,0:04:12.26,0:04:15.56,Default,,0000,0000,0000,,Or if I wanted to rewrite that,\NI could just write that as, the
Dialogue: 0,0:04:15.56,0:04:18.94,Default,,0000,0000,0000,,magnitude of the displacement\Ntimes the magnitude of
Dialogue: 0,0:04:18.94,0:04:23.40,Default,,0000,0000,0000,,the force times the\Ncosine of theta.
Dialogue: 0,0:04:23.40,0:04:26.76,Default,,0000,0000,0000,,And I've done multiple videos\Nof this, in the linear algebra
Dialogue: 0,0:04:26.76,0:04:28.88,Default,,0000,0000,0000,,playlist, in the physics\Nplaylist, where I talk about
Dialogue: 0,0:04:28.88,0:04:31.58,Default,,0000,0000,0000,,the dot product and the cross\Nproduct and all of that, but
Dialogue: 0,0:04:31.58,0:04:40.47,Default,,0000,0000,0000,,this is the dot product\Nof the vectors d and f.
Dialogue: 0,0:04:40.47,0:04:43.70,Default,,0000,0000,0000,,So in general, if you're trying\Nto find the work for a constant
Dialogue: 0,0:04:43.70,0:04:46.73,Default,,0000,0000,0000,,displacement, and you have a\Nconstant force, you just take
Dialogue: 0,0:04:46.73,0:04:48.53,Default,,0000,0000,0000,,the dot product of\Nthose two vectors.
Dialogue: 0,0:04:48.53,0:04:51.33,Default,,0000,0000,0000,,And if the dot product is a\Ncompletely foreign concept to
Dialogue: 0,0:04:51.33,0:04:53.77,Default,,0000,0000,0000,,you, might want to watch, I\Nthink I've made multiple, 4
Dialogue: 0,0:04:53.77,0:04:56.38,Default,,0000,0000,0000,,or 5 videos on the dot\Nproduct, and its intuition,
Dialogue: 0,0:04:56.38,0:04:57.42,Default,,0000,0000,0000,,and how it compares.
Dialogue: 0,0:04:57.42,0:04:59.28,Default,,0000,0000,0000,,But just to give you a little\Nbit of that intuition right
Dialogue: 0,0:04:59.28,0:05:03.92,Default,,0000,0000,0000,,here, the dot product, when\NI take f dot d, or d dot f,
Dialogue: 0,0:05:03.92,0:05:08.44,Default,,0000,0000,0000,,what it's giving me is, I'm\Nmultiplying the magnitude, well
Dialogue: 0,0:05:08.44,0:05:10.13,Default,,0000,0000,0000,,I could just read this out.
Dialogue: 0,0:05:10.13,0:05:13.59,Default,,0000,0000,0000,,But the idea of the dot product\Nis, take how much of this
Dialogue: 0,0:05:13.59,0:05:16.80,Default,,0000,0000,0000,,vector is going in the same\Ndirection as this vector,
Dialogue: 0,0:05:16.80,0:05:18.50,Default,,0000,0000,0000,,in this case, this much.
Dialogue: 0,0:05:18.50,0:05:21.11,Default,,0000,0000,0000,,And then multiply\Nthe two magnitudes.
Dialogue: 0,0:05:21.11,0:05:22.41,Default,,0000,0000,0000,,And that's what we\Ndid right here.
Dialogue: 0,0:05:22.41,0:05:26.23,Default,,0000,0000,0000,,So the work is going to be the\Nforce vector, dot, taking the
Dialogue: 0,0:05:26.23,0:05:28.98,Default,,0000,0000,0000,,dot part of the force vector\Nwith the displacement vector,
Dialogue: 0,0:05:28.98,0:05:30.84,Default,,0000,0000,0000,,and this, of course,\Nis a scalar value.
Dialogue: 0,0:05:30.84,0:05:33.04,Default,,0000,0000,0000,,And we'll work out some\Nexamples in the future where
Dialogue: 0,0:05:33.04,0:05:34.36,Default,,0000,0000,0000,,you'll see that that's true.
Dialogue: 0,0:05:34.36,0:05:39.00,Default,,0000,0000,0000,,So this is all review of\Nfairly elementary physics.
Dialogue: 0,0:05:39.00,0:05:42.50,Default,,0000,0000,0000,,Now let's take a more\Ncomplex example, but it's
Dialogue: 0,0:05:42.50,0:05:43.67,Default,,0000,0000,0000,,really the same idea.
Dialogue: 0,0:05:43.67,0:05:45.87,Default,,0000,0000,0000,,Let's define a vector field.
Dialogue: 0,0:05:45.87,0:05:48.66,Default,,0000,0000,0000,,
Dialogue: 0,0:05:48.66,0:05:51.37,Default,,0000,0000,0000,,So let's say that I have a\Nvector field f, and we're
Dialogue: 0,0:05:51.37,0:05:54.05,Default,,0000,0000,0000,,going to think about what\Nthis means in a second.
Dialogue: 0,0:05:54.05,0:05:58.89,Default,,0000,0000,0000,,It's a function of x and y, and\Nit's equal to some scalar
Dialogue: 0,0:05:58.89,0:06:04.49,Default,,0000,0000,0000,,function of x and y times the\Ni-unit vector, or the
Dialogue: 0,0:06:04.49,0:06:08.76,Default,,0000,0000,0000,,horizontal unit vector, plus\Nsome other function, scalar
Dialogue: 0,0:06:08.76,0:06:14.25,Default,,0000,0000,0000,,function of x and y, times the\Nvertical unit vector.
Dialogue: 0,0:06:14.25,0:06:15.58,Default,,0000,0000,0000,,So what would something\Nlike this be?
Dialogue: 0,0:06:15.58,0:06:17.46,Default,,0000,0000,0000,,This is a vector field.
Dialogue: 0,0:06:17.46,0:06:20.21,Default,,0000,0000,0000,,This is a vector field\Nin 2-dimensional space.
Dialogue: 0,0:06:20.21,0:06:21.33,Default,,0000,0000,0000,,We're on the x-y plane.
Dialogue: 0,0:06:21.33,0:06:31.19,Default,,0000,0000,0000,,
Dialogue: 0,0:06:31.19,0:06:35.84,Default,,0000,0000,0000,,Or you could even say, on R2.
Dialogue: 0,0:06:35.84,0:06:37.69,Default,,0000,0000,0000,,Either way, I don't want\Nto get too much into
Dialogue: 0,0:06:37.69,0:06:39.23,Default,,0000,0000,0000,,the mathiness of it.
Dialogue: 0,0:06:39.23,0:06:40.59,Default,,0000,0000,0000,,But what does this do?
Dialogue: 0,0:06:40.59,0:06:47.27,Default,,0000,0000,0000,,Well, if I were to draw my x-y\Nplane, so that is my, again,
Dialogue: 0,0:06:47.27,0:06:49.07,Default,,0000,0000,0000,,having trouble drawing\Na straight line.
Dialogue: 0,0:06:49.07,0:06:50.61,Default,,0000,0000,0000,,All right, there we go.
Dialogue: 0,0:06:50.61,0:06:54.05,Default,,0000,0000,0000,,That's my y-axis, and\Nthat's my x-axis.
Dialogue: 0,0:06:54.05,0:06:56.36,Default,,0000,0000,0000,,I'm just drawing the first\Nquadrant, and but you could
Dialogue: 0,0:06:56.36,0:06:59.45,Default,,0000,0000,0000,,go negative in either\Ndirection, if you like.
Dialogue: 0,0:06:59.45,0:07:01.26,Default,,0000,0000,0000,,What does this thing do?
Dialogue: 0,0:07:01.26,0:07:02.35,Default,,0000,0000,0000,,Well, it's essentially\Nsaying, look.
Dialogue: 0,0:07:02.35,0:07:06.80,Default,,0000,0000,0000,,You give me any x, any y, you\Ngive any x, y in the x-y plane,
Dialogue: 0,0:07:06.80,0:07:09.97,Default,,0000,0000,0000,,and these are going to end\Nup with some numbers, right?
Dialogue: 0,0:07:09.97,0:07:12.66,Default,,0000,0000,0000,,When you put x, y here, you're\Ngoing to get some value, when
Dialogue: 0,0:07:12.66,0:07:14.31,Default,,0000,0000,0000,,you put x, y here, you're\Ngoing to get some value.
Dialogue: 0,0:07:14.31,0:07:16.98,Default,,0000,0000,0000,,So you're going to get some\Ncombination of the i-
Dialogue: 0,0:07:16.98,0:07:18.07,Default,,0000,0000,0000,,and j-unit vectors.
Dialogue: 0,0:07:18.07,0:07:19.77,Default,,0000,0000,0000,,So you're going to\Nget some vector.
Dialogue: 0,0:07:19.77,0:07:23.02,Default,,0000,0000,0000,,So what this does, it defines a\Nvector that's associated with
Dialogue: 0,0:07:23.02,0:07:24.81,Default,,0000,0000,0000,,every point on x-y plane.
Dialogue: 0,0:07:24.81,0:07:28.78,Default,,0000,0000,0000,,So you could say, if I take\Nthis point on the x-y plane,
Dialogue: 0,0:07:28.78,0:07:32.48,Default,,0000,0000,0000,,and I would pop it into this,\NI'll get something times i plus
Dialogue: 0,0:07:32.48,0:07:34.73,Default,,0000,0000,0000,,something times j, and when you\Nadd those 2, maybe I get a
Dialogue: 0,0:07:34.73,0:07:37.13,Default,,0000,0000,0000,,vector that something\Nlike that.
Dialogue: 0,0:07:37.13,0:07:38.10,Default,,0000,0000,0000,,And you could do that\Non every point.
Dialogue: 0,0:07:38.10,0:07:39.19,Default,,0000,0000,0000,,I'm just taking random samples.
Dialogue: 0,0:07:39.19,0:07:41.42,Default,,0000,0000,0000,,Maybe when I go here,\Nthe vector looks
Dialogue: 0,0:07:41.42,0:07:42.28,Default,,0000,0000,0000,,something like that.
Dialogue: 0,0:07:42.28,0:07:44.91,Default,,0000,0000,0000,,Maybe when I go here, the\Nvictor looks like this.
Dialogue: 0,0:07:44.91,0:07:47.56,Default,,0000,0000,0000,,Maybe when I go here, the\Nvector looks like that.
Dialogue: 0,0:07:47.56,0:07:50.35,Default,,0000,0000,0000,,And maybe when I go up here,\Nthe vector goes like that.
Dialogue: 0,0:07:50.35,0:07:52.32,Default,,0000,0000,0000,,I'm just randomly\Npicking points.
Dialogue: 0,0:07:52.32,0:07:57.09,Default,,0000,0000,0000,,It defines a vector on all of\Nthe x, y coordinates where
Dialogue: 0,0:07:57.09,0:08:00.92,Default,,0000,0000,0000,,these scalar functions\Nare properly defined.
Dialogue: 0,0:08:00.92,0:08:02.37,Default,,0000,0000,0000,,And that's why it's\Ncalled a vector field.
Dialogue: 0,0:08:02.37,0:08:06.58,Default,,0000,0000,0000,,It defines what a potential,\Nmaybe, force would be,
Dialogue: 0,0:08:06.58,0:08:11.43,Default,,0000,0000,0000,,or some other type of\Nforce, at any point.
Dialogue: 0,0:08:11.43,0:08:14.35,Default,,0000,0000,0000,,At any point, if you happen\Nto have something there.
Dialogue: 0,0:08:14.35,0:08:15.90,Default,,0000,0000,0000,,Maybe that's what\Nthe function is.
Dialogue: 0,0:08:15.90,0:08:17.75,Default,,0000,0000,0000,,And I could keep doing\Nthis forever, and
Dialogue: 0,0:08:17.75,0:08:18.79,Default,,0000,0000,0000,,filling in all the gaps.
Dialogue: 0,0:08:18.79,0:08:19.66,Default,,0000,0000,0000,,But I think you get the idea.
Dialogue: 0,0:08:19.66,0:08:24.79,Default,,0000,0000,0000,,It associates a vector with\Nevery point on x-y plane.
Dialogue: 0,0:08:24.79,0:08:29.01,Default,,0000,0000,0000,,Now, this is called a vector\Nfield, so it probably makes a
Dialogue: 0,0:08:29.01,0:08:30.95,Default,,0000,0000,0000,,lot of sense that this could\Nbe used to describe
Dialogue: 0,0:08:30.95,0:08:31.87,Default,,0000,0000,0000,,any type of field.
Dialogue: 0,0:08:31.87,0:08:33.41,Default,,0000,0000,0000,,It could be a\Ngravitation field.
Dialogue: 0,0:08:33.41,0:08:36.84,Default,,0000,0000,0000,,It could be an electric field,\Nit could be a magnetic field.
Dialogue: 0,0:08:36.84,0:08:39.63,Default,,0000,0000,0000,,And this could be essentially\Ntelling you how much force
Dialogue: 0,0:08:39.63,0:08:43.19,Default,,0000,0000,0000,,there would be on some\Nparticle in that field.
Dialogue: 0,0:08:43.19,0:08:44.66,Default,,0000,0000,0000,,That's exactly what\Nthis would describe.
Dialogue: 0,0:08:44.66,0:08:48.95,Default,,0000,0000,0000,,Now, let's say that in this\Nfield, I have some particle
Dialogue: 0,0:08:48.95,0:08:51.61,Default,,0000,0000,0000,,traveling on x-y plane.
Dialogue: 0,0:08:51.61,0:08:58.62,Default,,0000,0000,0000,,Let's say it starts there, and\Nby virtue of all of these crazy
Dialogue: 0,0:08:58.62,0:09:03.85,Default,,0000,0000,0000,,forces that are acting on it,\Nand maybe it's on some tracks
Dialogue: 0,0:09:03.85,0:09:06.90,Default,,0000,0000,0000,,or something, so it won't\Nalways move exactly in the
Dialogue: 0,0:09:06.90,0:09:09.36,Default,,0000,0000,0000,,direction that the field\Nis trying to move it at.
Dialogue: 0,0:09:09.36,0:09:14.03,Default,,0000,0000,0000,,Let's say it moves in a path\Nthat moves something like this.
Dialogue: 0,0:09:14.03,0:09:17.71,Default,,0000,0000,0000,,And let's say that this path,\Nor this curve, is defined by
Dialogue: 0,0:09:17.71,0:09:22.01,Default,,0000,0000,0000,,a position vector function.
Dialogue: 0,0:09:22.01,0:09:25.15,Default,,0000,0000,0000,,So let's say that that's\Ndefined by r of t, which is
Dialogue: 0,0:09:25.15,0:09:33.78,Default,,0000,0000,0000,,just x of t times i plus y of\Nt times our unit factor j.
Dialogue: 0,0:09:33.78,0:09:35.13,Default,,0000,0000,0000,,That's r of t right there.
Dialogue: 0,0:09:35.13,0:09:37.73,Default,,0000,0000,0000,,Well, in order for this to be\Na finite path, this is true
Dialogue: 0,0:09:37.73,0:09:42.37,Default,,0000,0000,0000,,before t is greater than or\Nequal to a, and less
Dialogue: 0,0:09:42.37,0:09:45.64,Default,,0000,0000,0000,,than or equal to b.
Dialogue: 0,0:09:45.64,0:09:47.83,Default,,0000,0000,0000,,This is the path that the\Nparticle just happens to
Dialogue: 0,0:09:47.83,0:09:50.37,Default,,0000,0000,0000,,take, due to all of\Nthese wacky forces.
Dialogue: 0,0:09:50.37,0:09:54.27,Default,,0000,0000,0000,,So when the particle is right\Nhere, maybe the vector field
Dialogue: 0,0:09:54.27,0:09:56.96,Default,,0000,0000,0000,,acting on it, maybe it's\Nputting a force like that.
Dialogue: 0,0:09:56.96,0:09:59.52,Default,,0000,0000,0000,,But since the thing is on some\Ntype of tracks, it moves
Dialogue: 0,0:09:59.52,0:10:00.40,Default,,0000,0000,0000,,in this direction.
Dialogue: 0,0:10:00.40,0:10:03.83,Default,,0000,0000,0000,,And then when it's here, maybe\Nthe vector field is like that,
Dialogue: 0,0:10:03.83,0:10:05.74,Default,,0000,0000,0000,,but it moves in that direction,\Nbecause it's on some
Dialogue: 0,0:10:05.74,0:10:06.94,Default,,0000,0000,0000,,type of tracks.
Dialogue: 0,0:10:06.94,0:10:09.50,Default,,0000,0000,0000,,Now, everything I've done in\Nthis video is to build up
Dialogue: 0,0:10:09.50,0:10:11.18,Default,,0000,0000,0000,,to a fundamental question.
Dialogue: 0,0:10:11.18,0:10:13.91,Default,,0000,0000,0000,,What was the work done on\Nthe particle by the field?
Dialogue: 0,0:10:13.91,0:10:24.96,Default,,0000,0000,0000,,
Dialogue: 0,0:10:24.96,0:10:28.62,Default,,0000,0000,0000,,To answer that question, we\Ncould zoom in a little bit.
Dialogue: 0,0:10:28.62,0:10:31.10,Default,,0000,0000,0000,,I'm going to zoom in on\Nonly a little small
Dialogue: 0,0:10:31.10,0:10:34.71,Default,,0000,0000,0000,,snippet of our path.
Dialogue: 0,0:10:34.71,0:10:38.01,Default,,0000,0000,0000,,And let's try to figure out\Nwhat the work is done in a very
Dialogue: 0,0:10:38.01,0:10:40.47,Default,,0000,0000,0000,,small part of our path, because\Nit's constantly changing.
Dialogue: 0,0:10:40.47,0:10:42.19,Default,,0000,0000,0000,,The field is\Nchanging direction.
Dialogue: 0,0:10:42.19,0:10:43.63,Default,,0000,0000,0000,,my object is\Nchanging direction.
Dialogue: 0,0:10:43.63,0:10:47.78,Default,,0000,0000,0000,,So let's say when I'm here,\Nand let's say I move a
Dialogue: 0,0:10:47.78,0:10:49.74,Default,,0000,0000,0000,,small amount of my path.
Dialogue: 0,0:10:49.74,0:10:55.86,Default,,0000,0000,0000,,So let's say I move, this\Nis an infinitesimally
Dialogue: 0,0:10:55.86,0:10:58.50,Default,,0000,0000,0000,,small dr. Right?
Dialogue: 0,0:10:58.50,0:11:00.81,Default,,0000,0000,0000,,I have a differential, it's a\Ndifferential vector, infinitely
Dialogue: 0,0:11:00.81,0:11:02.63,Default,,0000,0000,0000,,small displacement.
Dialogue: 0,0:11:02.63,0:11:06.80,Default,,0000,0000,0000,,and let's say over the course\Nof that, the vector field is
Dialogue: 0,0:11:06.80,0:11:08.84,Default,,0000,0000,0000,,acting in this local\Narea, let's say it looks
Dialogue: 0,0:11:08.84,0:11:10.48,Default,,0000,0000,0000,,something like that.
Dialogue: 0,0:11:10.48,0:11:13.49,Default,,0000,0000,0000,,It's providing a force that\Nlooks something like that.
Dialogue: 0,0:11:13.49,0:11:16.64,Default,,0000,0000,0000,,So that's the vector field in\Nthat area, or the force
Dialogue: 0,0:11:16.64,0:11:18.75,Default,,0000,0000,0000,,directed on that particle right\Nwhen it's at that point.
Dialogue: 0,0:11:18.75,0:11:18.87,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:11:18.87,0:11:22.42,Default,,0000,0000,0000,,It's an infinitesimally small\Namount of time in space.
Dialogue: 0,0:11:22.42,0:11:24.44,Default,,0000,0000,0000,,You could say, OK, over that\Nlittle small point, we
Dialogue: 0,0:11:24.44,0:11:26.60,Default,,0000,0000,0000,,have this constant force.
Dialogue: 0,0:11:26.60,0:11:29.79,Default,,0000,0000,0000,,What was the work done\Nover this small period?
Dialogue: 0,0:11:29.79,0:11:32.33,Default,,0000,0000,0000,,You could say, what's the\Nsmall interval of work?
Dialogue: 0,0:11:32.33,0:11:36.12,Default,,0000,0000,0000,,You could say d work, or\Na differential of work.
Dialogue: 0,0:11:36.12,0:11:38.94,Default,,0000,0000,0000,,Well, by the same exact logic\Nthat we did with the simple
Dialogue: 0,0:11:38.94,0:11:43.81,Default,,0000,0000,0000,,problem, it's the magnitude of\Nthe force in the direction of
Dialogue: 0,0:11:43.81,0:11:48.55,Default,,0000,0000,0000,,our displacement times the\Nmagnitude of our displacement.
Dialogue: 0,0:11:48.55,0:11:52.80,Default,,0000,0000,0000,,And we know what that is, just\Nfrom this example up here.
Dialogue: 0,0:11:52.80,0:11:54.81,Default,,0000,0000,0000,,That's the dot product.
Dialogue: 0,0:11:54.81,0:11:58.34,Default,,0000,0000,0000,,It's the dot product of the\Nforce and our super-small
Dialogue: 0,0:11:58.34,0:11:59.48,Default,,0000,0000,0000,,displacement.
Dialogue: 0,0:11:59.48,0:12:07.86,Default,,0000,0000,0000,,So that's equal to the dot\Nproduct of our force and our
Dialogue: 0,0:12:07.86,0:12:09.87,Default,,0000,0000,0000,,super-small displacement.
Dialogue: 0,0:12:09.87,0:12:13.24,Default,,0000,0000,0000,,Now, just by doing this, we're\Njust figuring out the work
Dialogue: 0,0:12:13.24,0:12:16.44,Default,,0000,0000,0000,,over, maybe like a really\Nsmall, super-small dr. But
Dialogue: 0,0:12:16.44,0:12:18.82,Default,,0000,0000,0000,,what we want to do, is we\Nwant to sum them all up.
Dialogue: 0,0:12:18.82,0:12:21.87,Default,,0000,0000,0000,,We want to sum up all of the\Ndrs to figure out the total,
Dialogue: 0,0:12:21.87,0:12:25.09,Default,,0000,0000,0000,,all of the f dot drs to figure\Nout the total work done.
Dialogue: 0,0:12:25.09,0:12:27.51,Default,,0000,0000,0000,,And that's where the\Nintegral comes in.
Dialogue: 0,0:12:27.51,0:12:32.57,Default,,0000,0000,0000,,We will do a line integral\Nover-- I mean, you could
Dialogue: 0,0:12:32.57,0:12:33.91,Default,,0000,0000,0000,,think of it two ways.
Dialogue: 0,0:12:33.91,0:12:37.44,Default,,0000,0000,0000,,You could write just d dot w\Nthere, but we could say, we'll
Dialogue: 0,0:12:37.44,0:12:42.70,Default,,0000,0000,0000,,do a line integral along this\Ncurve c, could call that c
Dialogue: 0,0:12:42.70,0:12:46.41,Default,,0000,0000,0000,,or along r, whatever you\Nwant to say it, of dw.
Dialogue: 0,0:12:46.41,0:12:47.80,Default,,0000,0000,0000,,That'll give us the total work.
Dialogue: 0,0:12:47.80,0:12:49.50,Default,,0000,0000,0000,,So let's say, work\Nis equal to that.
Dialogue: 0,0:12:49.50,0:12:54.04,Default,,0000,0000,0000,,Or we could also write it over\Nthe integral, over the same
Dialogue: 0,0:12:54.04,0:13:00.50,Default,,0000,0000,0000,,curve of f of f dot dr.
Dialogue: 0,0:13:00.50,0:13:03.58,Default,,0000,0000,0000,,And this might seem like a\Nreally, you know, gee, this
Dialogue: 0,0:13:03.58,0:13:05.12,Default,,0000,0000,0000,,is really abstract, Sal.
Dialogue: 0,0:13:05.12,0:13:09.22,Default,,0000,0000,0000,,How do we actually calculate\Nsomething like this?
Dialogue: 0,0:13:09.22,0:13:13.13,Default,,0000,0000,0000,,Especially because we have\Neverything parameterized
Dialogue: 0,0:13:13.13,0:13:14.03,Default,,0000,0000,0000,,in terms of t.
Dialogue: 0,0:13:14.03,0:13:16.13,Default,,0000,0000,0000,,How do we get this\Nin terms of t?
Dialogue: 0,0:13:16.13,0:13:19.71,Default,,0000,0000,0000,,And if you just think about\Nit, what is f dot r?
Dialogue: 0,0:13:19.71,0:13:21.03,Default,,0000,0000,0000,,Or what is f dot dr?
Dialogue: 0,0:13:21.03,0:13:23.30,Default,,0000,0000,0000,,Well, actually, to answer\Nthat, let's remember
Dialogue: 0,0:13:23.30,0:13:25.83,Default,,0000,0000,0000,,what dr looked like.
Dialogue: 0,0:13:25.83,0:13:36.20,Default,,0000,0000,0000,,If you remember, dr/dt is equal\Nto x prime of t, I'm writing it
Dialogue: 0,0:13:36.20,0:13:39.12,Default,,0000,0000,0000,,like, I could have written dx\Ndt if I wanted to do, times the
Dialogue: 0,0:13:39.12,0:13:45.18,Default,,0000,0000,0000,,i-unit vector, plus y prime of\Nt, times the j-unit vector.
Dialogue: 0,0:13:45.18,0:13:49.32,Default,,0000,0000,0000,,And if we just wanted to dr, we\Ncould multiply both sides, if
Dialogue: 0,0:13:49.32,0:13:51.85,Default,,0000,0000,0000,,we're being a little bit more\Nhand-wavy with the
Dialogue: 0,0:13:51.85,0:13:53.47,Default,,0000,0000,0000,,differentials, not\Ntoo rigorous.
Dialogue: 0,0:13:53.47,0:13:58.48,Default,,0000,0000,0000,,We'll get dr is equal to x\Nprime of t dt times the unit
Dialogue: 0,0:13:58.48,0:14:05.07,Default,,0000,0000,0000,,vector i plus y prime of t\Ntimes the differential dt
Dialogue: 0,0:14:05.07,0:14:07.28,Default,,0000,0000,0000,,times the unit vector j.
Dialogue: 0,0:14:07.28,0:14:09.07,Default,,0000,0000,0000,,So this is our dr right here.
Dialogue: 0,0:14:09.07,0:14:12.11,Default,,0000,0000,0000,,
Dialogue: 0,0:14:12.11,0:14:16.28,Default,,0000,0000,0000,,And remember what our\Nvector field was.
Dialogue: 0,0:14:16.28,0:14:17.44,Default,,0000,0000,0000,,It was this thing up here.
Dialogue: 0,0:14:17.44,0:14:19.59,Default,,0000,0000,0000,,Let me copy and paste it.
Dialogue: 0,0:14:19.59,0:14:21.03,Default,,0000,0000,0000,,And we'll see that\Nthe dot product is
Dialogue: 0,0:14:21.03,0:14:23.36,Default,,0000,0000,0000,,actually not so crazy.
Dialogue: 0,0:14:23.36,0:14:26.71,Default,,0000,0000,0000,,So copy, and let me\Npaste it down here.
Dialogue: 0,0:14:26.71,0:14:31.13,Default,,0000,0000,0000,,
Dialogue: 0,0:14:31.13,0:14:33.82,Default,,0000,0000,0000,,So what's this integral\Ngoing to look like?
Dialogue: 0,0:14:33.82,0:14:37.60,Default,,0000,0000,0000,,This integral right here, that\Ngives the total work done by
Dialogue: 0,0:14:37.60,0:14:40.79,Default,,0000,0000,0000,,the field, on the particle,\Nas it moves along that path.
Dialogue: 0,0:14:40.79,0:14:44.09,Default,,0000,0000,0000,,Just super fundamental to\Npretty much any serious physics
Dialogue: 0,0:14:44.09,0:14:47.17,Default,,0000,0000,0000,,that you might eventually\Nfind yourself doing.
Dialogue: 0,0:14:47.17,0:14:48.17,Default,,0000,0000,0000,,So you could say, well gee.
Dialogue: 0,0:14:48.17,0:14:52.42,Default,,0000,0000,0000,,It's going to be the integral,\Nlet's just say from t is equal
Dialogue: 0,0:14:52.42,0:14:55.32,Default,,0000,0000,0000,,to a, to t is equal to b.
Dialogue: 0,0:14:55.32,0:14:58.31,Default,,0000,0000,0000,,Right? a is where we started\Noff on the path, t is equal
Dialogue: 0,0:14:58.31,0:14:59.79,Default,,0000,0000,0000,,to a to t is equal to b.
Dialogue: 0,0:14:59.79,0:15:01.76,Default,,0000,0000,0000,,You can imagine it as being\Ntimed, as a particle
Dialogue: 0,0:15:01.76,0:15:03.61,Default,,0000,0000,0000,,moving, as time increases.
Dialogue: 0,0:15:03.61,0:15:07.00,Default,,0000,0000,0000,,And then what is f dot dr?
Dialogue: 0,0:15:07.00,0:15:10.64,Default,,0000,0000,0000,,Well, if you remember from just\Nwhat the dot product is, you
Dialogue: 0,0:15:10.64,0:15:15.31,Default,,0000,0000,0000,,can essentially just take the\Nproduct of the corresponding
Dialogue: 0,0:15:15.31,0:15:17.74,Default,,0000,0000,0000,,components of your of\Nvector, and add them up.
Dialogue: 0,0:15:17.74,0:15:20.07,Default,,0000,0000,0000,,So this is going to be the\Nintegral from t equals a to t
Dialogue: 0,0:15:20.07,0:15:27.25,Default,,0000,0000,0000,,equals b, of p of p of x,\Nreally, instead of writing x,
Dialogue: 0,0:15:27.25,0:15:30.74,Default,,0000,0000,0000,,y, it's x of t, right? x as a\Nfunction of t, y as
Dialogue: 0,0:15:30.74,0:15:32.35,Default,,0000,0000,0000,,a function of t.
Dialogue: 0,0:15:32.35,0:15:33.69,Default,,0000,0000,0000,,So that's that.
Dialogue: 0,0:15:33.69,0:15:37.60,Default,,0000,0000,0000,,Times this thing right here,\Ntimes this component, right?
Dialogue: 0,0:15:37.60,0:15:39.30,Default,,0000,0000,0000,,We're multiplying\Nthe i-components.
Dialogue: 0,0:15:39.30,0:15:50.65,Default,,0000,0000,0000,,So times x prime of t d t, and\Nthen that plus, we're going
Dialogue: 0,0:15:50.65,0:15:52.37,Default,,0000,0000,0000,,to do the same thing\Nwith the q function.
Dialogue: 0,0:15:52.37,0:15:56.06,Default,,0000,0000,0000,,So this is q plus, I'll\Ngo to another line.
Dialogue: 0,0:15:56.06,0:15:57.76,Default,,0000,0000,0000,,Hopefully you realize I could\Nhave just kept writing, but
Dialogue: 0,0:15:57.76,0:15:59.02,Default,,0000,0000,0000,,I'm running out of space.
Dialogue: 0,0:15:59.02,0:16:09.96,Default,,0000,0000,0000,,Plus q of x of t, y of t, times\Nthe component of our dr. Times
Dialogue: 0,0:16:09.96,0:16:11.90,Default,,0000,0000,0000,,the y-component, or\Nthe j-component.
Dialogue: 0,0:16:11.90,0:16:15.53,Default,,0000,0000,0000,,y prime of t dt.
Dialogue: 0,0:16:15.53,0:16:16.62,Default,,0000,0000,0000,,And we're done!
Dialogue: 0,0:16:16.62,0:16:17.48,Default,,0000,0000,0000,,And we're done.
Dialogue: 0,0:16:17.48,0:16:19.30,Default,,0000,0000,0000,,This might still seem a little\Nbit abstract, but we're going
Dialogue: 0,0:16:19.30,0:16:23.02,Default,,0000,0000,0000,,to see in the next video,\Neverything is now in terms of
Dialogue: 0,0:16:23.02,0:16:25.48,Default,,0000,0000,0000,,t, so this is just a\Nstraight-up integration,
Dialogue: 0,0:16:25.48,0:16:27.17,Default,,0000,0000,0000,,with respect to dt.
Dialogue: 0,0:16:27.17,0:16:30.15,Default,,0000,0000,0000,,If we want, we could take the\Ndt's outside of the equation,
Dialogue: 0,0:16:30.15,0:16:32.27,Default,,0000,0000,0000,,and it'll look a little\Nbit more normal for you.
Dialogue: 0,0:16:32.27,0:16:34.64,Default,,0000,0000,0000,,But this is essentially\Nall that we have to do.
Dialogue: 0,0:16:34.64,0:16:38.08,Default,,0000,0000,0000,,And we're going to see some\Nconcrete examples of taking a
Dialogue: 0,0:16:38.08,0:16:43.23,Default,,0000,0000,0000,,line integral through a vector\Nfield, or using vector
Dialogue: 0,0:16:43.23,0:16:45.79,Default,,0000,0000,0000,,functions, in the next video.
Dialogue: 0,0:16:45.79,0:16:46.00,Default,,0000,0000,0000,,