0:00:00.000,0:00:00.330


0:00:00.330,0:00:03.110
One of the most fundamental[br]ideas in all of physics

0:00:03.110,0:00:05.385
is the idea of work.

0:00:05.385,0:00:08.450
Now when you first learn work,[br]you just say, oh, that's

0:00:08.450,0:00:10.120
just force times distance.

0:00:10.120,0:00:12.200
But then later on, when you[br]learn a little bit about

0:00:12.200,0:00:14.770
vectors, you realize that the[br]force isn't always going in

0:00:14.770,0:00:17.610
the same direction as[br]your displacement.

0:00:17.610,0:00:21.450
So you learn that work is[br]really the magnitude, let me

0:00:21.450,0:00:33.070
write this down, the magnitude[br]of the force, in the direction,

0:00:33.070,0:00:39.460
or the component of the force[br]in the direction

0:00:39.460,0:00:41.740
of displacement.

0:00:41.740,0:00:44.206
Displacement is just distance[br]with some direction.

0:00:44.206,0:00:49.970


0:00:49.970,0:00:55.290
Times the magnitude of the[br]displacement, or you could say,

0:00:55.290,0:00:56.695
times the distance displaced.

0:00:56.695,0:01:00.810


0:01:00.810,0:01:02.330
And the classic example.

0:01:02.330,0:01:06.250
Maybe you have an ice cube,[br]or some type of block.

0:01:06.250,0:01:08.740
I just ice so that there's[br]not a lot of friction.

0:01:08.740,0:01:12.510
Maybe it's standing on a bigger[br]lake or ice or something.

0:01:12.510,0:01:15.030
And maybe you're pulling on[br]that ice cube at an angle.

0:01:15.030,0:01:17.610
Let's say, you're pulling[br]at an angle like that.

0:01:17.610,0:01:20.820
That is my force, right there.

0:01:20.820,0:01:24.080
Let's say my force is[br]equal to-- well, that's

0:01:24.080,0:01:25.160
my force vector.

0:01:25.160,0:01:33.870
Let's say the magnitude of[br]my force vector, let's

0:01:33.870,0:01:35.310
say it's 10 newtons.

0:01:35.310,0:01:37.650
And let's say the direction of[br]my force vector, right, any

0:01:37.650,0:01:41.080
vector has to have a magnitude[br]and a direction, and the

0:01:41.080,0:01:44.920
direction, let's say it has a[br]30 degree angle, let's say a 60

0:01:44.920,0:01:47.770
degree angle, above horizontal.

0:01:47.770,0:01:49.560
So that's the direction[br]I'm pulling in.

0:01:49.560,0:01:52.600
And let's say I displace it.

0:01:52.600,0:01:55.930
This is all review, hopefully.

0:01:55.930,0:01:59.225
If you're displacing it, let's[br]say you displace it 5 newtons.

0:01:59.225,0:02:02.570
So let's say the displacement,[br]that's the displacement vector

0:02:02.570,0:02:10.290
right there, and the magnitude[br]of it is equal to 5 meters.

0:02:10.290,0:02:13.460
So you've learned from the[br]definition of work, you can't

0:02:13.460,0:02:16.940
just say, oh, I'm pulling with[br]10 newtons of force and

0:02:16.940,0:02:18.360
I'm moving it 5 meters.

0:02:18.360,0:02:22.560
You can't just multiply the 10[br]newtons times the 5 meters.

0:02:22.560,0:02:25.660
You have to find the magnitude[br]of the component going in the

0:02:25.660,0:02:29.050
same direction as[br]my displacement.

0:02:29.050,0:02:31.860
So what I essentially need to[br]do is, the length, if you

0:02:31.860,0:02:34.930
imagine the length of this[br]vector being 10, that's the

0:02:34.930,0:02:37.750
total force, but you need to[br]figure out the length of the

0:02:37.750,0:02:40.770
vector, that's the component of[br]the force, going in the same

0:02:40.770,0:02:43.460
direction as my displacement.

0:02:43.460,0:02:45.570
And a little simple[br]trigonometry, you know that

0:02:45.570,0:02:53.120
this is 10 times the cosine of[br]60 degrees, or that's equal to,

0:02:53.120,0:02:58.010
cosine of 60 degrees is 1/2, so[br]that's just equal to 5.

0:02:58.010,0:03:00.380
So this magnitude, the[br]magnitude of the force going

0:03:00.380,0:03:02.410
in the same direction of[br]the displacement in this

0:03:02.410,0:03:04.810
case, is 5 newtons.

0:03:04.810,0:03:07.500


0:03:07.500,0:03:09.850
And then you can[br]figure out the work.

0:03:09.850,0:03:19.560
You could say that the work is[br]equal to 5 newtons times, I'll

0:03:19.560,0:03:20.630
just write a dot for times.

0:03:20.630,0:03:22.290
I don't want you to think[br]it's cross product.

0:03:22.290,0:03:26.680
Times 5 meters, which is 25[br]newton meters, or you could

0:03:26.680,0:03:31.250
even say 25 Joules of[br]work have been done.

0:03:31.250,0:03:35.280
And this is all review of[br]somewhat basic physics.

0:03:35.280,0:03:36.720
But just think about[br]what happened, here.

0:03:36.720,0:03:37.430
What was the work?

0:03:37.430,0:03:39.190
If I write in the abstract.

0:03:39.190,0:03:42.550
The work is equal[br]to the 5 newtons.

0:03:42.550,0:03:46.700
That was the magnitude of my[br]force vector, so it's the

0:03:46.700,0:03:52.630
magnitude of my force vector,[br]times the cosine of this angle.

0:03:52.630,0:03:53.860
So you know, let's[br]call that theta.

0:03:53.860,0:03:55.010
Let's say it a[br]little generally.

0:03:55.010,0:03:58.150
So times the cosine[br]of the angle.

0:03:58.150,0:04:01.740
This is the amount of my force[br]in the direction of the

0:04:01.740,0:04:04.960
displacement, the cosine of the[br]angle between them, times the

0:04:04.960,0:04:06.800
magnitude of the displacement.

0:04:06.800,0:04:12.260
So times the magnitude[br]of the displacement.

0:04:12.260,0:04:15.560
Or if I wanted to rewrite that,[br]I could just write that as, the

0:04:15.560,0:04:18.940
magnitude of the displacement[br]times the magnitude of

0:04:18.940,0:04:23.400
the force times the[br]cosine of theta.

0:04:23.400,0:04:26.760
And I've done multiple videos[br]of this, in the linear algebra

0:04:26.760,0:04:28.880
playlist, in the physics[br]playlist, where I talk about

0:04:28.880,0:04:31.580
the dot product and the cross[br]product and all of that, but

0:04:31.580,0:04:40.470
this is the dot product[br]of the vectors d and f.

0:04:40.470,0:04:43.700
So in general, if you're trying[br]to find the work for a constant

0:04:43.700,0:04:46.730
displacement, and you have a[br]constant force, you just take

0:04:46.730,0:04:48.530
the dot product of[br]those two vectors.

0:04:48.530,0:04:51.330
And if the dot product is a[br]completely foreign concept to

0:04:51.330,0:04:53.770
you, might want to watch, I[br]think I've made multiple, 4

0:04:53.770,0:04:56.380
or 5 videos on the dot[br]product, and its intuition,

0:04:56.380,0:04:57.420
and how it compares.

0:04:57.420,0:04:59.280
But just to give you a little[br]bit of that intuition right

0:04:59.280,0:05:03.920
here, the dot product, when[br]I take f dot d, or d dot f,

0:05:03.920,0:05:08.440
what it's giving me is, I'm[br]multiplying the magnitude, well

0:05:08.440,0:05:10.130
I could just read this out.

0:05:10.130,0:05:13.590
But the idea of the dot product[br]is, take how much of this

0:05:13.590,0:05:16.800
vector is going in the same[br]direction as this vector,

0:05:16.800,0:05:18.500
in this case, this much.

0:05:18.500,0:05:21.110
And then multiply[br]the two magnitudes.

0:05:21.110,0:05:22.410
And that's what we[br]did right here.

0:05:22.410,0:05:26.230
So the work is going to be the[br]force vector, dot, taking the

0:05:26.230,0:05:28.980
dot part of the force vector[br]with the displacement vector,

0:05:28.980,0:05:30.840
and this, of course,[br]is a scalar value.

0:05:30.840,0:05:33.040
And we'll work out some[br]examples in the future where

0:05:33.040,0:05:34.360
you'll see that that's true.

0:05:34.360,0:05:39.000
So this is all review of[br]fairly elementary physics.

0:05:39.000,0:05:42.500
Now let's take a more[br]complex example, but it's

0:05:42.500,0:05:43.670
really the same idea.

0:05:43.670,0:05:45.873
Let's define a vector field.

0:05:45.873,0:05:48.660


0:05:48.660,0:05:51.371
So let's say that I have a[br]vector field f, and we're

0:05:51.371,0:05:54.050
going to think about what[br]this means in a second.

0:05:54.050,0:05:58.890
It's a function of x and y, and[br]it's equal to some scalar

0:05:58.890,0:06:04.490
function of x and y times the[br]i-unit vector, or the

0:06:04.490,0:06:08.760
horizontal unit vector, plus[br]some other function, scalar

0:06:08.760,0:06:14.250
function of x and y, times the[br]vertical unit vector.

0:06:14.250,0:06:15.580
So what would something[br]like this be?

0:06:15.580,0:06:17.460
This is a vector field.

0:06:17.460,0:06:20.210
This is a vector field[br]in 2-dimensional space.

0:06:20.210,0:06:21.330
We're on the x-y plane.

0:06:21.330,0:06:31.190


0:06:31.190,0:06:35.840
Or you could even say, on R2.

0:06:35.840,0:06:37.690
Either way, I don't want[br]to get too much into

0:06:37.690,0:06:39.230
the mathiness of it.

0:06:39.230,0:06:40.590
But what does this do?

0:06:40.590,0:06:47.270
Well, if I were to draw my x-y[br]plane, so that is my, again,

0:06:47.270,0:06:49.070
having trouble drawing[br]a straight line.

0:06:49.070,0:06:50.610
All right, there we go.

0:06:50.610,0:06:54.050
That's my y-axis, and[br]that's my x-axis.

0:06:54.050,0:06:56.360
I'm just drawing the first[br]quadrant, and but you could

0:06:56.360,0:06:59.450
go negative in either[br]direction, if you like.

0:06:59.450,0:07:01.260
What does this thing do?

0:07:01.260,0:07:02.350
Well, it's essentially[br]saying, look.

0:07:02.350,0:07:06.800
You give me any x, any y, you[br]give any x, y in the x-y plane,

0:07:06.800,0:07:09.970
and these are going to end[br]up with some numbers, right?

0:07:09.970,0:07:12.655
When you put x, y here, you're[br]going to get some value, when

0:07:12.655,0:07:14.310
you put x, y here, you're[br]going to get some value.

0:07:14.310,0:07:16.980
So you're going to get some[br]combination of the i-

0:07:16.980,0:07:18.070
and j-unit vectors.

0:07:18.070,0:07:19.770
So you're going to[br]get some vector.

0:07:19.770,0:07:23.020
So what this does, it defines a[br]vector that's associated with

0:07:23.020,0:07:24.810
every point on x-y plane.

0:07:24.810,0:07:28.780
So you could say, if I take[br]this point on the x-y plane,

0:07:28.780,0:07:32.480
and I would pop it into this,[br]I'll get something times i plus

0:07:32.480,0:07:34.730
something times j, and when you[br]add those 2, maybe I get a

0:07:34.730,0:07:37.130
vector that something[br]like that.

0:07:37.130,0:07:38.100
And you could do that[br]on every point.

0:07:38.100,0:07:39.190
I'm just taking random samples.

0:07:39.190,0:07:41.420
Maybe when I go here,[br]the vector looks

0:07:41.420,0:07:42.280
something like that.

0:07:42.280,0:07:44.910
Maybe when I go here, the[br]victor looks like this.

0:07:44.910,0:07:47.560
Maybe when I go here, the[br]vector looks like that.

0:07:47.560,0:07:50.350
And maybe when I go up here,[br]the vector goes like that.

0:07:50.350,0:07:52.320
I'm just randomly[br]picking points.

0:07:52.320,0:07:57.090
It defines a vector on all of[br]the x, y coordinates where

0:07:57.090,0:08:00.920
these scalar functions[br]are properly defined.

0:08:00.920,0:08:02.370
And that's why it's[br]called a vector field.

0:08:02.370,0:08:06.580
It defines what a potential,[br]maybe, force would be,

0:08:06.580,0:08:11.430
or some other type of[br]force, at any point.

0:08:11.430,0:08:14.350
At any point, if you happen[br]to have something there.

0:08:14.350,0:08:15.900
Maybe that's what[br]the function is.

0:08:15.900,0:08:17.750
And I could keep doing[br]this forever, and

0:08:17.750,0:08:18.790
filling in all the gaps.

0:08:18.790,0:08:19.660
But I think you get the idea.

0:08:19.660,0:08:24.790
It associates a vector with[br]every point on x-y plane.

0:08:24.790,0:08:29.010
Now, this is called a vector[br]field, so it probably makes a

0:08:29.010,0:08:30.950
lot of sense that this could[br]be used to describe

0:08:30.950,0:08:31.870
any type of field.

0:08:31.870,0:08:33.410
It could be a[br]gravitation field.

0:08:33.410,0:08:36.840
It could be an electric field,[br]it could be a magnetic field.

0:08:36.840,0:08:39.630
And this could be essentially[br]telling you how much force

0:08:39.630,0:08:43.190
there would be on some[br]particle in that field.

0:08:43.190,0:08:44.660
That's exactly what[br]this would describe.

0:08:44.660,0:08:48.950
Now, let's say that in this[br]field, I have some particle

0:08:48.950,0:08:51.610
traveling on x-y plane.

0:08:51.610,0:08:58.620
Let's say it starts there, and[br]by virtue of all of these crazy

0:08:58.620,0:09:03.850
forces that are acting on it,[br]and maybe it's on some tracks

0:09:03.850,0:09:06.900
or something, so it won't[br]always move exactly in the

0:09:06.900,0:09:09.360
direction that the field[br]is trying to move it at.

0:09:09.360,0:09:14.030
Let's say it moves in a path[br]that moves something like this.

0:09:14.030,0:09:17.710
And let's say that this path,[br]or this curve, is defined by

0:09:17.710,0:09:22.010
a position vector function.

0:09:22.010,0:09:25.150
So let's say that that's[br]defined by r of t, which is

0:09:25.150,0:09:33.780
just x of t times i plus y of[br]t times our unit factor j.

0:09:33.780,0:09:35.130
That's r of t right there.

0:09:35.130,0:09:37.730
Well, in order for this to be[br]a finite path, this is true

0:09:37.730,0:09:42.370
before t is greater than or[br]equal to a, and less

0:09:42.370,0:09:45.640
than or equal to b.

0:09:45.640,0:09:47.830
This is the path that the[br]particle just happens to

0:09:47.830,0:09:50.370
take, due to all of[br]these wacky forces.

0:09:50.370,0:09:54.270
So when the particle is right[br]here, maybe the vector field

0:09:54.270,0:09:56.960
acting on it, maybe it's[br]putting a force like that.

0:09:56.960,0:09:59.520
But since the thing is on some[br]type of tracks, it moves

0:09:59.520,0:10:00.400
in this direction.

0:10:00.400,0:10:03.830
And then when it's here, maybe[br]the vector field is like that,

0:10:03.830,0:10:05.740
but it moves in that direction,[br]because it's on some

0:10:05.740,0:10:06.940
type of tracks.

0:10:06.940,0:10:09.500
Now, everything I've done in[br]this video is to build up

0:10:09.500,0:10:11.180
to a fundamental question.

0:10:11.180,0:10:13.910
What was the work done on[br]the particle by the field?

0:10:13.910,0:10:24.960


0:10:24.960,0:10:28.620
To answer that question, we[br]could zoom in a little bit.

0:10:28.620,0:10:31.100
I'm going to zoom in on[br]only a little small

0:10:31.100,0:10:34.710
snippet of our path.

0:10:34.710,0:10:38.010
And let's try to figure out[br]what the work is done in a very

0:10:38.010,0:10:40.470
small part of our path, because[br]it's constantly changing.

0:10:40.470,0:10:42.190
The field is[br]changing direction.

0:10:42.190,0:10:43.630
my object is[br]changing direction.

0:10:43.630,0:10:47.780
So let's say when I'm here,[br]and let's say I move a

0:10:47.780,0:10:49.740
small amount of my path.

0:10:49.740,0:10:55.860
So let's say I move, this[br]is an infinitesimally

0:10:55.860,0:10:58.500
small dr. Right?

0:10:58.500,0:11:00.810
I have a differential, it's a[br]differential vector, infinitely

0:11:00.810,0:11:02.630
small displacement.

0:11:02.630,0:11:06.800
and let's say over the course[br]of that, the vector field is

0:11:06.800,0:11:08.840
acting in this local[br]area, let's say it looks

0:11:08.840,0:11:10.480
something like that.

0:11:10.480,0:11:13.490
It's providing a force that[br]looks something like that.

0:11:13.490,0:11:16.640
So that's the vector field in[br]that area, or the force

0:11:16.640,0:11:18.750
directed on that particle right[br]when it's at that point.

0:11:18.750,0:11:18.870
Right?

0:11:18.870,0:11:22.420
It's an infinitesimally small[br]amount of time in space.

0:11:22.420,0:11:24.440
You could say, OK, over that[br]little small point, we

0:11:24.440,0:11:26.600
have this constant force.

0:11:26.600,0:11:29.790
What was the work done[br]over this small period?

0:11:29.790,0:11:32.330
You could say, what's the[br]small interval of work?

0:11:32.330,0:11:36.120
You could say d work, or[br]a differential of work.

0:11:36.120,0:11:38.940
Well, by the same exact logic[br]that we did with the simple

0:11:38.940,0:11:43.810
problem, it's the magnitude of[br]the force in the direction of

0:11:43.810,0:11:48.550
our displacement times the[br]magnitude of our displacement.

0:11:48.550,0:11:52.800
And we know what that is, just[br]from this example up here.

0:11:52.800,0:11:54.810
That's the dot product.

0:11:54.810,0:11:58.340
It's the dot product of the[br]force and our super-small

0:11:58.340,0:11:59.480
displacement.

0:11:59.480,0:12:07.860
So that's equal to the dot[br]product of our force and our

0:12:07.860,0:12:09.870
super-small displacement.

0:12:09.870,0:12:13.240
Now, just by doing this, we're[br]just figuring out the work

0:12:13.240,0:12:16.440
over, maybe like a really[br]small, super-small dr. But

0:12:16.440,0:12:18.820
what we want to do, is we[br]want to sum them all up.

0:12:18.820,0:12:21.870
We want to sum up all of the[br]drs to figure out the total,

0:12:21.870,0:12:25.090
all of the f dot drs to figure[br]out the total work done.

0:12:25.090,0:12:27.510
And that's where the[br]integral comes in.

0:12:27.510,0:12:32.570
We will do a line integral[br]over-- I mean, you could

0:12:32.570,0:12:33.910
think of it two ways.

0:12:33.910,0:12:37.440
You could write just d dot w[br]there, but we could say, we'll

0:12:37.440,0:12:42.700
do a line integral along this[br]curve c, could call that c

0:12:42.700,0:12:46.410
or along r, whatever you[br]want to say it, of dw.

0:12:46.410,0:12:47.800
That'll give us the total work.

0:12:47.800,0:12:49.500
So let's say, work[br]is equal to that.

0:12:49.500,0:12:54.040
Or we could also write it over[br]the integral, over the same

0:12:54.040,0:13:00.500
curve of f of f dot dr.

0:13:00.500,0:13:03.580
And this might seem like a[br]really, you know, gee, this

0:13:03.580,0:13:05.120
is really abstract, Sal.

0:13:05.120,0:13:09.220
How do we actually calculate[br]something like this?

0:13:09.220,0:13:13.130
Especially because we have[br]everything parameterized

0:13:13.130,0:13:14.030
in terms of t.

0:13:14.030,0:13:16.130
How do we get this[br]in terms of t?

0:13:16.130,0:13:19.710
And if you just think about[br]it, what is f dot r?

0:13:19.710,0:13:21.030
Or what is f dot dr?

0:13:21.030,0:13:23.300
Well, actually, to answer[br]that, let's remember

0:13:23.300,0:13:25.830
what dr looked like.

0:13:25.830,0:13:36.200
If you remember, dr/dt is equal[br]to x prime of t, I'm writing it

0:13:36.200,0:13:39.120
like, I could have written dx[br]dt if I wanted to do, times the

0:13:39.120,0:13:45.180
i-unit vector, plus y prime of[br]t, times the j-unit vector.

0:13:45.180,0:13:49.320
And if we just wanted to dr, we[br]could multiply both sides, if

0:13:49.320,0:13:51.850
we're being a little bit more[br]hand-wavy with the

0:13:51.850,0:13:53.470
differentials, not[br]too rigorous.

0:13:53.470,0:13:58.480
We'll get dr is equal to x[br]prime of t dt times the unit

0:13:58.480,0:14:05.070
vector i plus y prime of t[br]times the differential dt

0:14:05.070,0:14:07.280
times the unit vector j.

0:14:07.280,0:14:09.070
So this is our dr right here.

0:14:09.070,0:14:12.110


0:14:12.110,0:14:16.280
And remember what our[br]vector field was.

0:14:16.280,0:14:17.440
It was this thing up here.

0:14:17.440,0:14:19.590
Let me copy and paste it.

0:14:19.590,0:14:21.030
And we'll see that[br]the dot product is

0:14:21.030,0:14:23.360
actually not so crazy.

0:14:23.360,0:14:26.710
So copy, and let me[br]paste it down here.

0:14:26.710,0:14:31.130


0:14:31.130,0:14:33.820
So what's this integral[br]going to look like?

0:14:33.820,0:14:37.600
This integral right here, that[br]gives the total work done by

0:14:37.600,0:14:40.790
the field, on the particle,[br]as it moves along that path.

0:14:40.790,0:14:44.090
Just super fundamental to[br]pretty much any serious physics

0:14:44.090,0:14:47.170
that you might eventually[br]find yourself doing.

0:14:47.170,0:14:48.170
So you could say, well gee.

0:14:48.170,0:14:52.420
It's going to be the integral,[br]let's just say from t is equal

0:14:52.420,0:14:55.320
to a, to t is equal to b.

0:14:55.320,0:14:58.310
Right? a is where we started[br]off on the path, t is equal

0:14:58.310,0:14:59.790
to a to t is equal to b.

0:14:59.790,0:15:01.760
You can imagine it as being[br]timed, as a particle

0:15:01.760,0:15:03.610
moving, as time increases.

0:15:03.610,0:15:07.000
And then what is f dot dr?

0:15:07.000,0:15:10.640
Well, if you remember from just[br]what the dot product is, you

0:15:10.640,0:15:15.310
can essentially just take the[br]product of the corresponding

0:15:15.310,0:15:17.740
components of your of[br]vector, and add them up.

0:15:17.740,0:15:20.070
So this is going to be the[br]integral from t equals a to t

0:15:20.070,0:15:27.246
equals b, of p of p of x,[br]really, instead of writing x,

0:15:27.246,0:15:30.740
y, it's x of t, right? x as a[br]function of t, y as

0:15:30.740,0:15:32.350
a function of t.

0:15:32.350,0:15:33.690
So that's that.

0:15:33.690,0:15:37.600
Times this thing right here,[br]times this component, right?

0:15:37.600,0:15:39.300
We're multiplying[br]the i-components.

0:15:39.300,0:15:50.650
So times x prime of t d t, and[br]then that plus, we're going

0:15:50.650,0:15:52.370
to do the same thing[br]with the q function.

0:15:52.370,0:15:56.060
So this is q plus, I'll[br]go to another line.

0:15:56.060,0:15:57.760
Hopefully you realize I could[br]have just kept writing, but

0:15:57.760,0:15:59.020
I'm running out of space.

0:15:59.020,0:16:09.960
Plus q of x of t, y of t, times[br]the component of our dr. Times

0:16:09.960,0:16:11.900
the y-component, or[br]the j-component.

0:16:11.900,0:16:15.530
y prime of t dt.

0:16:15.530,0:16:16.620
And we're done!

0:16:16.620,0:16:17.480
And we're done.

0:16:17.480,0:16:19.300
This might still seem a little[br]bit abstract, but we're going

0:16:19.300,0:16:23.020
to see in the next video,[br]everything is now in terms of

0:16:23.020,0:16:25.480
t, so this is just a[br]straight-up integration,

0:16:25.480,0:16:27.170
with respect to dt.

0:16:27.170,0:16:30.150
If we want, we could take the[br]dt's outside of the equation,

0:16:30.150,0:16:32.270
and it'll look a little[br]bit more normal for you.

0:16:32.270,0:16:34.640
But this is essentially[br]all that we have to do.

0:16:34.640,0:16:38.080
And we're going to see some[br]concrete examples of taking a

0:16:38.080,0:16:43.230
line integral through a vector[br]field, or using vector

0:16:43.230,0:16:45.790
functions, in the next video.

0:16:45.790,0:16:46.000