1
00:00:00,000 --> 00:00:00,330


2
00:00:00,330 --> 00:00:03,110
One of the most fundamental
ideas in all of physics

3
00:00:03,110 --> 00:00:05,385
is the idea of work.

4
00:00:05,385 --> 00:00:08,450
Now when you first learn work,
you just say, oh, that's

5
00:00:08,450 --> 00:00:10,120
just force times distance.

6
00:00:10,120 --> 00:00:12,200
But then later on, when you
learn a little bit about

7
00:00:12,200 --> 00:00:14,770
vectors, you realize that the
force isn't always going in

8
00:00:14,770 --> 00:00:17,610
the same direction as
your displacement.

9
00:00:17,610 --> 00:00:21,450
So you learn that work is
really the magnitude, let me

10
00:00:21,450 --> 00:00:33,070
write this down, the magnitude
of the force, in the direction,

11
00:00:33,070 --> 00:00:39,460
or the component of the force
in the direction

12
00:00:39,460 --> 00:00:41,740
of displacement.

13
00:00:41,740 --> 00:00:44,206
Displacement is just distance
with some direction.

14
00:00:44,206 --> 00:00:49,970


15
00:00:49,970 --> 00:00:55,290
Times the magnitude of the
displacement, or you could say,

16
00:00:55,290 --> 00:00:56,695
times the distance displaced.

17
00:00:56,695 --> 00:01:00,810


18
00:01:00,810 --> 00:01:02,330
And the classic example.

19
00:01:02,330 --> 00:01:06,250
Maybe you have an ice cube,
or some type of block.

20
00:01:06,250 --> 00:01:08,740
I just ice so that there's
not a lot of friction.

21
00:01:08,740 --> 00:01:12,510
Maybe it's standing on a bigger
lake or ice or something.

22
00:01:12,510 --> 00:01:15,030
And maybe you're pulling on
that ice cube at an angle.

23
00:01:15,030 --> 00:01:17,610
Let's say, you're pulling
at an angle like that.

24
00:01:17,610 --> 00:01:20,820
That is my force, right there.

25
00:01:20,820 --> 00:01:24,080
Let's say my force is
equal to-- well, that's

26
00:01:24,080 --> 00:01:25,160
my force vector.

27
00:01:25,160 --> 00:01:33,870
Let's say the magnitude of
my force vector, let's

28
00:01:33,870 --> 00:01:35,310
say it's 10 newtons.

29
00:01:35,310 --> 00:01:37,650
And let's say the direction of
my force vector, right, any

30
00:01:37,650 --> 00:01:41,080
vector has to have a magnitude
and a direction, and the

31
00:01:41,080 --> 00:01:44,920
direction, let's say it has a
30 degree angle, let's say a 60

32
00:01:44,920 --> 00:01:47,770
degree angle, above horizontal.

33
00:01:47,770 --> 00:01:49,560
So that's the direction
I'm pulling in.

34
00:01:49,560 --> 00:01:52,600
And let's say I displace it.

35
00:01:52,600 --> 00:01:55,930
This is all review, hopefully.

36
00:01:55,930 --> 00:01:59,225
If you're displacing it, let's
say you displace it 5 newtons.

37
00:01:59,225 --> 00:02:02,570
So let's say the displacement,
that's the displacement vector

38
00:02:02,570 --> 00:02:10,290
right there, and the magnitude
of it is equal to 5 meters.

39
00:02:10,290 --> 00:02:13,460
So you've learned from the
definition of work, you can't

40
00:02:13,460 --> 00:02:16,940
just say, oh, I'm pulling with
10 newtons of force and

41
00:02:16,940 --> 00:02:18,360
I'm moving it 5 meters.

42
00:02:18,360 --> 00:02:22,560
You can't just multiply the 10
newtons times the 5 meters.

43
00:02:22,560 --> 00:02:25,660
You have to find the magnitude
of the component going in the

44
00:02:25,660 --> 00:02:29,050
same direction as
my displacement.

45
00:02:29,050 --> 00:02:31,860
So what I essentially need to
do is, the length, if you

46
00:02:31,860 --> 00:02:34,930
imagine the length of this
vector being 10, that's the

47
00:02:34,930 --> 00:02:37,750
total force, but you need to
figure out the length of the

48
00:02:37,750 --> 00:02:40,770
vector, that's the component of
the force, going in the same

49
00:02:40,770 --> 00:02:43,460
direction as my displacement.

50
00:02:43,460 --> 00:02:45,570
And a little simple
trigonometry, you know that

51
00:02:45,570 --> 00:02:53,120
this is 10 times the cosine of
60 degrees, or that's equal to,

52
00:02:53,120 --> 00:02:58,010
cosine of 60 degrees is 1/2, so
that's just equal to 5.

53
00:02:58,010 --> 00:03:00,380
So this magnitude, the
magnitude of the force going

54
00:03:00,380 --> 00:03:02,410
in the same direction of
the displacement in this

55
00:03:02,410 --> 00:03:04,810
case, is 5 newtons.

56
00:03:04,810 --> 00:03:07,500


57
00:03:07,500 --> 00:03:09,850
And then you can
figure out the work.

58
00:03:09,850 --> 00:03:19,560
You could say that the work is
equal to 5 newtons times, I'll

59
00:03:19,560 --> 00:03:20,630
just write a dot for times.

60
00:03:20,630 --> 00:03:22,290
I don't want you to think
it's cross product.

61
00:03:22,290 --> 00:03:26,680
Times 5 meters, which is 25
newton meters, or you could

62
00:03:26,680 --> 00:03:31,250
even say 25 Joules of
work have been done.

63
00:03:31,250 --> 00:03:35,280
And this is all review of
somewhat basic physics.

64
00:03:35,280 --> 00:03:36,720
But just think about
what happened, here.

65
00:03:36,720 --> 00:03:37,430
What was the work?

66
00:03:37,430 --> 00:03:39,190
If I write in the abstract.

67
00:03:39,190 --> 00:03:42,550
The work is equal
to the 5 newtons.

68
00:03:42,550 --> 00:03:46,700
That was the magnitude of my
force vector, so it's the

69
00:03:46,700 --> 00:03:52,630
magnitude of my force vector,
times the cosine of this angle.

70
00:03:52,630 --> 00:03:53,860
So you know, let's
call that theta.

71
00:03:53,860 --> 00:03:55,010
Let's say it a
little generally.

72
00:03:55,010 --> 00:03:58,150
So times the cosine
of the angle.

73
00:03:58,150 --> 00:04:01,740
This is the amount of my force
in the direction of the

74
00:04:01,740 --> 00:04:04,960
displacement, the cosine of the
angle between them, times the

75
00:04:04,960 --> 00:04:06,800
magnitude of the displacement.

76
00:04:06,800 --> 00:04:12,260
So times the magnitude
of the displacement.

77
00:04:12,260 --> 00:04:15,560
Or if I wanted to rewrite that,
I could just write that as, the

78
00:04:15,560 --> 00:04:18,940
magnitude of the displacement
times the magnitude of

79
00:04:18,940 --> 00:04:23,400
the force times the
cosine of theta.

80
00:04:23,400 --> 00:04:26,760
And I've done multiple videos
of this, in the linear algebra

81
00:04:26,760 --> 00:04:28,880
playlist, in the physics
playlist, where I talk about

82
00:04:28,880 --> 00:04:31,580
the dot product and the cross
product and all of that, but

83
00:04:31,580 --> 00:04:40,470
this is the dot product
of the vectors d and f.

84
00:04:40,470 --> 00:04:43,700
So in general, if you're trying
to find the work for a constant

85
00:04:43,700 --> 00:04:46,730
displacement, and you have a
constant force, you just take

86
00:04:46,730 --> 00:04:48,530
the dot product of
those two vectors.

87
00:04:48,530 --> 00:04:51,330
And if the dot product is a
completely foreign concept to

88
00:04:51,330 --> 00:04:53,770
you, might want to watch, I
think I've made multiple, 4

89
00:04:53,770 --> 00:04:56,380
or 5 videos on the dot
product, and its intuition,

90
00:04:56,380 --> 00:04:57,420
and how it compares.

91
00:04:57,420 --> 00:04:59,280
But just to give you a little
bit of that intuition right

92
00:04:59,280 --> 00:05:03,920
here, the dot product, when
I take f dot d, or d dot f,

93
00:05:03,920 --> 00:05:08,440
what it's giving me is, I'm
multiplying the magnitude, well

94
00:05:08,440 --> 00:05:10,130
I could just read this out.

95
00:05:10,130 --> 00:05:13,590
But the idea of the dot product
is, take how much of this

96
00:05:13,590 --> 00:05:16,800
vector is going in the same
direction as this vector,

97
00:05:16,800 --> 00:05:18,500
in this case, this much.

98
00:05:18,500 --> 00:05:21,110
And then multiply
the two magnitudes.

99
00:05:21,110 --> 00:05:22,410
And that's what we
did right here.

100
00:05:22,410 --> 00:05:26,230
So the work is going to be the
force vector, dot, taking the

101
00:05:26,230 --> 00:05:28,980
dot part of the force vector
with the displacement vector,

102
00:05:28,980 --> 00:05:30,840
and this, of course,
is a scalar value.

103
00:05:30,840 --> 00:05:33,040
And we'll work out some
examples in the future where

104
00:05:33,040 --> 00:05:34,360
you'll see that that's true.

105
00:05:34,360 --> 00:05:39,000
So this is all review of
fairly elementary physics.

106
00:05:39,000 --> 00:05:42,500
Now let's take a more
complex example, but it's

107
00:05:42,500 --> 00:05:43,670
really the same idea.

108
00:05:43,670 --> 00:05:45,873
Let's define a vector field.

109
00:05:45,873 --> 00:05:48,660


110
00:05:48,660 --> 00:05:51,371
So let's say that I have a
vector field f, and we're

111
00:05:51,371 --> 00:05:54,050
going to think about what
this means in a second.

112
00:05:54,050 --> 00:05:58,890
It's a function of x and y, and
it's equal to some scalar

113
00:05:58,890 --> 00:06:04,490
function of x and y times the
i-unit vector, or the

114
00:06:04,490 --> 00:06:08,760
horizontal unit vector, plus
some other function, scalar

115
00:06:08,760 --> 00:06:14,250
function of x and y, times the
vertical unit vector.

116
00:06:14,250 --> 00:06:15,580
So what would something
like this be?

117
00:06:15,580 --> 00:06:17,460
This is a vector field.

118
00:06:17,460 --> 00:06:20,210
This is a vector field
in 2-dimensional space.

119
00:06:20,210 --> 00:06:21,330
We're on the x-y plane.

120
00:06:21,330 --> 00:06:31,190


121
00:06:31,190 --> 00:06:35,840
Or you could even say, on R2.

122
00:06:35,840 --> 00:06:37,690
Either way, I don't want
to get too much into

123
00:06:37,690 --> 00:06:39,230
the mathiness of it.

124
00:06:39,230 --> 00:06:40,590
But what does this do?

125
00:06:40,590 --> 00:06:47,270
Well, if I were to draw my x-y
plane, so that is my, again,

126
00:06:47,270 --> 00:06:49,070
having trouble drawing
a straight line.

127
00:06:49,070 --> 00:06:50,610
All right, there we go.

128
00:06:50,610 --> 00:06:54,050
That's my y-axis, and
that's my x-axis.

129
00:06:54,050 --> 00:06:56,360
I'm just drawing the first
quadrant, and but you could

130
00:06:56,360 --> 00:06:59,450
go negative in either
direction, if you like.

131
00:06:59,450 --> 00:07:01,260
What does this thing do?

132
00:07:01,260 --> 00:07:02,350
Well, it's essentially
saying, look.

133
00:07:02,350 --> 00:07:06,800
You give me any x, any y, you
give any x, y in the x-y plane,

134
00:07:06,800 --> 00:07:09,970
and these are going to end
up with some numbers, right?

135
00:07:09,970 --> 00:07:12,655
When you put x, y here, you're
going to get some value, when

136
00:07:12,655 --> 00:07:14,310
you put x, y here, you're
going to get some value.

137
00:07:14,310 --> 00:07:16,980
So you're going to get some
combination of the i-

138
00:07:16,980 --> 00:07:18,070
and j-unit vectors.

139
00:07:18,070 --> 00:07:19,770
So you're going to
get some vector.

140
00:07:19,770 --> 00:07:23,020
So what this does, it defines a
vector that's associated with

141
00:07:23,020 --> 00:07:24,810
every point on x-y plane.

142
00:07:24,810 --> 00:07:28,780
So you could say, if I take
this point on the x-y plane,

143
00:07:28,780 --> 00:07:32,480
and I would pop it into this,
I'll get something times i plus

144
00:07:32,480 --> 00:07:34,730
something times j, and when you
add those 2, maybe I get a

145
00:07:34,730 --> 00:07:37,130
vector that something
like that.

146
00:07:37,130 --> 00:07:38,100
And you could do that
on every point.

147
00:07:38,100 --> 00:07:39,190
I'm just taking random samples.

148
00:07:39,190 --> 00:07:41,420
Maybe when I go here,
the vector looks

149
00:07:41,420 --> 00:07:42,280
something like that.

150
00:07:42,280 --> 00:07:44,910
Maybe when I go here, the
victor looks like this.

151
00:07:44,910 --> 00:07:47,560
Maybe when I go here, the
vector looks like that.

152
00:07:47,560 --> 00:07:50,350
And maybe when I go up here,
the vector goes like that.

153
00:07:50,350 --> 00:07:52,320
I'm just randomly
picking points.

154
00:07:52,320 --> 00:07:57,090
It defines a vector on all of
the x, y coordinates where

155
00:07:57,090 --> 00:08:00,920
these scalar functions
are properly defined.

156
00:08:00,920 --> 00:08:02,370
And that's why it's
called a vector field.

157
00:08:02,370 --> 00:08:06,580
It defines what a potential,
maybe, force would be,

158
00:08:06,580 --> 00:08:11,430
or some other type of
force, at any point.

159
00:08:11,430 --> 00:08:14,350
At any point, if you happen
to have something there.

160
00:08:14,350 --> 00:08:15,900
Maybe that's what
the function is.

161
00:08:15,900 --> 00:08:17,750
And I could keep doing
this forever, and

162
00:08:17,750 --> 00:08:18,790
filling in all the gaps.

163
00:08:18,790 --> 00:08:19,660
But I think you get the idea.

164
00:08:19,660 --> 00:08:24,790
It associates a vector with
every point on x-y plane.

165
00:08:24,790 --> 00:08:29,010
Now, this is called a vector
field, so it probably makes a

166
00:08:29,010 --> 00:08:30,950
lot of sense that this could
be used to describe

167
00:08:30,950 --> 00:08:31,870
any type of field.

168
00:08:31,870 --> 00:08:33,410
It could be a
gravitation field.

169
00:08:33,410 --> 00:08:36,840
It could be an electric field,
it could be a magnetic field.

170
00:08:36,840 --> 00:08:39,630
And this could be essentially
telling you how much force

171
00:08:39,630 --> 00:08:43,190
there would be on some
particle in that field.

172
00:08:43,190 --> 00:08:44,660
That's exactly what
this would describe.

173
00:08:44,660 --> 00:08:48,950
Now, let's say that in this
field, I have some particle

174
00:08:48,950 --> 00:08:51,610
traveling on x-y plane.

175
00:08:51,610 --> 00:08:58,620
Let's say it starts there, and
by virtue of all of these crazy

176
00:08:58,620 --> 00:09:03,850
forces that are acting on it,
and maybe it's on some tracks

177
00:09:03,850 --> 00:09:06,900
or something, so it won't
always move exactly in the

178
00:09:06,900 --> 00:09:09,360
direction that the field
is trying to move it at.

179
00:09:09,360 --> 00:09:14,030
Let's say it moves in a path
that moves something like this.

180
00:09:14,030 --> 00:09:17,710
And let's say that this path,
or this curve, is defined by

181
00:09:17,710 --> 00:09:22,010
a position vector function.

182
00:09:22,010 --> 00:09:25,150
So let's say that that's
defined by r of t, which is

183
00:09:25,150 --> 00:09:33,780
just x of t times i plus y of
t times our unit factor j.

184
00:09:33,780 --> 00:09:35,130
That's r of t right there.

185
00:09:35,130 --> 00:09:37,730
Well, in order for this to be
a finite path, this is true

186
00:09:37,730 --> 00:09:42,370
before t is greater than or
equal to a, and less

187
00:09:42,370 --> 00:09:45,640
than or equal to b.

188
00:09:45,640 --> 00:09:47,830
This is the path that the
particle just happens to

189
00:09:47,830 --> 00:09:50,370
take, due to all of
these wacky forces.

190
00:09:50,370 --> 00:09:54,270
So when the particle is right
here, maybe the vector field

191
00:09:54,270 --> 00:09:56,960
acting on it, maybe it's
putting a force like that.

192
00:09:56,960 --> 00:09:59,520
But since the thing is on some
type of tracks, it moves

193
00:09:59,520 --> 00:10:00,400
in this direction.

194
00:10:00,400 --> 00:10:03,830
And then when it's here, maybe
the vector field is like that,

195
00:10:03,830 --> 00:10:05,740
but it moves in that direction,
because it's on some

196
00:10:05,740 --> 00:10:06,940
type of tracks.

197
00:10:06,940 --> 00:10:09,500
Now, everything I've done in
this video is to build up

198
00:10:09,500 --> 00:10:11,180
to a fundamental question.

199
00:10:11,180 --> 00:10:13,910
What was the work done on
the particle by the field?

200
00:10:13,910 --> 00:10:24,960


201
00:10:24,960 --> 00:10:28,620
To answer that question, we
could zoom in a little bit.

202
00:10:28,620 --> 00:10:31,100
I'm going to zoom in on
only a little small

203
00:10:31,100 --> 00:10:34,710
snippet of our path.

204
00:10:34,710 --> 00:10:38,010
And let's try to figure out
what the work is done in a very

205
00:10:38,010 --> 00:10:40,470
small part of our path, because
it's constantly changing.

206
00:10:40,470 --> 00:10:42,190
The field is
changing direction.

207
00:10:42,190 --> 00:10:43,630
my object is
changing direction.

208
00:10:43,630 --> 00:10:47,780
So let's say when I'm here,
and let's say I move a

209
00:10:47,780 --> 00:10:49,740
small amount of my path.

210
00:10:49,740 --> 00:10:55,860
So let's say I move, this
is an infinitesimally

211
00:10:55,860 --> 00:10:58,500
small dr. Right?

212
00:10:58,500 --> 00:11:00,810
I have a differential, it's a
differential vector, infinitely

213
00:11:00,810 --> 00:11:02,630
small displacement.

214
00:11:02,630 --> 00:11:06,800
and let's say over the course
of that, the vector field is

215
00:11:06,800 --> 00:11:08,840
acting in this local
area, let's say it looks

216
00:11:08,840 --> 00:11:10,480
something like that.

217
00:11:10,480 --> 00:11:13,490
It's providing a force that
looks something like that.

218
00:11:13,490 --> 00:11:16,640
So that's the vector field in
that area, or the force

219
00:11:16,640 --> 00:11:18,750
directed on that particle right
when it's at that point.

220
00:11:18,750 --> 00:11:18,870
Right?

221
00:11:18,870 --> 00:11:22,420
It's an infinitesimally small
amount of time in space.

222
00:11:22,420 --> 00:11:24,440
You could say, OK, over that
little small point, we

223
00:11:24,440 --> 00:11:26,600
have this constant force.

224
00:11:26,600 --> 00:11:29,790
What was the work done
over this small period?

225
00:11:29,790 --> 00:11:32,330
You could say, what's the
small interval of work?

226
00:11:32,330 --> 00:11:36,120
You could say d work, or
a differential of work.

227
00:11:36,120 --> 00:11:38,940
Well, by the same exact logic
that we did with the simple

228
00:11:38,940 --> 00:11:43,810
problem, it's the magnitude of
the force in the direction of

229
00:11:43,810 --> 00:11:48,550
our displacement times the
magnitude of our displacement.

230
00:11:48,550 --> 00:11:52,800
And we know what that is, just
from this example up here.

231
00:11:52,800 --> 00:11:54,810
That's the dot product.

232
00:11:54,810 --> 00:11:58,340
It's the dot product of the
force and our super-small

233
00:11:58,340 --> 00:11:59,480
displacement.

234
00:11:59,480 --> 00:12:07,860
So that's equal to the dot
product of our force and our

235
00:12:07,860 --> 00:12:09,870
super-small displacement.

236
00:12:09,870 --> 00:12:13,240
Now, just by doing this, we're
just figuring out the work

237
00:12:13,240 --> 00:12:16,440
over, maybe like a really
small, super-small dr. But

238
00:12:16,440 --> 00:12:18,820
what we want to do, is we
want to sum them all up.

239
00:12:18,820 --> 00:12:21,870
We want to sum up all of the
drs to figure out the total,

240
00:12:21,870 --> 00:12:25,090
all of the f dot drs to figure
out the total work done.

241
00:12:25,090 --> 00:12:27,510
And that's where the
integral comes in.

242
00:12:27,510 --> 00:12:32,570
We will do a line integral
over-- I mean, you could

243
00:12:32,570 --> 00:12:33,910
think of it two ways.

244
00:12:33,910 --> 00:12:37,440
You could write just d dot w
there, but we could say, we'll

245
00:12:37,440 --> 00:12:42,700
do a line integral along this
curve c, could call that c

246
00:12:42,700 --> 00:12:46,410
or along r, whatever you
want to say it, of dw.

247
00:12:46,410 --> 00:12:47,800
That'll give us the total work.

248
00:12:47,800 --> 00:12:49,500
So let's say, work
is equal to that.

249
00:12:49,500 --> 00:12:54,040
Or we could also write it over
the integral, over the same

250
00:12:54,040 --> 00:13:00,500
curve of f of f dot dr.

251
00:13:00,500 --> 00:13:03,580
And this might seem like a
really, you know, gee, this

252
00:13:03,580 --> 00:13:05,120
is really abstract, Sal.

253
00:13:05,120 --> 00:13:09,220
How do we actually calculate
something like this?

254
00:13:09,220 --> 00:13:13,130
Especially because we have
everything parameterized

255
00:13:13,130 --> 00:13:14,030
in terms of t.

256
00:13:14,030 --> 00:13:16,130
How do we get this
in terms of t?

257
00:13:16,130 --> 00:13:19,710
And if you just think about
it, what is f dot r?

258
00:13:19,710 --> 00:13:21,030
Or what is f dot dr?

259
00:13:21,030 --> 00:13:23,300
Well, actually, to answer
that, let's remember

260
00:13:23,300 --> 00:13:25,830
what dr looked like.

261
00:13:25,830 --> 00:13:36,200
If you remember, dr/dt is equal
to x prime of t, I'm writing it

262
00:13:36,200 --> 00:13:39,120
like, I could have written dx
dt if I wanted to do, times the

263
00:13:39,120 --> 00:13:45,180
i-unit vector, plus y prime of
t, times the j-unit vector.

264
00:13:45,180 --> 00:13:49,320
And if we just wanted to dr, we
could multiply both sides, if

265
00:13:49,320 --> 00:13:51,850
we're being a little bit more
hand-wavy with the

266
00:13:51,850 --> 00:13:53,470
differentials, not
too rigorous.

267
00:13:53,470 --> 00:13:58,480
We'll get dr is equal to x
prime of t dt times the unit

268
00:13:58,480 --> 00:14:05,070
vector i plus y prime of t
times the differential dt

269
00:14:05,070 --> 00:14:07,280
times the unit vector j.

270
00:14:07,280 --> 00:14:09,070
So this is our dr right here.

271
00:14:09,070 --> 00:14:12,110


272
00:14:12,110 --> 00:14:16,280
And remember what our
vector field was.

273
00:14:16,280 --> 00:14:17,440
It was this thing up here.

274
00:14:17,440 --> 00:14:19,590
Let me copy and paste it.

275
00:14:19,590 --> 00:14:21,030
And we'll see that
the dot product is

276
00:14:21,030 --> 00:14:23,360
actually not so crazy.

277
00:14:23,360 --> 00:14:26,710
So copy, and let me
paste it down here.

278
00:14:26,710 --> 00:14:31,130


279
00:14:31,130 --> 00:14:33,820
So what's this integral
going to look like?

280
00:14:33,820 --> 00:14:37,600
This integral right here, that
gives the total work done by

281
00:14:37,600 --> 00:14:40,790
the field, on the particle,
as it moves along that path.

282
00:14:40,790 --> 00:14:44,090
Just super fundamental to
pretty much any serious physics

283
00:14:44,090 --> 00:14:47,170
that you might eventually
find yourself doing.

284
00:14:47,170 --> 00:14:48,170
So you could say, well gee.

285
00:14:48,170 --> 00:14:52,420
It's going to be the integral,
let's just say from t is equal

286
00:14:52,420 --> 00:14:55,320
to a, to t is equal to b.

287
00:14:55,320 --> 00:14:58,310
Right? a is where we started
off on the path, t is equal

288
00:14:58,310 --> 00:14:59,790
to a to t is equal to b.

289
00:14:59,790 --> 00:15:01,760
You can imagine it as being
timed, as a particle

290
00:15:01,760 --> 00:15:03,610
moving, as time increases.

291
00:15:03,610 --> 00:15:07,000
And then what is f dot dr?

292
00:15:07,000 --> 00:15:10,640
Well, if you remember from just
what the dot product is, you

293
00:15:10,640 --> 00:15:15,310
can essentially just take the
product of the corresponding

294
00:15:15,310 --> 00:15:17,740
components of your of
vector, and add them up.

295
00:15:17,740 --> 00:15:20,070
So this is going to be the
integral from t equals a to t

296
00:15:20,070 --> 00:15:27,246
equals b, of p of p of x,
really, instead of writing x,

297
00:15:27,246 --> 00:15:30,740
y, it's x of t, right? x as a
function of t, y as

298
00:15:30,740 --> 00:15:32,350
a function of t.

299
00:15:32,350 --> 00:15:33,690
So that's that.

300
00:15:33,690 --> 00:15:37,600
Times this thing right here,
times this component, right?

301
00:15:37,600 --> 00:15:39,300
We're multiplying
the i-components.

302
00:15:39,300 --> 00:15:50,650
So times x prime of t d t, and
then that plus, we're going

303
00:15:50,650 --> 00:15:52,370
to do the same thing
with the q function.

304
00:15:52,370 --> 00:15:56,060
So this is q plus, I'll
go to another line.

305
00:15:56,060 --> 00:15:57,760
Hopefully you realize I could
have just kept writing, but

306
00:15:57,760 --> 00:15:59,020
I'm running out of space.

307
00:15:59,020 --> 00:16:09,960
Plus q of x of t, y of t, times
the component of our dr. Times

308
00:16:09,960 --> 00:16:11,900
the y-component, or
the j-component.

309
00:16:11,900 --> 00:16:15,530
y prime of t dt.

310
00:16:15,530 --> 00:16:16,620
And we're done!

311
00:16:16,620 --> 00:16:17,480
And we're done.

312
00:16:17,480 --> 00:16:19,300
This might still seem a little
bit abstract, but we're going

313
00:16:19,300 --> 00:16:23,020
to see in the next video,
everything is now in terms of

314
00:16:23,020 --> 00:16:25,480
t, so this is just a
straight-up integration,

315
00:16:25,480 --> 00:16:27,170
with respect to dt.

316
00:16:27,170 --> 00:16:30,150
If we want, we could take the
dt's outside of the equation,

317
00:16:30,150 --> 00:16:32,270
and it'll look a little
bit more normal for you.

318
00:16:32,270 --> 00:16:34,640
But this is essentially
all that we have to do.

319
00:16:34,640 --> 00:16:38,080
And we're going to see some
concrete examples of taking a

320
00:16:38,080 --> 00:16:43,230
line integral through a vector
field, or using vector

321
00:16:43,230 --> 00:16:45,790
functions, in the next video.

322
00:16:45,790 --> 00:16:46,000