1
00:00:00,040 --> 00:00:02,460
The following content is
provided under a Creative

2
00:00:02,460 --> 00:00:03,870
Commons license.

3
00:00:03,870 --> 00:00:06,320
Your support will help
MIT OpenCourseWare

4
00:00:06,320 --> 00:00:10,560
continue to offer high quality
educational resources for free.

5
00:00:10,560 --> 00:00:13,300
To make a donation or
view additional materials

6
00:00:13,300 --> 00:00:17,210
from hundreds of MIT courses,
visit MIT OpenCourseWare

7
00:00:17,210 --> 00:00:19,650
at ocw.mit.edu.

8
00:00:53,118 --> 00:00:54,620
HERBERT GROSS: Hi.

9
00:00:54,620 --> 00:00:58,200
We've sort of arrive
at D-day in our course;

10
00:00:58,200 --> 00:01:01,530
that in a manner of speaking,
everything that we've done up

11
00:01:01,530 --> 00:01:04,140
to now has been a rehearsal.

12
00:01:04,140 --> 00:01:07,050
That today we are
going to come to grips

13
00:01:07,050 --> 00:01:09,870
with what the course
is all about: namely,

14
00:01:09,870 --> 00:01:14,800
a real-valued function of
several real variables.

15
00:01:14,800 --> 00:01:17,360
And before getting into
that, what I'd like to do

16
00:01:17,360 --> 00:01:20,830
is to review very briefly
what we've talked about

17
00:01:20,830 --> 00:01:23,500
so far in terms of
functions, since we've

18
00:01:23,500 --> 00:01:25,860
introduced the vector notation.

19
00:01:25,860 --> 00:01:27,730
Namely, what we've
mentioned is, is

20
00:01:27,730 --> 00:01:29,230
that our function
machine could have

21
00:01:29,230 --> 00:01:32,640
either a vector or a
scalar as its input,

22
00:01:32,640 --> 00:01:36,190
and either a vector or
a scalar as its output.

23
00:01:36,190 --> 00:01:39,210
Part one of our course
centered around the idea

24
00:01:39,210 --> 00:01:42,880
where both the input and
the outputs were scalars.

25
00:01:42,880 --> 00:01:46,210
Whereas, the previous
block of material

26
00:01:46,210 --> 00:01:49,670
concerned the case where
our input was a scalar,

27
00:01:49,670 --> 00:01:52,630
and our output was a vector.

28
00:01:52,630 --> 00:01:55,070
Today what we're
going to discuss

29
00:01:55,070 --> 00:02:00,910
is the situation that occurs
when our input is a vector,

30
00:02:00,910 --> 00:02:04,240
and our output is a scalar.

31
00:02:04,240 --> 00:02:09,180
And I call today's lecture
"n-dimensional Vector Spaces."

32
00:02:09,180 --> 00:02:13,250
Eventually, we'll discuss
vector spaces in more detail.

33
00:02:13,250 --> 00:02:18,110
For the time being, I
simply want to set the mood,

34
00:02:18,110 --> 00:02:20,460
and hopefully by the
time I'm through,

35
00:02:20,460 --> 00:02:24,190
show you a rather peaceful
coexistence between the worlds

36
00:02:24,190 --> 00:02:26,390
of the new mathematics,
and the worlds

37
00:02:26,390 --> 00:02:28,090
of traditional mathematics.

38
00:02:28,090 --> 00:02:30,220
In fact, in many of
our topics that we're

39
00:02:30,220 --> 00:02:32,690
going to tackle in
this block of material,

40
00:02:32,690 --> 00:02:36,150
we will give both
points of view.

41
00:02:36,150 --> 00:02:38,910
But to set the stage
properly-- to get

42
00:02:38,910 --> 00:02:40,730
into the idea of what
a vector space is all

43
00:02:40,730 --> 00:02:43,240
about-- and if that
word frightens you, just

44
00:02:43,240 --> 00:02:45,460
don't worry about
for a minute or two.

45
00:02:45,460 --> 00:02:48,450
Worry about it after that,
but let's just get started.

46
00:02:48,450 --> 00:02:50,490
Let's consider the
situation where

47
00:02:50,490 --> 00:02:54,660
I have a function machine
where my input is a vector

48
00:02:54,660 --> 00:02:57,920
and my output is a scalar.

49
00:02:57,920 --> 00:03:01,840
As an example, let me just
generically let v represent

50
00:03:01,840 --> 00:03:05,970
the vector whose
form is x*i plus y*j,

51
00:03:05,970 --> 00:03:08,580
using i and j components.

52
00:03:08,580 --> 00:03:16,140
Let me define f of v to be
pi times the square of the i

53
00:03:16,140 --> 00:03:18,200
component times the j component.

54
00:03:18,200 --> 00:03:20,210
And let's not worry
right now about why I

55
00:03:20,210 --> 00:03:22,360
picked this particular recipe.

56
00:03:22,360 --> 00:03:26,640
Let's simply observe that
once this recipe is chosen,

57
00:03:26,640 --> 00:03:33,040
f is a function which maps a
two-dimensional vector here,

58
00:03:33,040 --> 00:03:37,880
in i and j components, into
a number of pi x squared y.

59
00:03:37,880 --> 00:03:40,530
Just to illustrate
this recipe, notice

60
00:03:40,530 --> 00:03:43,880
that if our vector
were 3i plus 4j,

61
00:03:43,880 --> 00:03:47,240
the output of the f machine--
if this were the input--

62
00:03:47,240 --> 00:03:48,150
would be what?

63
00:03:48,150 --> 00:03:53,010
Pi times the square of the first
component-- the component of i,

64
00:03:53,010 --> 00:03:57,820
that's 3 squared-- times the
component of j, which is 4.

65
00:03:57,820 --> 00:04:00,790
And that leads to 36*pi.

66
00:04:00,790 --> 00:04:03,650
By the way, observe that
order does make a difference,

67
00:04:03,650 --> 00:04:07,040
namely if I reverse the
roles of the coefficients,

68
00:04:07,040 --> 00:04:12,610
and feed the vector 4i
plus 3j into my f machine,

69
00:04:12,610 --> 00:04:16,200
the output would be
pi times 4 squared--

70
00:04:16,200 --> 00:04:19,459
the square of the i
component-- times 3,

71
00:04:19,459 --> 00:04:22,290
and that would be 48*pi.

72
00:04:22,290 --> 00:04:26,240
Notice also that we have
allowed, already in our course,

73
00:04:26,240 --> 00:04:32,540
the abbreviation that a comma
b would represent a*i plus b*j,

74
00:04:32,540 --> 00:04:36,860
or a comma b comma c would
represent a*i plus b*j plus

75
00:04:36,860 --> 00:04:37,730
c*k.

76
00:04:37,730 --> 00:04:40,370
The idea is that if we
now apply this shorthand

77
00:04:40,370 --> 00:04:45,100
notation to these two vectors,
what we could say is what?

78
00:04:45,100 --> 00:04:50,680
f of this vector, in other
words, f of 3 comma 4 is 36*pi,

79
00:04:50,680 --> 00:04:54,305
whereas f of 4 comma 3 is 48*pi.

80
00:04:54,305 --> 00:04:56,470
And that's exactly
what we mean when

81
00:04:56,470 --> 00:04:59,900
we say that we may treat
a two-dimensional vector

82
00:04:59,900 --> 00:05:01,850
as an ordered pair.

83
00:05:01,850 --> 00:05:05,150
You see, it's, not only
a pair but the order

84
00:05:05,150 --> 00:05:06,720
does make a difference.

85
00:05:06,720 --> 00:05:09,310
Both in what the
vector is and what

86
00:05:09,310 --> 00:05:12,280
the output of the f machine is.

87
00:05:12,280 --> 00:05:13,210
OK?

88
00:05:13,210 --> 00:05:16,090
Hopefully, let's
say, so far so good,

89
00:05:16,090 --> 00:05:21,900
and let's tackle now a rather
completely different problem.

90
00:05:21,900 --> 00:05:24,440
What I'd like to do
now is the following.

91
00:05:24,440 --> 00:05:28,300
Let's consider the cylinder--
the right circular cylinder--

92
00:05:28,300 --> 00:05:33,530
the radius of whose base is
x, and whose height is y.

93
00:05:33,530 --> 00:05:40,910
Notice that the volume of this
cylinder is pi x squared y.

94
00:05:40,910 --> 00:05:46,080
And if I use the same f
that we used previously--

95
00:05:46,080 --> 00:05:48,380
in other words,
the same f that we

96
00:05:48,380 --> 00:05:52,540
were talking about over here--
notice that another way--

97
00:05:52,540 --> 00:05:54,410
see how is f defined?

98
00:05:54,410 --> 00:05:59,220
Given that the input was x
comma y the output was pi x

99
00:05:59,220 --> 00:06:04,380
squared y, notice that
the volume is f of x, y.

100
00:06:04,380 --> 00:06:08,150
In particular, if I want
the volume of the cylinder,

101
00:06:08,150 --> 00:06:13,700
the radius of whose base is
3, and whose height is 4--

102
00:06:13,700 --> 00:06:15,320
see x is 3 and y is 4--

103
00:06:15,320 --> 00:06:19,040
what I really want
is f of 3 comma 4.

104
00:06:19,040 --> 00:06:23,190
That's pi times 3
squared times 4.

105
00:06:23,190 --> 00:06:27,010
f of 3 comma 4 is 36*pi.

106
00:06:27,010 --> 00:06:28,730
Now I'd like to
pause for a second

107
00:06:28,730 --> 00:06:32,280
again, and return to
our earlier remark.

108
00:06:32,280 --> 00:06:39,400
Namely, if I look at this,
and if I look at this.

109
00:06:45,500 --> 00:06:49,060
Notice that these two
expressions are identical.

110
00:06:49,060 --> 00:06:52,370
I cannot tell the difference
between whether I'm looking

111
00:06:52,370 --> 00:06:56,370
at the vector 3i plus 4j,
or whether I'm looking

112
00:06:56,370 --> 00:06:59,920
at the cylinder, the
radius of whose base is 3,

113
00:06:59,920 --> 00:07:03,500
and whose height is 4; whether
I look at this equation,

114
00:07:03,500 --> 00:07:06,040
or whether I look
at this equation.

115
00:07:06,040 --> 00:07:09,830
The difference is that in the
first case somehow or other,

116
00:07:09,830 --> 00:07:12,190
it was quite natural
to think of 3 comma

117
00:07:12,190 --> 00:07:17,000
4 as being either an
ordered pair, or an arrow.

118
00:07:17,000 --> 00:07:19,090
In this case,
however, my contention

119
00:07:19,090 --> 00:07:22,300
is that when we
think of the radius

120
00:07:22,300 --> 00:07:24,620
of the base of the
cylinder, and the height,

121
00:07:24,620 --> 00:07:27,660
we do not tend to think
in terms of arrows,

122
00:07:27,660 --> 00:07:30,530
but rather in terms
of ordered pairs.

123
00:07:30,530 --> 00:07:36,410
In other words, the ordered pair
x comma y, in the expression

124
00:07:36,410 --> 00:07:40,990
f of x, y, need not be
viewed as an arrow, but as

125
00:07:40,990 --> 00:07:42,770
an ordered pair.

126
00:07:42,770 --> 00:07:46,870
And an ordered pair
is called a 2-tuple.

127
00:07:46,870 --> 00:07:49,320
This leads to a
generalization that I

128
00:07:49,320 --> 00:07:51,540
think is rather
important, and I think

129
00:07:51,540 --> 00:07:57,310
you will see in a moment,
where the idea of this approach

130
00:07:57,310 --> 00:08:00,410
comes into functions of
several real variables.

131
00:08:00,410 --> 00:08:05,520
The topic I have in mind is
something called an n-tuple.

132
00:08:05,520 --> 00:08:09,130
And let me read into that
rather gradually as follows.

133
00:08:09,130 --> 00:08:15,220
Without giving you a specific
physical example-- meaning

134
00:08:15,220 --> 00:08:18,290
I'll give you an illustration,
but leave the numerical amounts

135
00:08:18,290 --> 00:08:19,390
out.

136
00:08:19,390 --> 00:08:23,370
Quite possibly if I'm studying
temperature in a room,

137
00:08:23,370 --> 00:08:25,790
the temperature will
in general what?

138
00:08:25,790 --> 00:08:28,960
It will depend on what
position I'm at in the room,

139
00:08:28,960 --> 00:08:33,820
and also at what time I
measure the temperature.

140
00:08:33,820 --> 00:08:36,710
It's fair to assume that
in many applications

141
00:08:36,710 --> 00:08:39,809
the temperature is some
function of the four

142
00:08:39,809 --> 00:08:43,750
independent variables
x, y, z, and t,

143
00:08:43,750 --> 00:08:47,400
where x, y, and z are
the Cartesian coordinates

144
00:08:47,400 --> 00:08:52,020
of three-dimensional space,
and t represents time.

145
00:08:52,020 --> 00:08:56,740
What I'm driving at is I can
now visualize this in terms

146
00:08:56,740 --> 00:08:58,810
of my function machine again.

147
00:08:58,810 --> 00:09:03,420
Namely to compute T, I
think of feeding what?

148
00:09:03,420 --> 00:09:08,180
Specific values into the
machine for x, y, z, and t.

149
00:09:08,180 --> 00:09:13,440
The f machine then
performs on x, y, z, and t

150
00:09:13,440 --> 00:09:18,200
as indicated by f to compute t.

151
00:09:18,200 --> 00:09:21,050
The input of my f
machine in this case

152
00:09:21,050 --> 00:09:25,490
is what I'm going to call a
4-tuple for the time being.

153
00:09:25,490 --> 00:09:29,700
I need four values--
x, y, z, and t.

154
00:09:29,700 --> 00:09:31,620
Order does make a difference.

155
00:09:31,620 --> 00:09:36,060
For example, if I interchange
the x- and the y-coordinate,

156
00:09:36,060 --> 00:09:39,200
those x and y, what I'm doing
is I'm interchanging the x-

157
00:09:39,200 --> 00:09:41,560
and the y-coordinate
of the point in space,

158
00:09:41,560 --> 00:09:44,800
and that in general is going
to change the point in space.

159
00:09:44,800 --> 00:09:48,010
The point, however, is that
in this particular f machine,

160
00:09:48,010 --> 00:09:51,480
notice that my
output is a scalar.

161
00:09:51,480 --> 00:09:54,020
Namely the temperature
is a number,

162
00:09:54,020 --> 00:09:57,680
but the input is a 4-tuple.

163
00:09:57,680 --> 00:10:00,910
x, y, z, and t.

164
00:10:00,910 --> 00:10:03,100
Now the trouble
with using symbolism

165
00:10:03,100 --> 00:10:07,280
like x, y, z, and t, I
guess-- without going

166
00:10:07,280 --> 00:10:09,940
into a long philosophic
discussion--

167
00:10:09,940 --> 00:10:11,410
among other things,
as soon as you

168
00:10:11,410 --> 00:10:14,800
have 27 or more
independent variables,

169
00:10:14,800 --> 00:10:17,070
you run out of letters
of the alphabet.

170
00:10:17,070 --> 00:10:20,730
As a result, it is
quite common for one

171
00:10:20,730 --> 00:10:23,430
to adopt a new notation.

172
00:10:23,430 --> 00:10:26,040
instead of saying
let (x, y, z, t)

173
00:10:26,040 --> 00:10:30,410
be a 4-tuple, what one
usually does is chooses

174
00:10:30,410 --> 00:10:34,065
one symbol-- say x-- and
then uses subscripts.

175
00:10:34,065 --> 00:10:38,570
Namely, a general 4-tuple
would have the form what?

176
00:10:38,570 --> 00:10:43,560
x_1 comma x_2 comma x_3
comma x_4, where x_1,

177
00:10:43,560 --> 00:10:46,150
x_2, x_3 and x_4 are numbers.

178
00:10:46,150 --> 00:10:48,940
An expression like this
is called the 4-tuple.

179
00:10:48,940 --> 00:10:51,280
What is this a
generalization of?

180
00:10:51,280 --> 00:10:55,010
The 4-tuple is a generalization
of the one-dimensional,

181
00:10:55,010 --> 00:10:57,910
two-dimensional, and
three-dimensional arrow,

182
00:10:57,910 --> 00:11:00,450
so to speak, where we
could think of what?

183
00:11:00,450 --> 00:11:06,200
The vector x_1*i as just needing
one number to specify it.

184
00:11:06,200 --> 00:11:11,930
The vector x_1*i plus x_2*j
could've been used to do what?

185
00:11:11,930 --> 00:11:15,580
It could've been abbreviated
by the 2-tuple (x_1, x_2).

186
00:11:15,580 --> 00:11:20,150
And the vector x_1*i plus
x_2*j plus x_3*k could've been

187
00:11:20,150 --> 00:11:24,470
abbreviated by the
3-tuple (x_1, x_2, x_3).

188
00:11:24,470 --> 00:11:26,610
It is conventional
in one-, two-,

189
00:11:26,610 --> 00:11:30,240
or three-dimensional
space to use x, y, and z,

190
00:11:30,240 --> 00:11:32,680
instead of x_1, x_2, and x_3.

191
00:11:32,680 --> 00:11:34,390
But that's just a convention.

192
00:11:34,390 --> 00:11:37,141
I think that it's because we
learnt it that way that we do

193
00:11:37,141 --> 00:11:37,640
it.

194
00:11:37,640 --> 00:11:39,900
In general, I
think the subscript

195
00:11:39,900 --> 00:11:44,440
notation is much nicer, but
in general the idea is what?

196
00:11:44,440 --> 00:11:50,240
Given an ordered array of
n numbers, x_1 up to x_n,

197
00:11:50,240 --> 00:11:52,590
we call that an n-tuple.

198
00:11:52,590 --> 00:11:55,120
And my friend and
colleague John Fitch

199
00:11:55,120 --> 00:11:56,780
mentioned to me
that if n is odd,

200
00:11:56,780 --> 00:11:58,770
like one, three,
five, or seven, then

201
00:11:58,770 --> 00:12:01,120
it's known as an odd-tuple.

202
00:12:01,120 --> 00:12:03,290
Which isn't a very
funny story, that's

203
00:12:03,290 --> 00:12:06,020
why I told you John told
me that particular story.

204
00:12:06,020 --> 00:12:10,200
But the whole idea is
this is an n-tuple.

205
00:12:10,200 --> 00:12:12,500
And the whole idea
again is what?

206
00:12:12,500 --> 00:12:15,110
That an n-tuple makes
sense, even when

207
00:12:15,110 --> 00:12:17,310
n is greater than three.

208
00:12:17,310 --> 00:12:20,210
The whole name of
the game of functions

209
00:12:20,210 --> 00:12:26,200
of several real variables-- in
terms of modern mathematics,

210
00:12:26,200 --> 00:12:28,800
in terms of the
language of n-tuples--

211
00:12:28,800 --> 00:12:32,360
is that a real-valued
function of several--

212
00:12:32,360 --> 00:12:35,950
where by several you mean more
than one-- real variables is

213
00:12:35,950 --> 00:12:39,360
simply a function in
which the input is

214
00:12:39,360 --> 00:12:43,730
an n-tuple and the
output is a number.

215
00:12:43,730 --> 00:12:44,570
OK?

216
00:12:44,570 --> 00:12:46,530
That's what this whole
thing is all about.

217
00:12:46,530 --> 00:12:50,530
And because of that, when we
then abbreviate the n-tuple,

218
00:12:50,530 --> 00:12:53,150
we use x with a bar under it.

219
00:12:53,150 --> 00:12:54,570
Let's call it x-bar.

220
00:12:54,570 --> 00:12:57,360
Rather than x with
the arrow over it,

221
00:12:57,360 --> 00:13:00,070
since arrows may
be inappropriate.

222
00:13:00,070 --> 00:13:01,770
Now what do I mean
by inappropriate?

223
00:13:01,770 --> 00:13:04,180
Well I mean that even
in the case of one, two,

224
00:13:04,180 --> 00:13:06,530
or three dimensions,
you might be thinking

225
00:13:06,530 --> 00:13:10,670
of, say, the radius and
the height of a cylinder,

226
00:13:10,670 --> 00:13:12,500
rather than as an arrow.

227
00:13:12,500 --> 00:13:14,520
And in more than
three dimensions--

228
00:13:14,520 --> 00:13:17,670
for most of us at least--
it's difficult to visualize

229
00:13:17,670 --> 00:13:19,820
what we would mean by an arrow.

230
00:13:19,820 --> 00:13:22,460
So we just use the
bar underneath.

231
00:13:22,460 --> 00:13:25,520
Now again, the major
point is, notice this--

232
00:13:25,520 --> 00:13:27,160
I keep saying the major point.

233
00:13:27,160 --> 00:13:29,720
I guess there's a lot of
major points about this.

234
00:13:29,720 --> 00:13:34,120
Remember that we did not
call arrows "vectors".

235
00:13:34,120 --> 00:13:38,630
We did not call arrows "vectors"
until we defined a structure

236
00:13:38,630 --> 00:13:39,290
on the arrows.

237
00:13:39,290 --> 00:13:40,250
Remember what we did?

238
00:13:40,250 --> 00:13:42,790
We told what it meant for
two arrows to be equal,

239
00:13:42,790 --> 00:13:45,350
we told how we added
two arrows, and we

240
00:13:45,350 --> 00:13:48,230
told how we multiplied
an arrow by a scalar.

241
00:13:48,230 --> 00:13:53,650
In a similar way, we will
not call n-tuples a structure

242
00:13:53,650 --> 00:13:57,970
until we tell how to
equate a pair of n-tuples,

243
00:13:57,970 --> 00:14:00,270
how to add a pair,
and how to multiply

244
00:14:00,270 --> 00:14:01,960
an n-tuple by a number.

245
00:14:01,960 --> 00:14:04,670
By the way, the structure
that we wind up with

246
00:14:04,670 --> 00:14:08,490
is then called an
n-dimensional vector space,

247
00:14:08,490 --> 00:14:12,270
or more concisely, n-space.

248
00:14:12,270 --> 00:14:14,480
And the idea works
like this-- let's

249
00:14:14,480 --> 00:14:16,690
pick a particular value of n.

250
00:14:16,690 --> 00:14:18,270
Lets just call it n.

251
00:14:18,270 --> 00:14:23,200
And let S sub n be the
set of all n-tuples x_1

252
00:14:23,200 --> 00:14:25,410
up to x_n, in other
words, the set of what?

253
00:14:25,410 --> 00:14:29,850
All n-tuples of
numbers x_1 up to x_n.

254
00:14:29,850 --> 00:14:31,730
Let's pick two
particular members

255
00:14:31,730 --> 00:14:34,830
of S sub n, which we'll
call a-bar and b-bar.

256
00:14:34,830 --> 00:14:37,090
Where a-bar is simply
an abbreviation

257
00:14:37,090 --> 00:14:42,860
for the n-tuple a_1 up to a_n,
where the a_1, a_2 up to a_n,

258
00:14:42,860 --> 00:14:44,640
et cetera, are real numbers.

259
00:14:44,640 --> 00:14:48,100
And b-bar is an abbreviation
for the n-tuple b_1,

260
00:14:48,100 --> 00:14:52,580
et cetera, b_n, where
b_1 up through b_n

261
00:14:52,580 --> 00:14:54,870
are also real numbers.

262
00:14:54,870 --> 00:14:58,120
Now again, here's where
structure comes into play.

263
00:14:58,120 --> 00:15:01,810
We have already defined
an n-tuple arithmetic

264
00:15:01,810 --> 00:15:04,370
in terms of arrows
for the case when

265
00:15:04,370 --> 00:15:07,640
n is either one, two, or three.

266
00:15:07,640 --> 00:15:10,910
Based on what happens when
n is one, two, or three,

267
00:15:10,910 --> 00:15:13,480
we invent the
following definitions.

268
00:15:13,480 --> 00:15:17,570
First of all, we
invent the definition

269
00:15:17,570 --> 00:15:22,580
that a-bar equals b-bar means
that the components-- meaning

270
00:15:22,580 --> 00:15:23,080
what?

271
00:15:23,080 --> 00:15:26,230
The individual members of
the n-tuple-- the components

272
00:15:26,230 --> 00:15:30,680
of a-bar are equal to the
components of b-bar, component

273
00:15:30,680 --> 00:15:31,570
by component.

274
00:15:31,570 --> 00:15:34,730
In other words, a_1
is equal to b_1,

275
00:15:34,730 --> 00:15:36,710
a_2 is equal to b_2, et cetera.

276
00:15:36,710 --> 00:15:37,980
All the way up to what?

277
00:15:37,980 --> 00:15:40,117
a_n is equal to b_n.

278
00:15:40,117 --> 00:15:41,950
Now, in other words
again, what we're saying

279
00:15:41,950 --> 00:15:44,310
is that for two
n-tuples to be equal,

280
00:15:44,310 --> 00:15:48,420
by definition, they should be
equal component by component,

281
00:15:48,420 --> 00:15:50,550
and this is
motivated by the fact

282
00:15:50,550 --> 00:15:52,300
that we already know
that we've accepted

283
00:15:52,300 --> 00:15:55,580
this structural definition
for the case of arrows.

284
00:15:55,580 --> 00:16:01,950
Similarly, given two n-tuples
a-bar and b-bar, to add them,

285
00:16:01,950 --> 00:16:06,150
let me define that to be the
n-tuple that I get by adding

286
00:16:06,150 --> 00:16:08,140
component by component.

287
00:16:08,140 --> 00:16:11,480
In other words, to find the
first component of a-bar plus

288
00:16:11,480 --> 00:16:14,950
b-bar, I add the first
component of a-bar

289
00:16:14,950 --> 00:16:17,080
to the first component of b-bar.

290
00:16:17,080 --> 00:16:20,970
Noticing of course,
that this is what?

291
00:16:20,970 --> 00:16:21,690
Be careful here.

292
00:16:21,690 --> 00:16:23,530
This is one number.

293
00:16:23,530 --> 00:16:25,800
a_1 plus b_1 is one number.

294
00:16:25,800 --> 00:16:27,750
a_2 plus b_2 is a number.

295
00:16:27,750 --> 00:16:30,020
a_n plus b_n is a number.

296
00:16:30,020 --> 00:16:32,550
In other words, notice
that by this definition,

297
00:16:32,550 --> 00:16:37,040
the sum of two n-tuples
is again an n-tuple.

298
00:16:37,040 --> 00:16:41,780
And finally, to multiply
a scalar by an n-tuple,

299
00:16:41,780 --> 00:16:44,330
I will agree to
define that definition

300
00:16:44,330 --> 00:16:51,050
to mean that you multiply the
n-tuple component by component

301
00:16:51,050 --> 00:16:54,500
by that particular
scalar, or number.

302
00:16:54,500 --> 00:16:57,360
Notice again that
all I have done here

303
00:16:57,360 --> 00:17:02,930
is I have obtained these
three structural definitions

304
00:17:02,930 --> 00:17:07,020
from the equivalent
situations of arrows.

305
00:17:07,020 --> 00:17:10,740
And since everything that was
true about arrows followed

306
00:17:10,740 --> 00:17:13,310
from these three
basic definitions,

307
00:17:13,310 --> 00:17:18,450
any set of n-tuples that obeys
this particular structure will

308
00:17:18,450 --> 00:17:20,930
also behave like the arrows did.

309
00:17:20,930 --> 00:17:23,869
And that's why we call
it a vector space.

310
00:17:23,869 --> 00:17:27,770
They behave like vectors even
though they can no longer

311
00:17:27,770 --> 00:17:30,060
be viewed as arrows.

312
00:17:30,060 --> 00:17:32,010
And again there are
creative people who

313
00:17:32,010 --> 00:17:33,780
view these things as arrows.

314
00:17:33,780 --> 00:17:35,750
I remember feeling very
intimidated one day

315
00:17:35,750 --> 00:17:37,760
by my undergraduate
professor the first time

316
00:17:37,760 --> 00:17:39,210
I learned vector spaces.

317
00:17:39,210 --> 00:17:42,084
I said, how do you visualize
an n-dimensional vector space?

318
00:17:42,084 --> 00:17:44,250
And in full seriousness,
without batting an eyelash,

319
00:17:44,250 --> 00:17:46,890
he says, "I visualize it
like a porcupine with a bunch

320
00:17:46,890 --> 00:17:48,460
of quills coming out of it."

321
00:17:48,460 --> 00:17:51,020
And I knew that he knew what
was visualizing it like,

322
00:17:51,020 --> 00:17:52,850
but didn't help me one bit.

323
00:17:52,850 --> 00:17:55,800
I'm saying, if you can
visualize this things as arrows,

324
00:17:55,800 --> 00:17:56,650
be my guest.

325
00:17:56,650 --> 00:17:58,030
Feel free to do so.

326
00:17:58,030 --> 00:18:01,710
If you can't, notice that
every one of these definitions

327
00:18:01,710 --> 00:18:05,130
stands on its own two feet.

328
00:18:05,130 --> 00:18:08,220
Subject to the condition that
when n is one, two, or three,

329
00:18:08,220 --> 00:18:12,130
we happen to have a very nice
geometric interpretation.

330
00:18:12,130 --> 00:18:15,010
By the way, I may have
given you the impression

331
00:18:15,010 --> 00:18:18,320
that vector spaces were
invented because of functions

332
00:18:18,320 --> 00:18:19,640
of several variables.

333
00:18:19,640 --> 00:18:22,280
Rather, the impression I
would like to leave you with

334
00:18:22,280 --> 00:18:25,210
is, that in terms of motivating
vector spaces, in terms

335
00:18:25,210 --> 00:18:27,610
of this course, that
was the motivation

336
00:18:27,610 --> 00:18:29,050
that we elected to use.

337
00:18:29,050 --> 00:18:31,660
That the mathematician
talked about vector spaces

338
00:18:31,660 --> 00:18:33,890
in many a different
context from what

339
00:18:33,890 --> 00:18:35,994
we might even dream possible.

340
00:18:35,994 --> 00:18:38,160
In other words, I don't
have to think of temperature

341
00:18:38,160 --> 00:18:41,460
being a function of the four
variables x, y, z, and t.

342
00:18:41,460 --> 00:18:44,070
Let me give you a different
kind of non-trivial example

343
00:18:44,070 --> 00:18:49,230
of a four-space that doesn't
even bring functions into play.

344
00:18:49,230 --> 00:18:52,940
Let's suppose I invent
the abbreviation,

345
00:18:52,940 --> 00:18:56,210
I write the 4-tuple
(a_0, a_1, a_2,

346
00:18:56,210 --> 00:19:02,720
a_3) to denote the cubic
polynomial a_0 plus a_1*x plus

347
00:19:02,720 --> 00:19:05,540
a_2 x squared plus a_3 x cubed.

348
00:19:05,540 --> 00:19:09,670
Notice that I can use these
as a place value system.

349
00:19:09,670 --> 00:19:12,230
The first member tells
me my constant term,

350
00:19:12,230 --> 00:19:14,260
the second member tells
me the coefficient

351
00:19:14,260 --> 00:19:17,660
of x, the third member tells me
the coefficient of x squared,

352
00:19:17,660 --> 00:19:20,890
and the fourth number gives
me the coefficient of x cubed.

353
00:19:20,890 --> 00:19:24,790
Notice that for two polynomials
to be identically equal,

354
00:19:24,790 --> 00:19:27,280
they must be equal, what?

355
00:19:27,280 --> 00:19:29,110
Coefficient by coefficient.

356
00:19:29,110 --> 00:19:30,870
That means what?

357
00:19:30,870 --> 00:19:32,700
Component by component.

358
00:19:32,700 --> 00:19:34,740
How do we add two polynomials?

359
00:19:34,740 --> 00:19:38,080
We add them coefficient
by coefficient.

360
00:19:38,080 --> 00:19:39,200
We add like terms.

361
00:19:39,200 --> 00:19:42,830
In other words, given two
polynomials, we add them what?

362
00:19:42,830 --> 00:19:44,230
Component by component.

363
00:19:44,230 --> 00:19:47,080
We add the two constant terms
together, the two coefficients

364
00:19:47,080 --> 00:19:49,000
of x together, the
two coefficients

365
00:19:49,000 --> 00:19:52,050
of x squared together,
the two coefficients of x

366
00:19:52,050 --> 00:19:52,730
cubed together.

367
00:19:52,730 --> 00:19:53,630
You see?

368
00:19:53,630 --> 00:19:56,320
How do we multiply a
polynomial by a scalar?

369
00:19:56,320 --> 00:19:59,420
We multiply each
term by the scalar.

370
00:19:59,420 --> 00:20:02,340
That, in turn, is equivalent
to multiplying each coefficient

371
00:20:02,340 --> 00:20:03,610
by that scalar.

372
00:20:03,610 --> 00:20:06,120
And that says, in terms
of n-tuple notation,

373
00:20:06,120 --> 00:20:10,100
that we have multiplied each
component by that scalar.

374
00:20:10,100 --> 00:20:12,450
The set of polynomials
of degree n

375
00:20:12,450 --> 00:20:16,680
forms a very nice vector space
in terms of our definition

376
00:20:16,680 --> 00:20:18,150
of a vector space.

377
00:20:18,150 --> 00:20:21,670
Now, of course, the danger
is that one gets the idea

378
00:20:21,670 --> 00:20:25,250
that any set of n-tuples can
be viewed as a vector space.

379
00:20:25,250 --> 00:20:27,080
An n-dimensional vector space.

380
00:20:27,080 --> 00:20:29,050
But this we have to
be careful about.

381
00:20:29,050 --> 00:20:31,380
Remember, it is
not the n-tuples,

382
00:20:31,380 --> 00:20:33,380
it is structure that they obey.

383
00:20:33,380 --> 00:20:36,600
Let me give you sort of a
simple example over here.

384
00:20:36,600 --> 00:20:38,520
Let me consider the
following situation.

385
00:20:38,520 --> 00:20:40,520
First of all, let me just
emphasize a statement;

386
00:20:40,520 --> 00:20:42,450
I just made it, let me
just read it with you.

387
00:20:42,450 --> 00:20:46,640
n-tuples are not
automatically n-spaces.

388
00:20:46,640 --> 00:20:52,600
For example, let me invent the
2-tuple a comma b to represent

389
00:20:52,600 --> 00:20:54,940
the number a plus b.

390
00:20:54,940 --> 00:20:58,390
For example, if I define
the 2-tuple a comma

391
00:20:58,390 --> 00:21:00,980
b to be an abbreviation
for a plus b,

392
00:21:00,980 --> 00:21:05,127
What would 4 comma 5 denote?

393
00:21:05,127 --> 00:21:06,460
Remember the 2-tuple means what?

394
00:21:06,460 --> 00:21:08,910
To get the value of
the 2-tuple is just

395
00:21:08,910 --> 00:21:10,760
the sum of the components.

396
00:21:10,760 --> 00:21:16,610
If I add a and b, in this
case, 4 plus 5 happens to be 9.

397
00:21:16,610 --> 00:21:21,360
How about the 2-tuple 6 comma
3, what value would that have?

398
00:21:21,360 --> 00:21:24,210
That would also
have the value 9.

399
00:21:24,210 --> 00:21:27,720
Therefore numerically,
the 2-tuple 4 comma 5

400
00:21:27,720 --> 00:21:30,870
is equal to the
2-tuple 6 comma 3.

401
00:21:30,870 --> 00:21:33,740
Yet notice that the
first component is not

402
00:21:33,740 --> 00:21:35,690
equal to the first
component here.

403
00:21:35,690 --> 00:21:38,630
In other words 4
is not equal to 6.

404
00:21:38,630 --> 00:21:42,610
Nor is 5 is equal
to 3, but if I were

405
00:21:42,610 --> 00:21:44,950
to choose this
definition of equality,

406
00:21:44,950 --> 00:21:47,940
I could not say that
these 2-tuples form

407
00:21:47,940 --> 00:21:51,770
a two-dimensional vector
space, because it violates

408
00:21:51,770 --> 00:21:54,910
the first definition for
a vector space, namely

409
00:21:54,910 --> 00:21:59,710
the definition of what it means
for two vectors to be equal.

410
00:21:59,710 --> 00:22:01,790
Since we're going to
let most of our material

411
00:22:01,790 --> 00:22:06,150
be covered by the exercises
and the supplementary notes,

412
00:22:06,150 --> 00:22:09,850
and this is just to be an
overview, let's move on now.

413
00:22:09,850 --> 00:22:13,020
Let's assume that we now know
what n-dimensional vector

414
00:22:13,020 --> 00:22:14,200
spaces are like.

415
00:22:14,200 --> 00:22:17,850
We now know that we can view
functions of several variables

416
00:22:17,850 --> 00:22:22,510
as functions that map n-tuples
into numbers, and as a result,

417
00:22:22,510 --> 00:22:25,600
it now makes sense to
talk about things like:

418
00:22:25,600 --> 00:22:28,620
suppose you were given the
n-tuple (x_1, x_2, x_3,

419
00:22:28,620 --> 00:22:33,420
x_3) and suppose that under
f, that n-tuple was mapped

420
00:22:33,420 --> 00:22:39,430
into x_1 cubed plus x_2
plus x_3 squared plus 2*x_4.

421
00:22:39,430 --> 00:22:46,230
For example, if I were to
replace x_1 by 1, x_2 by 3,

422
00:22:46,230 --> 00:22:52,170
x_3 by 1, and x_4 by 2, I would
arrive at the result what?

423
00:22:52,170 --> 00:22:57,130
1 cubed plus 3 plus 1
squared plus 2 times 2,

424
00:22:57,130 --> 00:22:59,130
and I can compute that output.

425
00:22:59,130 --> 00:23:01,370
Now the question that
comes up in calculus is,

426
00:23:01,370 --> 00:23:03,250
can we talk about limits here?

427
00:23:03,250 --> 00:23:08,280
Instead of computing what f of
1 comma 3 comma 1 comma 2 is,

428
00:23:08,280 --> 00:23:13,000
can I compute the limit of this
thing as (x_1, x_2, x_3, x_4)

429
00:23:13,000 --> 00:23:17,390
approaches 1 comma
3 comma 1 comma 2?

430
00:23:17,390 --> 00:23:21,320
I think intuitively it's clear
that since equality means

431
00:23:21,320 --> 00:23:24,590
that you must have equality
component by component,

432
00:23:24,590 --> 00:23:27,850
to say that this
approaches this means

433
00:23:27,850 --> 00:23:30,280
that the first component
here must approach

434
00:23:30,280 --> 00:23:31,810
the first component here.

435
00:23:31,810 --> 00:23:33,420
The second component
here approaches

436
00:23:33,420 --> 00:23:35,540
is the second component
here, et cetera.

437
00:23:35,540 --> 00:23:37,960
In other words, this
could be replaced

438
00:23:37,960 --> 00:23:41,440
by the four separate linear
one-dimensional limit

439
00:23:41,440 --> 00:23:47,400
problems: x_1 approaches 1, x_2
approaches 3, x_3 approaches 1,

440
00:23:47,400 --> 00:23:49,090
and x_4 approaches 2.

441
00:23:49,090 --> 00:23:51,360
And we would then be
tempted to say what?

442
00:23:51,360 --> 00:23:59,290
We will replace x_1 by 1,
x_2 by 3, x_3 by 1, x_4 by 2,

443
00:23:59,290 --> 00:24:01,610
see what happens
to this expression.

444
00:24:01,610 --> 00:24:04,750
And we would then be tempted to
say that this particular limit

445
00:24:04,750 --> 00:24:07,390
was equal to 9.

446
00:24:07,390 --> 00:24:11,020
Now the interesting point is
this-- that traditionally,

447
00:24:11,020 --> 00:24:15,510
this particular problem was
tackled long before anyone

448
00:24:15,510 --> 00:24:17,700
invented vector spaces.

449
00:24:17,700 --> 00:24:19,830
Or at least long
before anybody was

450
00:24:19,830 --> 00:24:23,290
serious about vector spaces.

451
00:24:23,290 --> 00:24:25,490
People did say,
why can't we reduce

452
00:24:25,490 --> 00:24:27,680
the study of
four-dimensional space

453
00:24:27,680 --> 00:24:30,140
to four separate studies
of one-dimensional space?

454
00:24:30,140 --> 00:24:33,880
In other words, let x_1 approach
1, x_2 approach 3, in that case

455
00:24:33,880 --> 00:24:35,030
you're allowing what?

456
00:24:35,030 --> 00:24:40,650
Four separate one-dimensional
limits to be taking place here.

457
00:24:40,650 --> 00:24:43,540
But the insight that
modern math gave

458
00:24:43,540 --> 00:24:46,720
us was that we can now go back
to our traditional definition

459
00:24:46,720 --> 00:24:47,220
of limit.

460
00:24:47,220 --> 00:24:50,570
Remember what was our old
structural definition of limit?

461
00:24:50,570 --> 00:24:53,140
Way back from the
first time we had it.

462
00:24:53,140 --> 00:24:56,850
The limit of f of x as x
approaches a equals L means,

463
00:24:56,850 --> 00:25:00,410
given epsilon greater than
0, we can find delta greater

464
00:25:00,410 --> 00:25:04,680
than zero, such that whenever
the absolute value of x minus a

465
00:25:04,680 --> 00:25:06,960
is greater than 0
but less than delta,

466
00:25:06,960 --> 00:25:10,670
the absolute value of f of x
minus L is less than epsilon

467
00:25:10,670 --> 00:25:12,740
Now here's what
the new math said.

468
00:25:12,740 --> 00:25:15,130
The modern approach
said look, let's just

469
00:25:15,130 --> 00:25:16,760
take our old
structural definition--

470
00:25:16,760 --> 00:25:19,580
the same as before-- and
vectorize everything.

471
00:25:19,580 --> 00:25:22,010
Notice in this situation
that we're dealing with,

472
00:25:22,010 --> 00:25:24,670
the input is where we have
the several variables.

473
00:25:24,670 --> 00:25:28,230
The input is the
n-tuple, the vector, OK?

474
00:25:28,230 --> 00:25:30,340
And the output is the scalar.

475
00:25:30,340 --> 00:25:33,100
So f is a scalar, L is a scalar.

476
00:25:33,100 --> 00:25:36,670
But x and a are
vectors, so every place

477
00:25:36,670 --> 00:25:41,340
I see an x and an a, I have
to put the bar underneath.

478
00:25:41,340 --> 00:25:44,380
And now I read this definition,
and all of a sudden,

479
00:25:44,380 --> 00:25:46,940
as so often has happened
in our course up to now,

480
00:25:46,940 --> 00:25:49,480
I come to something that
I've never seen before.

481
00:25:49,480 --> 00:25:53,220
Namely, as soon
as I look at this.

482
00:25:53,220 --> 00:25:57,200
This made very good sense
when these were arrows.

483
00:25:57,200 --> 00:25:59,760
Namely, we talked
about this earlier

484
00:25:59,760 --> 00:26:01,540
in our course and
one of our lectures.

485
00:26:01,540 --> 00:26:04,410
That to say the two arrows
were near each other

486
00:26:04,410 --> 00:26:06,790
was to say that their
difference was small,

487
00:26:06,790 --> 00:26:09,290
and that in turn said if
the arrows were placed tail

488
00:26:09,290 --> 00:26:12,480
to tail, we could make the
distance between their heads

489
00:26:12,480 --> 00:26:14,130
as small as we wish.

490
00:26:14,130 --> 00:26:17,650
Now the price that we have
to pay for higher dimensions

491
00:26:17,650 --> 00:26:20,010
is that if we have more
dimensions than what

492
00:26:20,010 --> 00:26:23,240
we can draw arrows in, the
problem that we're faced

493
00:26:23,240 --> 00:26:25,910
is that we have not
defined what you

494
00:26:25,910 --> 00:26:31,380
mean by the magnitude of x
minus a, where x and a happen

495
00:26:31,380 --> 00:26:33,710
to be n-tuples.

496
00:26:33,710 --> 00:26:36,480
And here again, we come
back to our structure.

497
00:26:36,480 --> 00:26:40,120
But now for the first time the
structure is not redundant.

498
00:26:40,120 --> 00:26:42,530
Let me tell you
what I mean by that.

499
00:26:42,530 --> 00:26:44,920
In the one-dimensional
case we define

500
00:26:44,920 --> 00:26:49,860
the magnitude of x minus a to
be the square root of x_1 minus

501
00:26:49,860 --> 00:26:52,220
a_1 squared, where
the vector x was

502
00:26:52,220 --> 00:26:57,640
the 1-tuple x_1, and the
vector a was the n-tuple a_1.

503
00:26:57,640 --> 00:26:59,710
In the two-dimensional
case, we said,

504
00:26:59,710 --> 00:27:03,910
OK, let's define the
magnitude of the vector

505
00:27:03,910 --> 00:27:07,590
x minus a to be x_1
minus a_1 squared

506
00:27:07,590 --> 00:27:10,370
plus x_2 minus a_2 squared.

507
00:27:10,370 --> 00:27:11,820
And in the
three-dimensional case,

508
00:27:11,820 --> 00:27:16,460
we said, let's define the
magnitude of x-bar minus a-bar

509
00:27:16,460 --> 00:27:19,740
to be the square root
of x_1 minus a_1 squared

510
00:27:19,740 --> 00:27:24,260
plus x_2 minus a_2 squared
plus x_3 minus a_3 squared.

511
00:27:24,260 --> 00:27:28,690
At that time, I kept saying,
notice that these recipes do

512
00:27:28,690 --> 00:27:32,620
not depend on a picture, that
these are numerical results

513
00:27:32,620 --> 00:27:34,060
that we can compute
without having

514
00:27:34,060 --> 00:27:35,550
to draw a picture at all.

515
00:27:35,550 --> 00:27:39,460
What happened of course was that
in the one-dimensional case,

516
00:27:39,460 --> 00:27:42,890
in the two-dimensional case,
in the three-dimensional case,

517
00:27:42,890 --> 00:27:46,360
it was easier to
visualize the picture.

518
00:27:46,360 --> 00:27:49,240
Now here's where the
real kicker comes in--

519
00:27:49,240 --> 00:27:52,270
and this is the real crucial
point-- structurally,

520
00:27:52,270 --> 00:27:54,780
can't you see what's
happening over here?

521
00:27:54,780 --> 00:27:58,360
Can't you see how I can now
define the absolute value

522
00:27:58,360 --> 00:28:04,850
of the vector x-bar minus a-bar,
even if n is greater than 3,

523
00:28:04,850 --> 00:28:06,820
in such a way that
the definition will

524
00:28:06,820 --> 00:28:11,620
make sense and still mimic
everything that we're doing?

525
00:28:11,620 --> 00:28:14,100
I hope you are a
step ahead of me

526
00:28:14,100 --> 00:28:17,730
on this except for some new
notation I introduced here.

527
00:28:17,730 --> 00:28:20,290
It turns out that in
the modern math book,

528
00:28:20,290 --> 00:28:24,960
one distinguishes between
the absolute value

529
00:28:24,960 --> 00:28:28,290
of a number, and the
magnitude of a vector,

530
00:28:28,290 --> 00:28:34,140
and it is frequently traditional
to introduce a double bar

531
00:28:34,140 --> 00:28:39,010
on each side to represent the
magnitude of the difference

532
00:28:39,010 --> 00:28:40,140
between two vectors,

533
00:28:40,140 --> 00:28:42,700
which I claim behaves
like a distance.

534
00:28:42,700 --> 00:28:44,510
Let me show you
what I mean by that.

535
00:28:44,510 --> 00:28:48,820
Let me define the magnitude
of the n-tuple x-bar

536
00:28:48,820 --> 00:28:53,400
minus the n-tuple a-bar,
written this way, to be

537
00:28:53,400 --> 00:28:57,110
the positive square root
of x_1 minus a_1 squared,

538
00:28:57,110 --> 00:29:02,710
plus et cetera, plus
x_n minus a_n squared.

539
00:29:02,710 --> 00:29:04,720
The thing that I would
like you to notice here

540
00:29:04,720 --> 00:29:08,850
is that since each of these
numbers are non-negative--

541
00:29:08,850 --> 00:29:10,850
see they're squares
of real numbers--

542
00:29:10,850 --> 00:29:14,600
the only way this can be 0--
well the only way that the sum

543
00:29:14,600 --> 00:29:17,860
of squares of non-negative
numbers can be 0

544
00:29:17,860 --> 00:29:20,080
is for each of the
numbers to be 0.

545
00:29:20,080 --> 00:29:23,860
Consequently, the only way the
magnitude of x-bar minus a-bar

546
00:29:23,860 --> 00:29:28,710
can equal 0 is if x_1 equals
a_1, x_2 equals a_2, et cetera,

547
00:29:28,710 --> 00:29:31,060
and x_n equals a_n.

548
00:29:31,060 --> 00:29:35,900
In this vein, notice that
the geometric phrase x-bar

549
00:29:35,900 --> 00:29:39,870
near a-bar still makes sense.

550
00:29:39,870 --> 00:29:41,760
It doesn't make
sense pictorially,

551
00:29:41,760 --> 00:29:44,955
because we can't draw the
arrows if n is greater than 3.

552
00:29:44,955 --> 00:29:46,330
But notice that
what we're saying

553
00:29:46,330 --> 00:29:49,630
is that for x to be
near a, all we're saying

554
00:29:49,630 --> 00:29:53,140
is that the magnitude--
defined this way--

555
00:29:53,140 --> 00:29:57,420
the magnitude of x-bar
minus a-bar is small.

556
00:29:57,420 --> 00:30:00,010
When you're adding
up positive squares,

557
00:30:00,010 --> 00:30:03,700
the only way the sum can be
small is if each of the factors

558
00:30:03,700 --> 00:30:04,640
are small.

559
00:30:04,640 --> 00:30:07,580
But notice what these factors
are, except for the square,

560
00:30:07,580 --> 00:30:11,440
it's the difference between
x_1 and a_1, x_2 and a_2,

561
00:30:11,440 --> 00:30:13,730
et cetera, x_n and a_n.

562
00:30:13,730 --> 00:30:17,320
In other words, to say that
x-bar is near a-bar means

563
00:30:17,320 --> 00:30:21,400
that x_1 is near a_1, x_2
is near a_2, et cetera,

564
00:30:21,400 --> 00:30:24,170
and x_n is near a_n,
which is exactly

565
00:30:24,170 --> 00:30:25,730
the traditional approach.

566
00:30:25,730 --> 00:30:28,960
And in fact, except for the
fact that we can capitalize

567
00:30:28,960 --> 00:30:31,340
on structure,
notice that once we

568
00:30:31,340 --> 00:30:34,840
define the magnitude of
the difference between two

569
00:30:34,840 --> 00:30:37,170
n-tuples-- do you
notice that by the way?

570
00:30:37,170 --> 00:30:41,490
The magnitude of the difference
of two n-tuples is a number.

571
00:30:41,490 --> 00:30:45,240
Notice now if we replace
this fancy phrase--

572
00:30:45,240 --> 00:30:47,830
which we didn't know
the meaning of before,

573
00:30:47,830 --> 00:30:52,340
but which we now know--
by its new definition,

574
00:30:52,340 --> 00:30:56,420
we obtain the traditional
definition of limit.

575
00:30:56,420 --> 00:31:01,840
Namely the limit of f of x_1 up
to x_n, as x_1 approaches a_1,

576
00:31:01,840 --> 00:31:03,630
et cetera, and x_n
approaches a_n,

577
00:31:03,630 --> 00:31:07,830
equals L means that given
epsilon greater than 0,

578
00:31:07,830 --> 00:31:10,930
we can find delta
greater than 0 such

579
00:31:10,930 --> 00:31:14,040
that whenever the square
root of x_1 minus a_1

580
00:31:14,040 --> 00:31:18,630
squared plus et cetera x_n minus
a_n squared is less than delta

581
00:31:18,630 --> 00:31:21,460
but greater than 0, then
the magnitude-- you see,

582
00:31:21,460 --> 00:31:23,330
these numbers
here-- the magnitude

583
00:31:23,330 --> 00:31:28,010
of f of x_1 up to x_n minus
L is less than epsilon.

584
00:31:28,010 --> 00:31:30,300
In, other words,
this definition here

585
00:31:30,300 --> 00:31:33,940
happens to be the
traditional definition.

586
00:31:33,940 --> 00:31:34,670
OK?

587
00:31:34,670 --> 00:31:38,340
But the point is that the
traditional definition

588
00:31:38,340 --> 00:31:42,330
has exactly the same structure
as the modern definition.

589
00:31:42,330 --> 00:31:44,820
And as a result, to make
fun of the traditional math

590
00:31:44,820 --> 00:31:47,230
because it's not as
pretty as the modern math

591
00:31:47,230 --> 00:31:49,070
is the wrong thing to say.

592
00:31:49,070 --> 00:31:51,340
It's like the fellow who
once asked me at a PTA

593
00:31:51,340 --> 00:31:54,680
meeting, "how much is 8 plus
7 in the new mathematics?"

594
00:31:54,680 --> 00:31:56,550
That part hasn't changed.

595
00:31:56,550 --> 00:31:59,380
The beauty of using
the n-tuple notation

596
00:31:59,380 --> 00:32:03,370
was that it allowed us to
use the previous structure

597
00:32:03,370 --> 00:32:04,180
of limits.

598
00:32:04,180 --> 00:32:07,350
So that we can get all of our
theorems, all of our formulas

599
00:32:07,350 --> 00:32:10,470
and what have you, to go
through word for word,

600
00:32:10,470 --> 00:32:14,640
even though the higher the
dimension, the more complex

601
00:32:14,640 --> 00:32:15,950
our computations are.

602
00:32:15,950 --> 00:32:19,600
But structurally, It
essentially boiled down to,

603
00:32:19,600 --> 00:32:21,990
after you've seen
one-dimensional space,

604
00:32:21,990 --> 00:32:23,160
you've seen them all.

605
00:32:23,160 --> 00:32:26,100
That was the big innovation
with the modern approach

606
00:32:26,100 --> 00:32:28,100
to n-dimensional vector spaces.

607
00:32:28,100 --> 00:32:30,540
And to help put this
in proper perspective,

608
00:32:30,540 --> 00:32:34,100
next time I shall
introduce the calculus

609
00:32:34,100 --> 00:32:38,000
of several real variables
in terms of the more

610
00:32:38,000 --> 00:32:39,430
traditional approach.

611
00:32:39,430 --> 00:32:44,220
But again, until
next time, good bye.

612
00:32:44,220 --> 00:32:46,590
Funding for the
publication of this video

613
00:32:46,590 --> 00:32:51,470
was provided by the Gabriella
and Paul Rosenbaum foundation.

614
00:32:51,470 --> 00:32:55,640
Help OCW continue to provide
free and open access to MIT

615
00:32:55,640 --> 00:33:03,350
courses by making a donation
at MIT ocw.mit.edu/donate.