5
DERIVATIVES I N N-DIMENSIONAL VECTOR SPACES
A
Introduction
,
, . -
-'
*
-..:
The approach of o u r textbook ( a s d&ioped
i n t h e previous two u n i t s )
i n emphasizing t h e g e o m e t r i c a l a s p e c t s of p a r t i a l d e r i v a t i v e s ,
d i r e c t i o n a l d e r i v a t i v e s , and t h e g r a d i e n t i s , i n many ways, an
e x c e l l e n t approach. W e g e t a p i c t u r e of what appears t o be happening,
and, e s p e c i a l l y f o r t h e beginning s t u d e n t , t h i s i s much more
meaningful than a more p r e c i s e and g e n e r a l d i s c u s s i o n t h a t would
apply t o a l l v e c t o r spaces, r e g a r d l e s s of dimension. I n f a c t , i f we
want t o use t h e l a t t e r approach, it i s w i s e t o use t h e former f i r s t .
This i s i n accord w i t h many such d e c i s i o n s which have a l r e a d y been
made .in o u r p r e v i o u s t r e a t m e n t of mathematics. A s an example, w i t h i n
t h e framework of o u r p r e s e n t course, n o t i c e how c a r e f u l l y we exp l o i t e d t h e geometric a s p e c t s of v e c t o r s a s arrows b e f o r e we even
mentjoned n-dimensional v e c t o r spaces. And, i n t h i s r e s p e c t , r e c a l l
how s o p h i s t i c a t e d many of o u r "arrow" r e s u l t s were i n t h e i r own r i g h t .
That i s , . g r a n t e d t h a t arrows were s i m p l e r than n-tuples, when a l l we
had w e r e arrows, t h e r e w e r e s t i l l some r a t h e r d i f f i c u l t i d e a s f o r u s
t o assimilate.
I n any e v e n t , o u r aim i n t h i s c h a p t e r i s t o continue w i t h i n t h e
s p i r i t of t h e game of mathematics which, thus f a r , has served us s o
w e l l . Namely, we have introduced n-dimensional v e c t o r spaces s o t h a t
w e could u t i l i z e t h e s i m i l a r i t y i n form between t h e previously
s t u d i e d f u n c t i o n s of t h e form f ( x ) and o u r new f u n c t i o n s of t h e form
f ( x ) . For example, when i t came time t o d e f i n e c o n t i n u i t y and l i m i t s
f o r f u n c t i o n s of s e v e r a l r e a l v a r i a b l e s , w e rewrote f ( x l ,
xn) a s
...,
f ( 5 ) and t h e n mimicked t h e f (x) s i t u a t i o n by copying d e f i n i t i o n s
verbatim, s u b j e c t only t o making the a p p r o p r i a t e " v e c t o r i z a t i o n s . "
Rather than review this procedure, l e t us i n s t e a d move t o t h e n e x t
p l a t e a u . Using t h e approach developed i n Chapter 4 of t h e s e
supplementary n o t e s , once w e had d e f i n e d what l i m i t s and c o n t i n u i t y
meant f o r any f u n c t i o n , f :E"+E, o u r n e x t l o g i c a l s t e p would have been
t o mimic o u r d e f i n i t i o n of f ' ( a ) , and t h u s "induce" a meaning of f ' (5)
where a e ~ n .
I
.
'
.
,
.
Our aim i n t h i s c h a p t e r i s t o s e e where such an a t t e m p t would have
l e d us and, once t h i s i s done, t o s e e how our "new" concept of
d i f f e r e n t i a t i o n i s r e l a t e d t o t h e more t r a d i t i o n a l concept of
d i f f e r e n t i a t i o n a s i t was p r e s e n t e d i n t h e previous two u n i t s .
B
A New
Type of Q u o t i e n t
From t h e d e f i n i t i o n
£1 1.
=
1
" a + ~ xAX) - f ( a )
1
it seems n a t u r a l t h a t i f now f:En+E and afzEn, we should d e f i n e f ' (a)
where (2) i s o b t a i n e d from (1) by t h e a p p r o p r i a t e " v e c t o r i z a t i o n s .
"
I n a r r i v i n g a t (21, t h e most s u b t l e p o i n t i s t o observe t h a t we must
- s i n c e , among o t h e r t h i n g s , a+Ax h a s no meaning
r e p l a c e Ax by Ax,
a ] t o a r e a l number
(i.e . , we have no r u l e f o r adding a v e c t o r [- i t i s c l e a r t h a t 0 must be replaced
[AX]). Once Ax i s r e p l a c e by Ax,
by 0.
I f w e n e x t look a t t h e bracketed e x p r e s s i o n i n ( 2 ) , w e g e t a r a t h e r
e l e g a n t i n s i g h t a s t o how new mathematical concepts a r e o f t e n born.
For, w h i l e a t f i r s t g l a n c e , t h e d e r i v a t i o n of (2) from (1) may s e e m
harmless, a second g l a n c e shows us t h a t w e have "invented" an
o p e r a t i o n which we have never encountered b e f o r e i n o u r study of
v e c t o r s ( e i t h e r a s arrows o r a s n - t u p l e s ) .
More s p e c i f i c a l l y , i n
t h e b r a c k e t e d e x p r e s s i o n , our numerator i s a r e a l number ( i . e . ,
while a i s a v e c t o r [n-tuple] , r e c a l l t h a t o u r n o t a t i o n f (5) i n d i c a t e s
t h a t our "output" i s a number), and o u r denominator i s a vector!
*
Thus, whether w e l i k e i t o r n o t , i f w e decide t o a c c e p t (2) a s a
C
s t a r t i n g p o i n t , w e a r e o b l i g e d t o i n v e s t i g a t e t h e meaning of
-
where c denotes an a r b i t r a r y b u t f i x e d r e a l number and v an a r b i t r a r y
b u t f i x e d v e c t o r . Notice, of course, t h a t we could e l e c t t o abandon
( 2 ) , b u t , i n t h e u s u a l s p i r i t of t h i n g s , it h a r d l y seems wise t o
abandon an approach t h a t has been f r u i t f u l simply because one
p o t e n t i a l problem a r r i s e s . Rather, it seems w i s e r f o r us t o t r y ,
a t l e a s t , t o f i n d a p r a c t i c a l d e f i n i t i o n of what i t means t o d i v i d e
a number by a v e c t o r .
Once we have agreed on t h i s c o u r s e of a c t i o n , we again f a l l back
on o u r p r e v i o u s experience and r e c a l l how we d e f i n e d d i v i s i o n i n t h e
c a s e of two numbers. We d e f i n e d
t o be t h a t number which when
m u l t i p l i e d by b y i e l d e d a. In terms o f a more computational form,
a was d e f i n e d by
* .
l
b "times"
.
9
;= a
With t h i s i n mind, it seems t h a t a v e r y n a t u r a l way t o mimic ( 3 ) i s
C
t o define by
v
-
,
"
;:i
-v " t i p e s t ' --vC =
c
. \
4
1
.
. .
." ,*.
.., ...
'*
s
(4)
T h i s immediately s u g g e s t s t h a t u n l e s s we want t o i n v e n t new forms of
m u l t i p l i c a t i o n , (and n o t h i n g p r e c l u d e s t h i s p o s s i b i l i t y , e x c e p t t h a t
we a l r e a d y have enough problems1 t h e "times" i n ( 4 ) must denote t h e
d o t product, f o r according t o ( 4 ) we must m u l t i p l y a v e c t o r (v)
- by
"something" t o o b t a i n a number, and of t h e types of m u l t i p l i c a t i o n w e
have d i s c u s s e d , only t h e d o t product allows u s t o m u l t i p l y a v e c t o r
by "something" ( i n f a c t , by a n o t h e r v e c t o r ) t o o b t a i n a number.
Thus, t h e "times" i n ( 4 ) must denote a d o t product. And, i t t h e r e f o r e
C
appears t h a t v must, i t s e l f , be a v e c t o r , s i n c e , a s we have j u s t
s a i d , t h e dot-product combines two v e c t o r s t o produce a number.
There i s , however, a l i t t l e complication t h a t p r e s e n t s i t s e l f here.
The problem a c t u a l l y e x i s t e d when we t a l k e d about t h e q u o t i e n t of
two numbers, b u t , e x c e p t i n t h e r a t h e r s p e c i a l c a s e i n which t h e
denominator was zero, t h e problem never occurred. The p o i n t we a r e
i s +he number which when m u l t i p l i e d
d r i v i n g a t i s t h a t when we say
by b y i e l d s a , " what g i v e s us t h e r i g h t t o say "t h e ? " Why c a n ' t
t h e r e be more than one such number, o r f o r t h a t m a t t e r , why n o t no
such number?
";
The language of s e t s provides u s with a n i c e e x p l a n a t i o n .
Suppose.:
we define 2
set of a l l numbers, c , such t h a t hxc = a.
b t o be t h e - .
is,
-.
~.
,
That
c
s e t a c o n s i s t s of the s i n g l e 'number a*
Then, u n l e s s b = 0'. t h e 6 = {c:3c=6) = {6;3) = (211, and hence t h e r e i s no harm i n
(e.g.,
where i n t h e l a t t e r c o n t e x t ,
confusing t h e set w i t h t h e number
i s used a s t h e f r a c t i o n which denotes a*.
E,
Mimicking ( 5 ) we may a l s o d e f i n e 2 t o be a set; namely
V C
The problem i s t h a t , u n l e s s 1= g and c # 0 ( i n which c a s e 2
= $), t h e
set v has i n f i n i t e l y many elements!
Before we document t h i s l a s t remark, l e t u s make s u r e t h a t it i s
-
-
c l e a r t o you t h a t t h e s e t d e s c r i b e d i n (6) i s a well-defined s e t
r e g a r d l e s s of t h e dimension of t h e v e c t o r space. I n o t h e r words,
r e c a l l t h a t i n t h e l a s t c h a p t e r of t h e s e supplementary n o t e s , we
g e n e r a l i z e d t h e d e f i n i t i o n of a d o t product s o t h a t i t e x i s t e d i n
any dimensional space. By way of a b r i e f review, i f
= (xl,
xn
and
y = (yl
,...,yn)
x
then x - y = xlyl+
...,
...+ xnyn'
4
Thus, by way of an example, suppose c = 5 and v = (1,2,3,4)~E
Then by (61,
.
*
I f b - 0 , we h a v e s e e n t h a t i f a#O t h e n =I:
s i n c e no number t i m e s
0 c a n y i e l d a n o n - z e r o number, and i f b=O and a=O, we h a v e s e e n t h a t
a
0
( i . e . , ),
is zero.
i s the set of
numbers, s i n c e any number t i m e s z e r o
.
Notice t h a t while it may be d i f f i c u l t t o t h i n k of
pictorially,
(1,2,3,4 )
( 7 ) shows u s t h a t it i s a "very r e a l " set, a t l e a s t i n t h e sense t h a t
i t i s t h e s o l u t i o n s e t of t h e "very r e a l * l i n e a r equation i n f o u r
unknowns
I n f a c t , it might seem more n a t u r a l t o you i f we r e s t a t e d t h i s l a s t
remark i n s o r t of a 'reverse" way. Suppose i n a t r a d i t i o n a l math
course w e w e r e asked t o f i n d a l l s o l u t i o n s of t h e equation given by
( 8 ) . Among o t h e r t h i n g s , we may pick v a l u e s f o r x 2 , x3, and x4
completely a t random, and once t h e s e random c h o i c e s a r e made, xl
can t h e n be u n i q u e l y determined from (8) simply by l e t t i n g
x 1 = 5 - 2x2 - 3x3 - 4x4. I n modern language, t h i s means t h a t t h e
s o l u t i o n s e t of (8) has i n f i n i t e l y many elements, and, by (7),
a n o t h e r name f o r t h i s i n f i n i t e s o l u t i o n s e t i s
5
(1,2,3,4)
I n s t i l l o t h e r words, while w e may n o t be used t o t h i n k i n g of t h e
s o l u t i o n s of an e q u a t i o n l i k e (8) a s being p o i n t s i n a 4-dimensional
v e c t o r space, t h e f a c t i s t h a t , conceptually, t h e i d e a i s sound. W e
admit, however, t h a t i n t h e s p i r i t of t h e t e x t , t h e r e i s probably
more s a t i s f a c t i o n i f w e t h i n k of t h e s p e c i a l c a s e s i n which we may
*
view o u r v e c t o r s ( n - t u p l e s ) a s arrows and s e e what t h e geometric
i m p l i c a t i o n s a r e . For t h e sake of a b i t of s i m p l i c i t y (and we s h a l l
show i n a moment t h a t t h i s r e s t r i c t i o n i n no way l o s e s any g e n e r a l i t y ) ,
let u s assume t h a t i n (6),
denotes a u n i t v e c t o r (because i f v
is a u n i t vector,
i s simply t h e p r o j e c t i o n o f x i n the direction
i s n o t a u n i t v e c t o r , w e must merely come t o g r i p s
of v while i f
w i t h t h e more computational f a c t t h a t x 0-v i s a s c a l a r m u l t i p l e of ',
t h e p r o j e c t i o n of 5 i n t h e d i r e c t i o n of x).
x * ~
v
So, under t h e assumption t h a t v i s a u n i t v e c t o r and t h a t our v e c t o r s
a r e now arrows ( i n t h e diagrams which follow, we view our v e c t o r s
a s p l a n a r a r r o w s ) , we may view
v
a s t h e s e t of a l l v e c t o r s whose p r o j e c t i o n i n t h e d i r e c t i o n of
is c.
C l e a r l y , t h e r e a r e i n f i n i t e l y many such v e c t o r s and t h i s i s amplified
i n Figure 1.
S
-
Thus pkl, E?R~ e t c . a l l have
-k
the property t h a t v*PRn = C ,
when Rn i s any p o i n t on L.
2.
Therefore, according t o (61,
any v e c t o r of t h e form
is
an acceptable value of
3.
%
IP%I
= c. Then i f R i s
any p o i n t on t h e l i n e L through
+
Q perpendicular t o
PR*v
- = c,
since v is a u n i t vector.
1.
Let
v,
(Figure I )
To see Figure 1 ih terms of a more s p e c i f i c example, consider t h e
s e t defined by
~.
2
*=
+
+
{v:
v - 1* = 23
1 -+
*
But v * i~s t h e p r o j e c t i o n of
t
i n t h e d i r e c t i o n of t h e x-axis.
+
-*-t
Thus,
For any such v, v - 1 = 2; hence,
any such f q u a l i f i e s a s a
member of 2
.
1 (Figure 2 )
Our r e s u l t s a r e n o t s e r i o u s l y a f f e c t e d i f
V -
key l i e s i n t h e f a c t t h a t i f v
-# 0,
d i r e c t i o n of
v.
I n this e v e n t we
lkll
have:
v
is not a u n i t vector.
is a u n i t vector
:.
ThusI from ( 9 ) w e see t h a t i n g e n e r a l , f o r v
-# 0,
v e c t o r s whose p r o j e c t i o n i n t h e d i r e c t i o n of
s p e c i a l case t h a t
I n t h e e v e n t 1=
i s a u n i t vector,
0, Ikll= O
Ikibl,
and accordingly
v
*
The
i n the
\.('
i s the s e t o.k9'all
'
V C
is , and 2n t h e
and
leu
C
-
i s simply c.
lkll
C
C
= -,
0 which
llvll
i s the
"forbidden" q u o t i e n t of two numbers. A s a s m a l l a s i d e , observe t h a t
if v = g,
behaves a s i t s numeral c o u n t e r p a r t . Namely,
5-
bC = 15: zag= c}.
-
x -0 = 0 [i.e.,
But -
(Xl , . . . ~ n ) * ( O C
= 01; hence, i f c # 0, 8
=
-e n t i r e v e c t o r space.
= O+...+O
I, while i f c
,...,0 )
= xlO+
...+xnO
C
= 0, 5 i s , t h e 7 - - -
Now, w e f e e l t h a t o u r d e r i v a t i o n of 2 which culminated i n ( 9 ) i s
V -,, .shows
t h a t w e have made necessary
very s a t i s f y i n g , e s p e c i a l l y s i:%. n--c e +it
. -<'.
*
.
headway f o r d e f i n i n g
a-a-
$
*
Again, w h i l e o u r
has been
that
t h e s p i r i t of t h e
use the Euclidean
Ikl]
t a l k about
v
-
-.
re11
b'j A -
p r e s e n t i l l u s t r a t i o n i s i n terms o f a r r o w s , n o t i c e
I n keeping with
defined f o r a l l vector spaces.
m e t r i c u s e d i n t h e " a r r o w " c a s e s , we d o a g r e e t o m e t r i c r a t h e r t h a n t h e M i n k o w s k i m e t r i c when w e
M o r e o v e r , a s we s a w i n U n i t 1 o f t h i s B l o c k ,
x*y<Iblllbll
if
we m u s t u e e t h e
we w a n t t o u s e t h e g e n e r a l r e s u l t t h a t
E u c l i d e a n m e t r i c s i n c e t h e r e s u l t i s n o t t r u e f o r t h e Minkowski m e t r i c .
f (atAx)
- - -f ( 5 )
Ax
lim
Ax+O
--
Y e t t h e r e i s something about ( 9 ) which t e n d s t o make t h e mathematician
c a u t i o u s . S p e c i f i c a l l y , w e would most l i k e l y want t o be a b l e t o
t h i n k o f C a s a s i n g l e v e c t o r r a t h e r t h a n a s an i n f i n i t e set of
v e c t o r s (zn much t h e same way t h a t w e need n o t d i s t i n g u i s h between
and t h e s e t a when b # 0 , e x c e p t t h a t then t h e choice
t h e number
was easy s i n c e t h e set a has o n l y a s i n g l e e l e m e n t ) .
2
But how s h a l l we go about t h e p r o c e s s of s e l e c t i n g a s i n g l e , w e l l d e f i n e d member of t h e s e t d e s c r i b e d by ( 9 ) ? Well, t h e one t h i n g t h a t
-member of the set s h a r e s i n common i s t h a t its p r o j e c t i o n i n the
each
d i r e c t i o n o f 1 i s C *
I f this i s t h e only p r o p e r t y t h a t i s of
- .
Itll
i n t e r e s t t o u s , why n o t choose t h a t m e m b e r o f
**
- which
5,
-
A f t e r a l l , i n t h e expression
d i r e c t i o n o f v?
v e c t o r which is s p e c i f i c a l l y named.
a l r e a d y has t h e
i s t h e only
*W h i l e ,
a d m i t t e d l y , i t i s e a s i e r t o t h i n k i n terms o f a r r o w s t h a n t o
t h i n k a b s t r a c t l y , k e e p i n mind from o u r d i s c u s s i p n i n C h a p t e r 4 of
these notes t h a t "direction" is defined f o r
spaces.
Namely,
"x h a s t h e same d i r e c t i o n a s
s i m p l y means t h a t 5 i s a s c a l a r
m u l t i p l e o f 1, o r , i n n - t u p l e n o t a t i o n , g i v e n y = (v1,
v ), t h e n
n
t h e s e t o f a l l v e c t o r s w h i c h h a v e t h e same d i r e c t i o n a s 1 i s d e f i n e d
t o b e {tl:t
a r e a l number}, i . e . , { ( t v l ,
t v ) : t a n y r e a l number}.
n
all
v"
** While
f
...,
h a s i n f i n i t e l y many members
still rather selective.
Namely,
membership i s
i f w e i n s i s t on a p a r t i c u l a r d i r e c t i o n
and s e n s e t h e r e i s o n e and o n l y one
this
same
(and
same
...,
( i f 1 + 2) t h e
~fE
C
;
such
t h a t x.1 = c . P i c t o r i a l l y ,
i s e a s y t o s e e , f o r a l t h o u g h i n f i n i t e l y many v e c t o r s h a v e t h e
p r o j e c t i o n i n a g i v e n d i r e c t i o n , n o two v e c t o r s w h o s e d i r e c t i o n s
s e n s e ) a r e e q u a l b u t whose m a g n i t u d e s a r e u n e q u a l c a n h a v e t h e
projection.
More a n a l y t i c a l l y , i n t e r m s o f a s p e c i f i c e x a m p l e
i s an i n f i n i t e s e t .
I f we now s e l e c t a l l members o f t h e s e t w h i c h
h a v e t h e s a m e d i r e c t i o n a s ( 3 , 1 , 2 , 3 , 4 ) we mean t B a t o u r members h a v e
t h e form ( 3 t , t , 2 t , 3 t , 4 t ) , and h e n c e
Therefore,
(6,2,4,6,8)
i s t h e o n l y member of
t h e same d i r e c t i o n (and s e n s e ) a s ( 3 , 1 , 2 , 3 , 4 ) .
78
(3,1,2,3,4)
which h a s
- is the u n i t
ltll th e v e c t o r - would be
Since
C;
v
v e c t o r i n t h e d i r e c t i o n of v, we have t h a t
written as
With t h i s a s motivation, w e nuw d e f i n e
-C by
V
and o u r p r e v i o u s d i s c u s s i o n i n s u r e s t h a t by ( 1 1 1 ,
AS a computational review,' we may compute
-v-l!
=
1
1 , so
vWv=
C
-
- using
.
c.
the f a c t that
M12=c.
that v
.
The D i r e c t i o n a l D e r i v a t i v e i n n-Dimensions
L e s t w e l o s e s i g h t of t h e f o r e s t because of t h e t r e e s , l e t u s summarize
t h e main computational p o i n t s of t h e p r e v i k s s e c t i o n . I n an attempt
t o d e f i n e f ' (a)
- by mimicking t h e d e f i n i t i o n of f ' ( a ) w e a r e forced t o
*Here
Namely,
jVl2+.
-
w e see a s t r o n g r e a s o n why we a r e u s i n g t h e E u c l i d e a n m e t r i c .
i f l=(vl,.
.. + v n 2
.. , v n )
then
1-1
- lk1l2.
v12+.
..+vn 2 .
Now
(el]
o n l y i f r e are using t h e E u c l i d e a n metric.
t h e E u c l i d e a n m e t r i c i(*1
w o u l d s t i l l b e v12+.
..+vn2
I n o t h e r w o r d s , v.1-
...
IbI1. 2
Hence for
( F o r t h e Minkowski m e t r i c ,
y-1
s i n c e the dot product is defined without.
(kjl w o u l d
H o w e v e r , now,
r e f e r e n c e t o any m e t r i c .
2
Certainly,
hence
= max(vl ,
, v n 2}.
that
lbi12
=
1
b e max{ vl
1 , . .I v n l 1
t h e r e i s no r e a s o n t o e x p e c t
n e e d n o t b e t r u e f o r t h e Minkowski m e t r i c . )
" i n v e n t " a d e f i n i t i o n of what it meant t o d i v i d e a number by a v e c t o r .
A f t e r much experimentation i n terms of the l o g i c a l consequences, we
accepted a s t h e main d e f i n i t i o n :
'
'I
I £ c i s any r e a l number and v i s any non-zero v e c t o r then
C
-
v e c t o r which i s i n t h e d i r e c t i o n of v and whose magnitude
Ts
&
i s that..,
-.C
*
tkl
I f w e now r e t u r n t o t h e d e f i n i t i o n of f ' (a ) a s given by (2) , w e f i n d
t h a t w e have s o l v e d one problem b u t have c r e a t e d another. That i s ,
w e became s o worried about what it meant t o d i v i d e a number by a
v e c t o r t h a t w e went no f u r t h e r with ( 2 ) b u t r a t h e r began an immediate
i n v e s t i g a t i o n i n t o how t h i s q u o t i e n t should be defined. Having solved
t h i s problem, w e a r e now ready t o d i s c o v e r t h a t a new major problem
looms b e f o r e us. Namely, we have p r e v i o u s l y agreed ( a n d - f o r good
lim t h e meaning was t h a t n o t o n l y should
r e a s o n s ) t h a t when w e wrote Ax+O
--
t h e l i m i t e x i s t b u t i t s value must n o t depend on t h e d i r e c t i o n by
which Ax approached 0.
On t h e o t h e r hand, one conseguence of our
d e f i n i t i o n of t h e q u o t i e n t of a number d i v i d e d by a v e c t o r i s t h a t
t h e v e c t o r which is denoted by t h e q u o t i e n t changes a s
changes
d i r e c t i o n ! That i s , t h e q u o t i e n t was d e f i n e d a s a v e c t o r i n t h e
s o t h a t , i f Ax is not rigidly specified, the quotient
d i r e c t i o n of
Ax
Ax,
i s , i n a s e n s e , undefined s i n c e we have no way of determining t h e
d i r e c t i o n of t h e q u o t i e n t .
Thus, it appears t h a t a n o t h e r refinement i s r e q u i r e d b e f o r e we can
work w i t h ( 2 ) , and it i s t h i s refinement t h a t motlvates t h e meaning
of a d i r e c t i o n a l d e r i v a t i v e . More s p e c i f i c a l l y , i t w i l l happen q u i t e
i n g e n e r a l t h a t t h e l i m i t i n (2) w i l l depend on t h e d i r e c t i o n by which
Ax approaches 0, f o r not only does changing t h e d i r e c t i o n of Ax
a f f e c t t h e denominator of our q u o t i e n t , i t a f f e c t s the numerator a s
w i l l , i n g e n e r a l , depend an the d i r e c t i o n of Ax.
w e l l s i n c e f (a +
Thus, it would appear t h a t i f w e h e l d t o (2) without some m o d i f i c a t i o n ,
f ' (a ) would never e x i s t s i n c e t h e l i m i t which d e f i n e s f ' (5) w i l l n o t
Az)
*
Actually,
i t i s p o s s i b l e t h a t e i s n e g a t i v e i n which e a s e
c a n n o t b e a magnitude ( s i n c e m a g n i t u d e s a r e n o n - n e g a t i v e ) .
s h o u l d s a y i s t h a t t h e magnitude i s
h a s t h e o p p o s i t e s e n s e o f 1.
V
C
1c1
and
1k11
C
1 1I
What we
that if c i s negative -
e x i s t ( s i n c e f o r t h e l i m i t t o e x i s t i t s value must n o t depend upon
how Ax+O).
-W e s h a l l t r y t o c l a r i f y t h i s p o i n t i n a few moments by
means of a s p e c i f i c example (with o t h e r examples being s u p p l i e d i n
t h e ~ x e r c i s e s ) ,b u t f i r s t we p r e f e r t o remove t h e new p i t f a l l .
I
W e now a g r e e t o remove any ambiguity from ( 2 ) by s p e c i f y i n g a
( I t i s o f t e n conventional t o s p e c i f y a d i r e c t i o n
particular direction.
i n n-dimensional space i n terms of a u n i t v e c t o r . Q u i t e o f t e n , i n
f a c t , a u n i t v e c t o r i s c a l l e d a d i r e c t i o n . For example, i n terms of
f +
i s a u n i t v e c t o r i n the d i r e c t i o n of
a p l a n a r example,
3 f + 4
I n t h e modern v e r n a c u l a r , w e would s a y
f +
is a
d i r e c t i o n , and any s c a l a r m u l t i p l e of this v e c t o r would be s a i d t o have
denote a s p e c i f i c
I n any e v e n t , i f w e l e t
t h e same d i r e c t i o n . )
u,
d i r e c t i o n , w e may t h i n k o f Ax a s always being i n the d i r e c t i o n of - means t h a t t h e d i r e c t i o n of Ax- s t a y s
and t h a t i n t h i s c o n t e x t Ax+g
f i x e d b u t i t s magnitude approaches 0.
$
3.
$3
5
:3
u
t,i
.
- .
The n e x t q u e s t i o n i s t h a t of f i n d i n g a way t o change t h e n o t a t i o n i n
( 2 ) t o r e f l e c t t h i s idea.
To t h i s end, w e observe t h a t once t h e
d i r e c t i o n g i s f i x e d , a l l o t h e r v e c t o r s i n t h i s d i r e c t i o n a r e of t h e
- which
form t u where t i s a r e a l number. Thus, r a t h e r than w r i t e Ax,
With
c a r r i e s t h e connotation 0 f . a varying d i r e c t i o n , w e w r i t e
t h i s i n mind, t h e l e f t s i d e of (2) becomes ambiguous s i n c e it does
For t h i s reason, we agree t o r e w r i t e , =
n o t i n d i c a t e t h e d i r e c t i o n u_.
tu.
-
-
.. .
f 1 ( a ) a s f U 1(5) and w e c a l l t h i s t h e d e r i v a t i v e of f a t 5 i n t h e direction
E. A t t h e same
t i m e , r e p l a c i n g Ax by tg c o n v e r t s t h e r i g h t s i d e of (2) i n t o
Istt~);.mo
- - [f
tu
-
I
I
_. .
2
J
.
L
f f ~ )
and since u is a u n i t v e c t o r ,
.
?
.
.
(121
1
,
t;+g
i f and only i f t + O . Moreover,
are t o have t h e same s e n s e , t M ; hence, t+O may be
s i n c e t: and
r e p l a c e d by t + 0 + . Thus, (12) may be r e w r i t t e n a s
-
b
-.
-. .
.,T,
4..
%
-..
-.
-,
-
.'
-1
.-
1,.
-.-,-.q
v.
k8
.
b,
-
..i
The bracketed e x p r e s s i o n i n (13) denotes t h e v e c t o r i n t h e d i r e c t i o n
of (u ) whose magnitude i s
. ,
.
:
I
%
I!
and t h i s i n t u r n i s
a'
ntull
Then s i n c e
= I tl UgIl (and s i n c e t > O ,
v i r t u e of being a u n i t _. v .e c t o r , w e have
,
I
,
-
It1 =. t ) and
L
= 1 by.
,
1
:
.
=
8
x
IkII
*.'
*-
lim
f (a+tu)
-
-
F
.
f (5)
r e p r e s e n t s the i n s t a n t a n e o u s r a t e of change of f ( x ) with r e s p e c t t o
ltfll i n the d i r e c t i o n u at x
a.
-= I f we make t h e s e changes ( 2 ) becomes
Equation (15) summarizes r a t h e r n i c e l y the i d e a t h a t f l u ( % ) is a
u whose magnitude i s t h e d e r i v a t i v e of
vector i n the direction (of) ..
x = a i n t h e d i r e c t i o n u.
f (x)
- with r e s p e c t t o Ilxll
- at -
I
I-t
The bracketed e x p r e s s i o n on t h e r i g h t s i d e of (14) r e p r e s e n t s t h e
- i n the direction u with respect t o
average r a t e of change o f f (x)
as x v a r i e s from
= a to x = a+tu.
- Thus
;
Thus, it i s n a t u r a l t o view f U u ( a )a s a d i r e c t i o n a l d e r i v a t i v e , t h a t
is, a derivative i n the directzon u . Notice t h a t i f w e use t h e
g e n e r a l i z e d d e f i n i t i o n of d i r e c t i o n t h a t a p p l i e s t o a l l dimensional
v e c t o r s p a c e s , then (15) i s v a l i d f o r any dimensional space, and, i n
p a r t i c u l a r , i f we restrict our a t t e n t i o n t o t h e s p e c i a l case of
p l a n a r arrows, it should be easy t o s e e t h a t (15) i s e q u i v a l e n t t o
t h e d e f i n i t i o n of a d i r e c t i o n a l d e r i v a t i v e a s given i n t h e t e x t and
s t u d i e d a s p a r t of t h e p r e v i o u s u n i t ,
'
Perhaps a s p e c i f i c example i n which we compute a d i r e c t i o n a l d e r i v a t i v e
i n t h e p l a n e using each of t h e two methods and then show t h a t t h e
answers a r e t h e same, might be of more b e n e f i t than an attempt t o
make an a b s t r a c t , formal proof. To t h i s end, l e t us consider t h e
-
-I
problem of f i n d i n g t h e d i r e c t i o n a l d e r i v a t i v e of t h e s u r f a c e w u x4y
a t t h e p o i n t (2,3,12) i n t h e d i r e c i t o n of t h e v e c t o r 3: + 4 3 . Using
L
t h e t r a d i t i o n a l approach, w e have t h a t f ( x ) = f ( x , y ) = x y , whence
-
,.
.
Therefore, f x ( 2 , 3 ) = 12 and
f x ( x , y ) = 2xy and f (x,y) = x
Y
f y ( 2 , 3 ) = 4 . Accordingly, t h e g r a d i e n t of f a t (2.3) i s 121 + 4J,
3-t
4t
Since t h e
and a u n i t v e c t o r i n t h e g i v e n d i r e c t i o n i s 51 + SJ.
d e s i r e d . d i r e c t i o n a 1 d e r i v a t i v e i s t h e d o t product of t h e g r a d i e n t and
t h e given u n i t v e c t o r , we o b t a i n a s t h e d i r e c t i o n a l d e r i v a t i v e
I f we now u s e t h e method of
this s e c t i o n ,
we have
.
and
.
-
Then, s i n c e f ( a ) = f ( 2 , 3 ) = 12 we have
f (a
-
+
ts)
-
f ( 5 ) = 52t/5
+
2
3t
+
36t3/125,
and
I£ w e now l e t t+0+, w e o b t a i n from (15)
which i s a v e c t o r whose magnitude i s 52/5 and whose d i r e c t i o n i s
4-t
u = -3-t
51 + ~3 , and t h i s , of course, h a s t h e same d i r e c t i o n a s 3: + 43.
W e t h u s s e e t h a t w e o b t a i n t h e same answer i n both cases.
..
.
L e t us a l s o observe t h a t t h e method of t h i s c h a p t e r does n o t r e q u i r e : .
t h a t we be aware of t h e concept of t h e g r a d i e n t ( t h a t i s , i n d e r i v i n g
(16) we never used anything b u t t h e e x p r e s s i o n s implied i n ( 1 5 ) , and
c e r t a i n l y no n o t i o n of t h e g r a d i e n t i s p r e s e n t t h e r e ) . Not only d i d
w e n o t need t h e g r a d i e n t , b u t w e had no need t o t a l k about p a r t i a l J . .
d e r i v a t i v e s . This i s a s i t should be, s i n c e t h e p a r t i a l d e r i v a t i v e s
a r e merely d e r i v a t i v e s w i t h r e s p e c t t o some h i g h l y s e l e c t i v e d i r e c t i o n s .
)
(15) i n any v e c t o r space, .
I n t h i s r e s p e c t we can compute f V u ( gfrom
without r e g a r d t o e i t h e r a gradieKt o r p a r t i a l d e r i v a t i v e s . Such
a d d i t i o n a l examples a r e l e f t f o r t h e e x e r c i s e s . Before ending t h i s
s e c t i o n , however, w e f e e l it might provide a new i n s i g h t t o calc'ulus
of a s i n g l e r e a l v a r i a b l e if we apply t h e d i s c u s s i o n of t h i s s e c t i o n
t o a 1-dimensional v e c t o r space.
I;
To t h i s end, observe t h a t i n 1-dimensional space our v e c t o r s (1-tuples)
a may be i d e n t i f i e d with
a r e simply r e a l numbers. Thus, t h e v e c t o r t h e number, a. Moreover, i n 1-dimensional space t h e r e a r e only t h e
-+ (Geometrically, t h e space i s t h e x-axis
two u n i t v e c t o r s , f and -1.
a s c a l a r m u l t i p l e of f.
and along t h e x-axis any v e c t o r
is
Now, r e c a l l t h a t one way of saying t h a t f 8 ( a ) e x i s t e d was t o say t h a t
both
AXM+
1,
exist
[
f
A
(, X,
£(a)
]
and
lim
AX+O-
[
f (a+Axkx- f ( a )
I
a r e equal.
But
is a d i r e c t i o n a l d e r i v a t i v e . I t i s t h e d e r i v a t i v e i n t h e d i r e c t i o n
t h a t ~ x + dthrough p o s i t i v e values, t h a t i s , from r i g h t - t o - l e f t , and
t h i s , i n t u r n , i s t h e d i r e c t i o n -1. I n o t h e r words, using t h e
n o t a t i o n of t h i s s e c t i o n ,
and Thus, from (17) our new language s a y s t h a t f' (a) e x i s t s i n 1-dimens5onal
space i f and only i f t h e two d i r e c t i o n a l d e r i v a t i v e s (where both
d i r e c t i o n s d i f f e r only i n sensej e x i s t and a r e equal.
The Derivative i n n-Space
Our attempts t o d e f i n e t h e d e r i v a t i v e of a function f:En+E by mimicking
t h e 1-dimensional case has l e d t o t h e "invention" of t h e d i r e c t i o n a l
d e r i v a t i v e . The t r o u b l e with t h e d i r e c t i o n a l d e r i v a t i v e i s t h a t i t
may e x i s t i n some d i r e c t i o n s b u t n o t i n o t h e r s . T h i s i s e a s y t o
p i c t u r e i n t h e c a s e f : E ~ + E s i n c e then t h e graph of f i s a s u r f a c e ,
and what w e a r e then s a y i n g is t h a t some c r o s s s e c t i o n s of t h i s
s u r f a c e through a g i v e n p o i n t may be smooth while o t h e r s a r e n ' t ,
I f w e s t i l l t h i n k of t h e d e r i v a t i v e a s denoting "smoothness"
(without worrying a b o u t what t h i s means i n high dimensions), t h e n
u n l e s s t h e d i r e c t i o n a l d e r i v a t i v e e x i s t s i n e v e r y d i r e c t i o n (i.e. ,
every s l i c e through t h e p o i n t i s smooth) t h e s u r f a c e w i l l n o t be
smooth. S t a t e d more p o s i t i v e l y , we d e f i n e a s u r f a c e t o be smooth
a t a p o i n t i f t h e d i r e c t i o n a l d e r i v a t i v e e x i s t s i n each d i r e c t i o n
a t t h e given p o i n t .
With t h i s a s m o t i v a t i o n , i t i s an easy s t e p t o g e n e r a l i z e t h e d e f i n i t i o n
(even i f t h e geometric i n t e r p r e t a t i o n may no longef a p p l y ) . Namely,
we s a y t h a t ~ : E " + E i s d i f f e r e n t i a b l e a t x = a i f and only i f
e x i s t s i n every d i r e c t i o n , u.
f',(a)
While t h e above d i s c u s s i o n i s an adequate i n t u i t i v e m o t i v a t i o n , w e
can s u p p o r t t h e argument f o r o u r d e f i n i t i o n of d i f f e r e n t i a b i l i t y on
more computational grounds a s w e l l . Namely, w e would l i k e t h e
d e f i n i t i o n of a d e r i v a t i v e t o be independent of any p a r t i c u l a r choice
of d i r e c t i o n . Why i s t h i s so? Perhaps t h e following p i c t o r i a l
demonstration i n t h e c a s e n=2 w i l l shed some l i g h t on t h e answer.
Suppose f ( x , y ) i s d e f i n e d i n some neighborhood N of (a,b) and we
choose (c,d) t o be any o t h e r p o i n t i n t h i s neighborhood. To
emphasize t h a t ( a , b ) i s o u r f o c a l p o i n t , w e s h a l l rewrite ( c , d ) a s
(a+h, b+k) where h and k a r e c o n s t a n t s which depend on t h e choice of
the point (c,d).
[To be more s p e c i f i c , h i s what w e would o r d i n a r i l y
c a l l Ax and k i s Ay].
(Figure 3)
Somehow w e e x p e c t t h a t t h e d e r i v a t i v e of f , no m a t t e r how we u l t i m a t e l y
f ( a , b ) I . Now,
d e f i n e i t , must i n v o l v e Af [which denotes f (a+h, b+k)
f ( a , b ) i s a well-defined
once h and k a r e s p e c i f i e d , f (a+h, b+k)
number which i n no way depends on d i r e c t i o n . Moreover, when w e f i n a l l y
t a k e t h e a p p r o p r i a t e l i m i t , w e want an answer which w i l l n o t depend
on t h e p a t h which j o i n s (a+h, b+k) t o ( a , b ) t f o r i f t h e answer does
depend on t h e p a t h , t h e l i m i t does n o t e x i s t . To be s u r e , t h e usage
of f x ( a , b ) and f ( a , b ) u t i l i z e p a t h s such a s i n F i g u r e s 4a and 4b,
Y
b u t o u r answer must hold f o r a r b i t r a r y (and n o t n e c e s s a r i l y s t r a i g h t
. .>,l i n e ) p a t h s a s i n Figure 4c.
-
-
(a)
(Figure 4 )
_.
.
.
directional
(Figure 5)
(c)
. .
, '
The s p e c i a l r o l e of t h e
shown i n Figure 5.
(b)
-
p
.iY
'
.
9,
n
derivative i n t h i s context i s
1 .
That i s , once (a+h,b+k) i s chosen t h e d i r e c t i o n a l d e r i v a t i v e assumes
t h a t t h e p a t h is the s t r a i g h t l i n e which j o i n s (a+h,b+k) t o ( a , b ) .
I n t h i s s e n s e , once t h e s t r a i g h t l i n e i s determined, t h e p o i n t s
(x,y) f o r which w e e v a l u a t e f (x,y) a l l l i e on t h i s l i n e , s o t h a t i n
computing a d i r e c t i o n a l d e r i v a t i v e we do have t h e analog o f t h e
1-dimensional d e r i v a t i v e . The o n l y problem i s t h a t t h i s d e r i v a t i v e ,
a s w e have s e e n , v a r i e s w i t h d i r e c t i o n .
W e should a l s o p o i n t o u t t h a t while, a s i n Figure 4 , o u r p a t h s do n o t
have t o be s t r a i g h t l i n e s , w e may assume ( a t l e a s t i n t u i t i v e l y i n t h e
c a s e n=2) t h a t they are s t r a i g h t l i n e s . Namely, a s w e t a k e l i m i t s ,
w e a r e only i n t e r e s t e d i n what happens "near" the p o i n t ( a , b ) . Thus,
i f n e a r ( a , b ) t h e path which j o i n s (a+h,b+k) t o ( a , b ) i s n o t a
s t r a i g h t l i n e , w e may r e p l a c e it by t h e s t r a i g h t l i n e which i s
t a n g e n t t o t h e p a t h a t ( a , b ) , assuming of course, t h a t t h e p a t h i s
smooth.
(See F i g u r e 6 )
I n N1 t h e p a t h and the t a n g e n t
l i n e a r e e s s e n t i a l l y t h e same.
-\---
(Figure 6 )
Thus, t h i s d i s c u s s i o n , t o o , m o t i v a t e s why i n formulating t h e d e f i n i t i o n
of t h e d e r i v a t i v e of a f u n c t i o n of s e v e r a l ( n ) r e a l v a r i a b l e s we f i r s t
i n s i s t t h a t t h e d i r e c t i o n a l d e r i v a t i v e of f a t 5 e x i s t i n every
direction.
Now t h a t w e have given a few m o t i v a t i o n s f o r d e f i n i n g f t o be d i f f e r e n t i a b l e i f i t s d i r e c t i o n a l d e r i v a t i v e e x i s t s i n each d i r e c t i o n , t h e n e x t
s u b t l e p o i n t i s t h e choice of how the d e r i v a t i v e of f should be defined.
That i s , t h e r e i s a d i f f e r e n c e between saying t h a t f i s d i f f e r e n t i a b l e
and determining what t h e d e r i v a t i v e a c t u a l l y is. For reasons t h a t w e
hope w i l l be made c l e a r e r a s w e proceed, l e t u s d e f i n e t h e d e r i v a t i v e
x = a t o be t h e d i r e c t i o n a l d e r i v a t i v e of f a t a which has t h e
of f a t -
g r e a t e s t magnitude. Notice t h a t this i n t r o d u c e s the a d d i t i o n a l s u b t l e t y
t h a t t h e r e must be a d e r i v a t i v e of maximum magnitude. T h i s i s n o t a t
a l l s e l f - e v i d e n t (and indeed i t need n o t be t r u e , although w e s h a l l
n o t pursue t h i s m a t t e r h e r e ) s i n c e t h e r e a r e i n f i n i t e l y many d i r e c t i o n a l
d e r i v a t i v e s of f a t a , and f o r an i n f i n i t e s e t t h e r e need n o t be a
( f i n i t e ) upper bound.
While i t may n o t be c l e a r y e t why w e choose this d e f i n i t i o n , what
should be c l e a r i s t h a t s i n c e t h e d i r e c t i o n a l d e r i v a t i v e v a r i e s from
d i r e c t i o n t o d i r e c t i o n , w e must somehow o r o t h e r make a choice t h a t
p i c k s one of t h e s e v a l u e s from a l l o t h e r s ( i n much t h e same way a s w e
had t o choose one v e c t o r from t h e s e t of v e c t o r s , $1.
How s h a l l we
make t h i s choice? A s u s u a l , we s h a l l l e t t h e most-important
a p p l i c a t i o n of t h e concept determine t h e d e f i n i t i o n . I n this c a s e ,
w e f i n d t h a t w e s h a l l u s u a l l y be t r y i n g , i n one form o r another, t o
a ) l " s u f f i c i e n t l y s m a l l n r e g a r d l e s s of
make the e x p r e s s i o n I f ( 5 ) - f ( x approaches 5, b u t i f w e want. 1 f (x)
- - f (-a ) 1 t o be s u f f i c i e n t l y
how s m a l l , then w e need only i n s u r e t h a t i t i s s u f f i c i e n t l y small i n t h e
d i r e c t i o n i n which i t i s t h e g r e a t e s t .
I f w e now l e t f '(5)denote t h e d i r e c t i o n a l d e r i v a t i v e of f a t a
( A more whose magnitude i s maximum, this problem is taken c a r e o f .
formal way of s a y i n g this i s t h a t the d i r e c t i o n a l d e r i v a t i v e behaves l i k e an o r d i n a r y 1-dimensional d e r i v a t i v e i n t h e given d i r e c t i o n . That i s , i f f'U(z?l) e x i s t s then -
lim k = 0 and Ax i s i n the d i r e c t i o n , u. A l l w e a r e then saying
*x+O i s t h a t - ~ f i s maximum when f ' (a) i s maximum i n magnitude.
. w h e r e
-
A t any r a t e , t h i s completes o u r supplementary d i s c u s s i o n of t h e d e r i v a t i v e of a f u n c t i o n of s e v e r a l v a r i a b l e s . A s a review, s o t h a t we s e e where a l l t h e p i e c e s f i t i n t o p l a c e , l e t us observe t h a t i n Unit 3 w e d i s c u s s e d the g r a d i e n t and the d i r e c t i o n a l d e r i v a t i v e i n While i t was (using t h e 2-dimensional n o t a t i o n )
terms of f x and f
Y
n o t s p e c i f i c a l l y mentioned i n t h e t e x t , t h e concept of d i f f e r e n a and be
x = a r e q u i r e d t h a t f x and f e x i s t a t t i a b i l i t y of f a t Y
continuous t h e r e a s w e l l . .
MIT OpenCourseWare
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Resource: Calculus Revisited: Multivariable Calculus
Prof. Herbert Gross
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