3
AN INTRODUCTION TO VECTOR CALCULUS
A
-
Introduction
I n t h e same way t h a t w e s t u d i e d n u m e r i c a l c a l c u l u s a f t e r we l e a r n e d
n u m e r i c a l a r i t h m e t i c , w e c a n now s t u d y v e c t o r c a l c u l u s s i n c e we have
a l r e a d y s t u d i e d v e c t o r a r i t h m e t i c . Q u i t e simply (and t h i s w i l l be
e x p l o r e d i n t h e remaining s e c t i o n s of t h i s c h a p t e r ) , w e might have a
vector quantity t h a t v a r i e s with respect t o another variable, e i t h e r a
scalar or a vector.
I n t h i s c h a p t e r w e s h a l l b e most i n t e r e s t e d i n
t h e c a s e where w e have a v e c t o r which v a r i e s w i t h r e s p e c t t o a s c a l a r .
A r a t h e r s i m p l e p h y s i c a l s i t u a t i o n of t h i s c a s e would be t h e problem
where w e measure t h e f o r c e on an o b j e c t a t d i f f e r e n t times.
.
That i s ,
f o r c e ( a v e c t o r ) i s t h e n a f u n c t i o n of t i m e ( a s c a l a r )
I n t h i s cont e x t w e can a s k whether o u r f o r c e i s a c o n t i n u o u s f u n c t i o n of t i m e , a
d i f f e r e n t i a b l e f u n c t i o n o f t i m e , and s o f o r t h ,
A t t h e same t i m e ,
t h i s t y p e of e x p l o r a t i o n opens up new avenues of
i n v e s t i g a t i o n and s u p p l i e s u s w i t h e x c e l l e n t p r a c t i c a l r e a s o n s f o r exp l o r i n g t h e r a t h e r a b s t r a c t n o t i o n of h i g h e r dimensional s p a c e s . More
s p e c i f i c a l l y , l e t u s keep i n mind t h a t it i s a r a t h e r overly-simple
s i t u a t i o n i n r e a l l i f e when t h e q u a n t i t y under c o n s i d e r a t i o n depends
on o n l y one o t h e r v a r i a b l e . For example, w i t h r e s p e c t t o our above
i l l u s t r a t i o n of f o r c e depending upon t i m e , l e t u s n o t e t h a t i n a r e a l l i f e s i t u a t i o n t h e o b j e c t w e a r e s t u d y i n g h a s n o n - n e g l i g i b l e s i z e and
f o r t h i s r e a s o n a f o r c e a p p l i e d a t one p o i n t on t h e o b j e c t h a s d i f f e r -
e n t e f f e c t s a t o t h e r - p o i n t s . ~ h u s ,w e might f i n d t h a t o u r v e c t o r
f o r c e depends on t h e f o u r v a r i a b l e s x , y, z , and t . That i s , t h e
f o r c e may be a f u n c t i o n of b o t h p o s i t i o n ( t h r e e dimensions) and t i m e .
I n terms of a f u n c t i o n , o u r i n p u t i s a 4 - t u p l e ( t h a t i s , an o r d e r e d
a r r a y [sequence] of f o u r r e a l numbers x , y , z , and t ) . P i c t o r i a l l y ,
Now w e have a l r e a d y used 2 - t u p l e s t o a b b r e v i a t e v e c t o r s i n t h e p l a n e
and w e have used 3 - t u p l e s t o a b b r e v i a t e v e c t o r s i n s p a c e . T h a t i s , w e
have l e t ( a , b ) d e n o t e <i + gj and ( a , b , c ) d e n o t e c?i + g j + gk.
In
t h i s c o n t e x t , w e may t h i n k of a 4 - t u p l e a s d e n o t i n g a 4-dimensional
" v e c t o r , " even though we may n o t be a b l e t o v i s u a l i z e it i n t h e u s u a l
sense. We have e a r l i e r come t o g r i p s w i t h t h i s problem i n t h e form of
exponents. Namely, if we look a t b l , b2. and b 3 w e can i n t e r p r e t each
of t h e s e p i c t o r i a l l y v e r y n i c e l y . That i s , b 1 i s a l e n g t h of b u n i t s ,
2
3
b i s t h e a r e a of a square whose s i d e has l e n g t h b , and b i s t h e
volume of a cube whose s i d e has l e n g t h b . But while b 4 does n o t have
such a simple geometric i n t e r p r e t a t i o n , it i s j u s t a s meaningful t o
m u l t i p l y f o u r ( o r more) f a c t o r s of b a s i t i s t o m u l t i p l y one, two,
or three.
In o t h e r words, we a r e back t o a fundamental t o p i c of P a r t 1
of our c o u r s e - a p i c t u r e i s worth a thousand words i f you can draw
it.
-
I n any e v e n t , what w e a r e saying i s t h a t when we s t u d y f u n c t i o n s of n
v a r i a b l e s we a r e i n e f f e c t looking a t an n-dimensional space. I n f a c t
t h i s o f f e r s u s a r a t h e r s t r a i g h t - f o r w a r d , non-mystic i n t e r p r e t a t i o n a s
t o why t i m e i s o f t e n r e f e r r e d t o a s t h e f o u r t h dimension. Namely, i n
most p h y s i c a l s i t u a t i o n s t h e v a r i a b l e we a r e measuring ( a s we d i s c u s s e d
e a r l i e r ) depends on p o s i t i o n and time. P o s i t i o n i n g e n e r a l i n v o l v e s
t h r e e dimensions and time i s then t h e f o u r t h dimension ( v a r i a b l e ) .
Notice t h a t i n t h i s r e s p e c t we can have a s c a l a r (as w e l l a s a v e c t o r )
f u n c t i o n of n-dimensional v e c t o r s (meaning t h e i n p u t c o n s i s t s of n
v a r i a b l e s w h i l e t h e o u t p u t i s a number). For example, we might be
studying temperature ( a s c a l a r ) a s a f u n c t i o n of p o s i t i o n and t i m e , i n
t h e sense t h a t temperature i n g e n e r a l does v a r y from p o i n t t o p o i n t
and a t t h e same p o i n t it u s u a l l y v a r i e s from t i m e t o time.
I n o r d e r n o t t o i n t r o d u c e t o o many new i d e a s a t once, w e s h a l l devote
t h e remainder of t h i s c h a p t e r o n l y t o t h e c a l c u l u s of 2 and 3dimensional v e c t o r s . Once we g a i n some f a m i l i a r i t y with t h i s b a s i c
concept, we s h a l l extend our f r o n t i e r s t o i n c l u d e t h e more g e n e r a l
s t u d y of n-dimensional v e c t o r s , T h i s t o p i c w i l l be d i s c u s s e d i n a
l a t e r c h a p t e r of supplementary n o t e s which w i l l be covered d u r i n g our
s t u d y of Block 3 .
b
Functions R e v i s i t e d
When we f i r s t introduced t h e concept of f u n c t i o n s , we mentioned t h a t a
f u n c t i o n was a r u l e which assigned t o members of one s e t members of
another s e t . W e p o i n t e d o u t t h a t t h e two s e t s involved could be
a r b i t r a r y b u t t h a t i n t h e study of r e a l v a r i a b l e s our a t t e n t i o n , b y
d e f i n i t i o n , i s confined t o t h e case i n which both s e t s a r e s u b s e t s of
t h e r e a l numbers.
W e a l s o developed t h e v i s u a l a i d of t h e f u n c t i o n machine, wherein i f
f :A+B,
w e had
f -machine (elements
(elements The p o i n t of t h i s c h a p t e r i s t h a t , i n t h e r e a l w o r l d , w e a r e o f t e n
c o n f r o n t e d w i t h s i t u a t i o n s i n which w e a r e d e a l i n g w i t h v e c t o r s r a t h e r
t h a n s c a l a r s , whereupon w e have t h e s i t u a t i o n t h a t e i t h e r t h e i n p u t o r
t h e o u t p u t of o u r f-machine could be v e c t o r s .
I n o u r e a r l i e r example, w e spoke of f o r c e ( a v e c t o r ) a s a f u n c t i o n of
t i m e ( a s c a l a r ) . I n t h i s c a s e o u r f-machine l o o k s l i k e
input
t (scalar)
f -machine
+
More c o n c r e t e l y , w e might have a f o r c e F d e f i n e d i n t h e xy-plane by
From (1) it i s c l e a r t h a t d i f f e r e n t v a l u e s of t h e s c a l a r t y i e l d d i f +
f e r e n t v a l u e s of t h e v e c t o r F. F o r i n s t a n c e , i f t = 3 t h e n
8 = 3 1 + $j Moreover, i t s h o u l d seem f a i r l y obvious t h a t w e might
l i k e t o a b b r e v i a t e t h e above i n f o r m a t i o n i n the form
.
Notice how t h e language i n ( 2 ) c l o s e l y p a r a l l e l s t h e n o t a t i o n t h a t w e
have a l r e a d y used f o r " r e g u l a r " f u n c t i o n s . The o n l y d i f f e r e n c e i s
t h a t b e f o r e w e were t a l k i n g a b o u t s c a l a r f u n c t i o n s ( i . e . t h e " o u t p u t s "
were s c a l a r s ) of s c a l a r v a r i a b l e s ( i . e . t h e " i n p u t s " were a l s o
s c a l a r s ) , w h i l e i n t h i s c a s e we a r e d e a l i n g w i t h v e c t o r f u n c t i o n s
( i . e . t h e o u t p u t s a r e v e c t o r s ) of s c a l a r v a r i a b l e s .
I n t h i s same v e i n , we might w r i t e
t o i n d i c a t e t h a t we have a s c a l a r f u n c t i o n ( i . e . t h e o u t p u t of f i s a
s c a l a r ) of a v e c t o r v a r i a b l e ( i . e . t h e i n p u t of t h e £-machine i s a
v e c t o r ) . A s an example i n " r e a l i i f e " ~f e q u a t i o n ( 3 ) , c o n s i d e r a
s i t u a t i o n i n which a l l we were i n t e r e s t e d i n was t h e magnitude of
-+
v a r i o u s f o r c e s being considered. I n t h i s c a s e , given a f o r c e F we
+
would compute i t s magnitude I F ( .
I n t h i s way our procedure i s t o t a k e
a v e c t o r ( f o r c e ) and c o n v e r t i t t o a s c a l a r (magnitude of t h e f o r c e ) .
Pictorially,
input
8 (vector)
1 8 ( -machine
.output
191 ( s c a l a r )
.
The f i n a l p o s s i b i l i t y i s t h a t we have
Equation ( 4 ) denotes a v e c t o r f u n c t i o n of a v e c t o r v a r i a b l e . An
example of t h i s s i t u a t i o n might be t h a t a t any g i v e n i n s t a n t we know
t h e f o r c e ( a v e c t o r ) being e x e r t e d on a p a r t i c l e from which we want t o
determine t h e a c c e l e r a t i o n of t h e p a r t i c l e ( a c c e l e r a t i o n i s a l s o a
v e c t o r - i n f a c t i n Newtonian physics t h e d i r e c t i o n of the a c c e l e r a t i o n
i s t h e same a s t h a t of t h e f o r c e , which means t h a t t h e "lawH f = ma
+
Again p i c t o r i a l l y ,
remains c o r r e c t i n t h e v e c t o r form f = mg)
.
input
3 (vector)
";-machine
"
output
+
a (vector)
/
I n summary t h e n , i f w e allow v e c t o r s a s w e l l a s s c a l a r s t o be our
v a r i a b l e s , t h e n t h e s t u d y of f u n c t i o n s of a s i n g l e v a r i a b l e reduces t o
one of f o u r t y p e s
I n t h i s b l o c k we s h a l l , when d e a l i n g w i t h v e c t o r f u n c t i o n s , assume
t h a t w e have a v e c t o r f u n c t i o n of a s c a l a r v a r i a b l e . T h a t i s , w e
s h a l l d e a l w i t h f u n c t i o n s of t h e form
Pictorially,
x
( s c a l a r1
-t
f -machine
(x,
(vector)
*
The key o b s e r v a t i o n t o b e made from t h e p i c t u r e above i s t h a t i f t h e
a r r o w s a r e o m i t t e d o u r diagram i s i d e n t i c a l w i t h o u r p r e v i o u s diagram
of a f u n c t i o n machine i n t h e s t u d y of r e a l v a r i a b l e s . Namely,
T h i s o b s e r v a t i o n i s t h e backbone of t h e n e x t s e c t i o n .
C
The "Game" Applied t o V e c t o r C a l c u l u s
I n d e f i n i n g l i m f ( x ) = L , w e used o u r i n t u i t i o n t o make s u r e t h a t w e
x+a
knew what w e wanted t h e e x p r e s s i o n t o mean, and we t h e n proceeded t o
make t h e d e f i n i t i o n more r i g o r o u s u s i n g E ' S and 6 ' s .
The r i g o r o u s way
seemed q u i t e f r i g h t e n i n g a t f i r s t , b u t , a f t e r a w h i l e , w e began t o
n o t i c e t h a t it o n l y s a i d i n p r e c i s e m a t h e m a t i c a l terms what we a l r e a d y
believed t o be t r u e i n t u i t i v e l y .
What may have gone u n n o t i c e d was t h a t once we had t h e r i g o r o u s d e f i n i t i o n , e v e r y proof we gave c o n c e r n i n g l i m i t s c o n s i s t e d of a p p l y i n g t h e
r u l e s of mathematics t o t h e mathematical d e f i n i t i o n of a l i m i t .
In
t e r m s of o u r game i d e a , we d e f i n e d what a l i m i t was, used t h e a c c e p t e d
r u l e s of n u m e r i c a l a r i t h m e t i c , and t h e n proved theorems a b o u t l i m i t s .
The key p o i n t h e r e i s t h a t i f it should happen t h a t our d e f i n i t i o n of
l i m i t i n our d i s c u s s i o n of v e c t o r f u n c t i o n s of a s c a l a r v a r i a b l e h a s
t h e same s t r u c t u r e a s our d e f i n i t i o n of l i m i t when w e were d e a l i n g
with s c a l a r f u n c t i o n s of s c a l a r v a r i a b l e s , and i f i t happens t h a t
every accepted r u l e of numerical a r i t h m e t i c t h a t was used i n t h e
e a r l i e r p r o o f s of our l i m i t theorems a l s o happens t o be a n accepted
r u l e f o r v e c t o r a r i t h m e t i c , . th e n , i n terms of t h e philosophy of our
game concept, e v e r y theorem about l i m i t s t h a t was v a l i d i n t h e s t u d y
of "numerical l i m i t s n w i l l remain v a l i d i n t h e study of " v e c t o r i a l
The purpose of t h i s s e c t i o n i s t o e x p l o r e t h i s i d e a i n more
limits."
computational d e t a i l .
How can we t e s t whether t h e v e c t o r s i t u a t i o n p a r a l l e l s t h e numerical
c a s e ? There i s a v e r y e l e g a n t y e t simple technique t h a t we s h a l l
employ h e r e . W e s h a l l begin by w r i t i n g t h e d e f i n i t i o n of l i m i t i n t h e
s c a l a r c a s e . We s h a l l t h e n go through t h e d e f i n i t i o n and i n s e r t arrows
over those symbols t h a t w i l l now denote v e c t o r s r a t h e r t h a n s c a l a r s
(and, a d m i t t e d l y , w e must t a k e c a r e i n determining what i s s t i l l a
s c a l a r and what has become a v e c t o r ) . I n t h i s way, we a r e s u r e t h a t
t h e b a s i c s t r u c t u r e cannot be d i f f e r e n t s i n c e t h e o n l y changes we made
were i n t h e names of t h e v a r i a b l e s . We t h e n check t o s e e i f t h e
r e s u l t i n g d e f i n i t i o n obtained from t h e s c a l a r d e f i n i t i o n by t h i s
"arrowizing" technique i s s t i l l meaningful. For example, i f we t r i e d
t o v e c t o r i z e t h e numerical statement t h a t a/b = c , we would o b t a i n
- + + -b
a / b = c which makes no s e n s e , s i n c e v e c t o r d i v i s i o n i s undefined.
(Of
c o u r s e , had we v e c t o r i z e d a l l b u t b , t h e r e s u l t i n g s t a t e m e n t would
make sense b u t t h i s i s n o t t h e p o i n t w e a r e t r y i n g t o make h e r e . )
I n any e v e n t , c a r r y i n g o u t o u r i n s t r u c t i o n s s o f a r , f i r s t w e w r i t e t h e
o r i g i n a l d e f i n i t i o n of l i m i t i n t h e s c a l a r c a s e , namely
l i m f (x) = L
x+a
means given any E > O , w e can f i n d 6>0 such t h a t whenever
then
Next, we r e w r i t e t h i s d e f i n i t i o n p u t t i n g i n arrows wherever a p p r o p r i a t e . R e c a l l t h a t we a r e d e a l i n g w i t h v e c t o r f u n c t i o n s of s c a l a r
v a r i a b l e s . Hence, x i s a s c a l a r . But f and L should be r e p r e s e n t e d
a s v e c t o r s s i n c e w e have a l r e a d y agreed t h a t we a r e d e a l i n g with vect o r f u n c t i o n s and t h i s i m p l i e s t h a t our o u t p u t , i n t h i s c a s e f ( x ) and
L, a r e a l s o v e c t o r s .
Notice, however, t h a t even though we have now
introduced v e c t o r s , E and 6 are s t i l l s c a l a r s . The reason f o r t h i s
i s t h a t E and 6 denote a b s o l u t e v a l u e s (magnitudes) of q u a n t i t i e s , and
t h e magnitude of both s c a l a r s and v e c t o r s a r e non-negative r e a l
numbers.
I n any e v e n t , i f w e now r e w r i t e our o l d d e f i n i t i o n with t h e
a p p r o p r i a t e "arrowi z a t i o n n w e o b t a i n
means given any E > O we can f i n d 6 > 0 such t h a t whenever
then
Our f i r s t o b s e r v a t i o n of t h i s d e f i n i t i o n should show t h a t i t i s mean+ +
i n g f u l . That i s , e v e r y p a r t of t h e e x p r e s s i o n i s d e f i n e d ( u n l i k e a / b ) .
Ou,r next t a s k i s t o see whether t h e d e f i n i t i o n c a p t u r e s what we would
-+
+
l i k e l i m f ( x ) = L t o mean. A f t e r a l l , what good i s it i f our d e f i n i x+a
t i o n i s meaningful b u t it d o e s n ' t c a p t u r e t h e meaning we have i n mind?
To t h i s end, it i s probably f a i r t o assume t h a t we would l i k e
+
+
-+
l i m f (x) = L t o mean t h a t i f x i s " s u f f i c i e n t l y c l o s e t o " a- then f ( x )
x+a
-+ i s " s u f f i c i e n t l y c l o s e t o " L. Notice t h a t f o r two v e c t o r s t o be " n e a r l y " e q u a l t h e i r d i f f e r e n c e must be small. I n o t h e r words, f o r any p a i r of v e c t o r s A and B , i f we assume t h a t t h e y o r i g i n a t e a t a common p o i n t then t h e i r d i f f e r e n c e i n magnitude i s t h e l e n g t h of t h e v e c t o r t h a t goes from t h e head of one t o t h e head of t h e o t h e r . C l e a r l y , t h e s m a l l e r t h i s v e c t o r i s i n magnitude t h e more " t o g e t h e r " a r e t h e heads of t h e two v e c t o r s . I n terms of our above d e f i n i t i o n , -+
-+
n o t i c e t h a t f ( x ) - L means p i c t o r i a l l y t h a t
The s m a l l e r t h e d i f f e r e n c e between
-+
t h e c l o s e r i s t h e head of f
f ( x ) and
+
t o t h e head of L which means t h a t
i s more n e a r l y e q u a l t o t.
2
(
.
.
I n o t h e r words, t h e n , i n terms of what t h e a n a l y t i c e x p r e s s i o n s mean
g e o m e t r i c a l l y , w e s h o u l d be convinced t h a t t h e f o r m a l d e f i n i t i o n a g r e e s
w i t h o u r i n t u i t i o n . Again, i n t e r m s o f a p i c t u r e
@
If x i s i n h e r e
/
>
x-axis
( C i r c l e of r a d i u s
J
@
of
E
c e n s e r e d a t t h e head of I.) -b
t h e n f ( x ) t e r m i n a t e s i n h e r e ( p r o v i d e d it o r i g i n a t e s a t t h e t a i l
2,.
So f a r , s o good, b u t i f w e want t o be a b l e t o c a p i t a l i z e on o u r game
i d e a , t h e c r u c i a l t e s t now l i e s ahead. R e c a l l t h a t a l l o u r f o r m u l a s
f o r d e r i v a t i v e s stemmed from c e r t a i n b a s i c p r o p e r t i e s of l i m i t s (such
a s t h e l i m i t of a sum i s t h e sum of t h e l i m i t s , and s o o n ) . These
p r o p e r t i e s of l i m i t s followed m a t h e m a t i c a l l y from c e r t a i n p r o p e r t i e s of
r e a l numbers. The k e y now i s t o show t h a t t h e p r o o f s a p p l y v e r b a t i m
when w e r e p l a c e t h e s c a l a r s by a p p r o p r i a t e v e c t o r s .
For example, what w e r e some of t h e p r o p e r t i e s w e used a b o u t a b s o l u t e
v a l u e s i n d e a l i n g w i t h o u r p r o o f s a b o u t l i m i t s ? W e l l , f o r one t h i n g ,
w e used t h e p r o p e r t y t h a t f o r any numbers a and b
The i m p o r t a n t t h i n g i s t h a t i f we now v e c t o r i z e (1) t o o b t a i n
we s e e t h a t ( 2 ) i s a l s o t r u e .
Pictorially,
The l e n g t h of one s i d e of a t r i a n g l e
c a n n o t exceed t h e sum of t h e l e n g t h s
of t h e o t h e r o t h e r two s i d e s .
a
W e a l s o used such f a c t s a s
la1 3 0 and la1 = 0
i f and o n l y i f a = 0 .
obtain
+
la1 3 0 and
121
I f w e now v e c t o r i z e ( 3 ) a p p r o p r i a t e l y , w e
= 0
i f and on'ly i f 2 = 8 . R e c a l l i n g t h a t 1; 1 means t h e magnitude of
i s t r i v i a l t o see t h a t ( 4 ) i s a l s o a t r u e s t a t e m e n t .
g,
it
W e must b e c a r e f u l n o t t o become t o o g l i b i n o u r p r e s e n t approach.
T h a t i s , w e must n o t f e e l t h a t i t i s a t r u i s m t h a t we c a n j u s t v e c t o r i z e e v e r y t h i n g i n s i g h t and o b t a i n t r u e v e c t o r s t a t e m e n t s from t r u e
s c a l a r s t a t e m e n t s . One p l a c e where w e must be v e r y c a r e f u l , f o r
example, i s when w e v e c t o r i z e any n u m e r i c a l s t a t e m e n t t h a t i n v o l v e s
p r o d u c t s . Consider t h e p r o p e r t y of a b s o l u t e v a l u e s t h a t f o r any
numbers a and b
I f we t r y t o v e c t o r i z e ( 5 ) by b r u t e f o r c e we f i n d f o r one t h i n g t h a t
t h e r e s u l t i n g e q u a t i o n may w e l l b e meaningless. F o r i n s t a n c e , t h e
statement
i s meaningless ( i . e . , undefined.) s i n c e w e have n o t y e t d e f i n e d an
That i s , t h e l e f t s i d e of-:
" o r d i n a r y n p r o d u c t of two v e c t o r s -)a$.
Equation ( 6 ) i s u n d e f i n e d . I f we i n t e r p r e t t h e m u l t i p l i c a t i o n a s
s c a l a r m u l t i p l i c a t i o n and v e c t o r i z e Equation ( 5 ) a c c o r d i n g l y , we
obtain
:
(where on t h e r i g h t s i d e we mean t h e u s u a l m u l t i p l i c a t i o n s i n c e both
1 ;Iand I g ]
a r e numbers)
.
The c r u c i a l p o i n t i s t h a t both forms of equation ( 8 ) a r e meaningful b u t they both happen t o be f a l s e . Namely, -
-+ +
a-b =
(;I
IS(
cos
Hence,
since
lcos<l<
1, ( 9 ) y i e l d s the i n t e r e s t i n g , b u t perhaps s h a t t e r i n g
result that
I n a s i m i l a r way,
laxbl =
If1 ( 1
If
b Isin'
. \;
1;1
lgl
(since Isin
I n o t h e r words, any l i m i t proof t h a t involved t h e r e s u l t t h a t t h e
a b s o l u t e v a l u e of a product i s t h e product of t h e a b s o l u t e v a l u e s
might w e l l be f a l s e i n a b s o l u t e v a l u e o r l i m i t problems involving
e i t h e r t h e d o t o r t h e c r o s s product of v e c t o r s .
Rather t h a n b e l a b o r t h i s p o i n t , l e t us i l l u s t r a t e our i d e a s by means
of a few examples; and t o emphasize t h e i r importance, l e t us u t i l i z e
a new s e c t i o n t o d i s c u s s them.
Some L i m i t Theorems f o r Vectors
Theorem 1
Suppose l i r n
x+a
Then l i r n
x+a
a.
f (x)
(f ( x ) +
+
+
+
= Ll and l i r n g ( x ) = L2
x+@
G(x)) =
Zl + t2
.
.
To prove t h i s theorem, w e w r i t e down t h e analogous proof i n t h e s c a l a r
( W e w i l l n o t motivate t h i s proof s i n c e t h i s was a l r e a d y done i n
case.
P a r t 1 of o u r course. However, it w i l l be an i n t e r e s t i n g check f o r
you t o s e e how " n a t u r a l n t h e proof now seems t o you.)
W e had :
Suppose l i r n f (x) = L1 and l i r n f (x) = L2
x+a
x+a
.
Then l i r n [ f ( x ) + g ( x ) l = L1 + L2
x+a
The proof went a s f o l l o w s
(1) Let h ( x ) = f (x)
+
g (x)
.
W e must show t h a t l i m h ( x ) = L1
x+a
(2) Let E > 0 be a r b i t r a r i l y given.
6 2 such t h a t
+
L~
.
Then t h e r e e x i s t numbers 61 and
and
(3)
Let 6 = min {61,62)
.
T h e r e f o r e , 0 < I x - a1 < 6
(4)
But I f ( x )
-
L1l
+
(5) T h e r e f o r e , 0 < Ix
+
Ig(x)
-
a1
If(x)
-
- ~~1
> I[f(x)
< 6 +
L1l
+
I[f(x)
-
Ig(x)
-
L1l
L1l
-
E
L21 <
E
E
+ [(?(XI- 1 2 1 1
+ [g(x)
-
.
(=T+2)
L21I <
E
.
-
a1 < 6
17) Therefore, 0 < Ix
Therefore, l i m h ( x ) = L1 + L2
x+a
.
+
Ih(x)
-
(LL
-
+
L~)I
< t
The key now i s t o observe t h a t e v e r y s t e p i n o u r proof remains v a l i d
when v e c t o r s a r e introduced. Why? W e l l , f o r example, t h e v a l i d i t y of
s t e p (1) r e q u i r e s t h a t t h e sum of two numbers be a number; o t h e r w i s e ,
h ( x ) = f (x) + g ( x ) would be meaningless. To c o n v e r t t h i s i n t o a v e c t o r
s t a t e m e n t we only r e q u i r e t h a t t h e sum of two v e c t o r s be a v e c t o r , and
t h i s w e know is t r u e . The v a l i d i t y of s t e p ( 2 ) depends o n l y on t h e
d e f i n i t i o n of l i m i t and w e have taken c a r e t o m i m i c t h e s c a l a r d e f i n i t i o n i n forming t h e v e c t o r d e f i n i t i o n . S t e p (3) remains v a l i d by
" d e f h u l t H s i n c e only s c a l a r p r o p e r t i e s a r e being used. That i s , both
+
I f ( x ) - LII and I$(x)
LII a r e s c a l a r s .
The v a l i d i t y of s t e p ( 4 )
hinges on t h e " t r i a n g l e i n e q u a l i t y ; " l a + bl 6 la1 + lbl, b u t t h i s , a s
w e have s e e n , i s a l s o v a l i d f o r v e c t o r s . S t e p ( 5 ) is merely s u b s t i t u t i o n of s c a l a r q u a n t i t i e s . A s f o r s t e p ( 6 ) , t h i s f o l l o w s from t h e com-
-
mutative and a s s o c i a t i v e p r o p e r t i e s of numerical a d d i t i o n , and, a s w e
have s e e n , t h e s e p r o p e r t i e s a r e a l s o obeyed i n v e c t o r a d d i t i o n .
~ n ' o t h e rwords, i f we n v e c t o r i z e " t h e s c a l a r p r o o f , t h e r e s u l t i n g proof
i s a v a l i d v e c t o r proof. When w e do t h i s ( j u s t f o r p r a c t i c e ) we o b t a i n :
(1) L e t g ( x ) =
3 (x) +
6 (x).
W e must show t h a t l i m
x+a
h (x)
=
tl + t2 .
,
(2)
Let
E
t h e r e e x i s t numbers 61 and
> 0 be a r b i t r a r i l y given.
6 2 such t h a t
and
.
Let 6 = min {61,62)
Therefore, 0 < I x - a / < 6
(3)
+
+
~ f ( x )- L 1 l
+
I;(x)
+
-121 <
E
(4)
I ~ ( x ) - +~~1
(5) Therefore, 0 < Ix
(7) Therefore, 0
-
I;(x)
+
B U ~
-
a1 < 6
Ix-a1
C
+
Therefore, l i r n B(x) = L1
x+a
i212 I [ % ( X I-
+
+
< 6 +
+
L2
.
i l l + [;(XI
I[%(x)
-
ill
lg(x)
-
( i l + i 2<)El
+ [;(XI
-
i21
I
.
- t 2 1 ~ <E .
I t i s , of c o u r s e , important t o n o t i c e t h a t our v e c t o r proof a s i t
stands i s self-contained.
That i s , it e x i s t s i n i t s own r i g h t even
It's just that
had w e n o t o b t a i n e d it from an e a r l i e r s c a l a r proof.
by observing t h e s i m i l a r i t y i n s t r u c t u r e , we may deduce t h e p r o p e r t i e s
of c e r t a i n v e c t o r f u n c t i o n s from t h e corresponding p r o p e r t i e s of t h e
s c a l a r f u n c t i o n s , and i n t h i s way, we again b e n e f i t from t h e p a r a l l e l
structure.
Theorem 2
+
I f l i m ? ( x ) = L1 and i f l i m g ( x ) =
x+a
x+a
t2, then
i f h(x) =
f (x)
a x )I
Before proceeding w i t h t h e proof of Theorem 2 , we should n o t e t h a t
t h e r e i s more than one way t o form a product of two v e c t o r s . We have
a r b i t r a r i l y e l e c t e d t o t a l k about t h e d o t product. A s i m i l a r proof
holds f o r t h e c r o s s product and w i l l be l e f t a s an e x e r c i s e . I t i s
worth n o t i n g t h a t t h e f a c t t h a t we can form a product of two v e c t o r s
i n more than one way w i l l cause c e r t a i n complications i n v e c t o r calcul u s t h a t d i d n o t e x i s t i n s c a l a r c a l c u l u s , s i n c e t h e r e , a product of
two numbers could be i n t e r p r e t e d i n only one way.
Proof
Had t h i s been w r i t t e n i n terms of s c a l a r s r a t h e r than v e c t o r s , o u r
proof would have taken t h e form
f (x) = [f (x)
-
L1]
+
L1
therefore, If (x) g(x)
-
L1 L21 = IL2[f (x)
+ [f(x)
-
-
+
L1l
Ll[g(x)
Lll [g(x)
-
L2]
- L~II
-
-
Since 1 Ll 1 and1 L2 1 are bounded and both 1 f (x)
Ll 1 and1 g (x)
L2 1 can
be made arbitrarily small by choosing x sufficiently near 5, the right
a. This guarantees that
side of (1) gets arbitrarily small as x
+
- L1 L2) = 0
lim (f(x) g(x)
x+a
lim f (x) g(x) = L1 L2
x+ a
.
If we now repeat this argument with the appropriate "vectorization,"
we have
Therefore, ftx)
3 (XI
=
113(XI - tl1
iCElot.ice t h a t w h i l e , ' f o r -exampl'e,
+
ILl[g(x)'
- t21
1
'11
-
L2]
I
.
I'
+
F. 1 1 g ( x )
221
-
L2
I
our
s t r i n g of i n e q u a l i t i e s r e q u i r e s only t h e weaker r e s u l t t h a t
T h i s , a s we s h a l l s o o n s e e , i s
ILl [ g ( x )
L2]
6
lg(x)
L2
+
v e r y i m p o r t a n t b e c a u s e o f t h e v e c t o r p r o p e r t i e s t h a t la
bl d
-
and
;1
x
I
i~~1
-
I.
1:1 I ~ I
61 < 121 161.
**We p u t t h i s s t e p i n t o e m p h a s i z e t h e f a c t t h a t t h e r u l e s o f o r d i a a r y
algebra carry over very nicely here t o dot products.
therefore,
If (x)
-+
g(x)
-
-+
Ll
-+
L21
- it2
[f (x) -
tl] + El
[;(XI
-
221
(This last step is independent of any vector arithmetic since all dot
products are numbers not vectors.)
61 ,< 121 161 (observe that we don't need
Now since 1;
our last inequality yields
(Notice that there are no dot products here since
are numbers.)
1;
9 ) = 121 1611,
1 t21 , I 'E(x) - El1
etc.
Our last equation has the same properties as (1) and so the desired
result follows.
Additional examples are left for the exercises. Our aim in this section
was simply to illustrate that our limit theorems for scalars do indeed
carry over, and in a rather natural way, to vectors.
E
Derivatives of Vector Functions of Scalar Variables
In the study of ordinary calculus, we mentioned that the study of
differentiation could be viewed as an application of the limit theorems.
That is, when we defined f' by
f' (xl) = lim
Ax+O
f(xl + Ax)
A-x
-
1
f (xl)
every property of f' was then computed by the use of the limit theorems
applied to (1)
.
The point is that we now do the same thing for vectors.
that the expression
T(xl + Ax)
-
T(xl)
We observe
is well-defined since it involves the quotient of a vector and a scalar
and this may ve viewed as scalar multiplication. That is,
scalar
vector
Without belaboring this point, this expression can indeed be thought
of as representing an average rate of change, and, if we then proceed
to the limit, we may think of this as an instantaneous rate of change.
With this as motivation, all we are saying is that it is meaningful to
define f ' by
f v (xl) =
$(x1 + Ax)
lim
AX +O
- f (xl)
1
AX
The key point now is that since the usual recipes for derivatives
follow from the basic properties of limits and since the same limit
theorems are true for vectors as for scalars, it follows that there
will be recipes for vector derivatives similar to those for scalar
derivatives.
For example, it is provable that the derivative of a sum is the sum
of the derivatives. The proof is again verbatim from the proof in the
scalar case.
Without bothering to copy the scalar proof, we write down the vector
proof with the hope that you can recognize that it is the scalar proof
with appropriate vectorization. We have
Let S(x) =
?(XI +
.
;(XI
[h(xl + Ax) - s(xl) ]
-b
Then
if'
(xl) =
lim
Ax+O
=
lim
Ax+O
bx..
+
.
(xl + AX)
Ax) +
+ Ax)
- f (xl)
[f (xl) +
- 1
AX
Ax+O
lim
Ax+O
-
(xl)I
AX
*(xl
=
.
[f
(xl + AX)
AX
-
x
]
g(xl + AX)
AX
+
lim
Ax+O
[$
1
- g(xl)
-t
(xl + Ax)
AX
-~(Ic~)
(since the limit of a sum is the sum of the limits,
as shown in the previous section).
3.16
I n a s i m i l a r way (and t h e d e t a i l s a r e l e f t a s e x e r c i s e s ) , t h e same
product r u l e a p p l i e s t o v e c t o r s . The only d i f f e r e n c e i s t h a t we now
have s e v e r a l t y p e s of products. For example, w e may have t h e product
of a s c a l a r f u n c t i o n w i t h a v e c t o r f u n c t i o n , o r we may have t h e d o t
product of two v e c t o r f u n c t i o n s , o r w e may have t h e c r o s s product of
two v e c t o r f u n c t i o n s .
I n any c a s e , we o b t a i n t h e r e s u l t s
( I n t h i s l a s t c a s e beware of changing t h e r o l e s of t h e f a c t o r s
+
+
a x b = - 6 x ;).I
We even have a form of t h e q u o t i e n t r u l e pravided t h a t our q u o t i e n t
i s a v e c t o r f u n c t i o n d i v i d e d by a s c a l a r . f u n c t i o n ( f o r , otherwise, t h e
q u o t i e n t i s n o t d e f i n e d ) . The formula i s
These formulas have a n i c e a p p l i c a t i o n t o C a r t e s i a n c o o r d i n a t e s . Suppose
F i s a v e c t o r f u n c t i o n of t h e s i n g l e r e a l v a r i a b l e t. For example, we
-k
may be c o n s i d e r i n g f o r c e a s a f u n c t i o n of time. Suppose F has t h e form
-b
and we d e s i r e t o compute d$/dt.
Since t h e d e r i v a t i v e of a sum i s t h e sum of t h e d e r i v a t i v e s , we have
2~:
:
y,L'
,,i
,..-
:..
=
I L*
3.y-
5,
-1.
'.L,
\
.
.
s ri-
*-
g .
US".
*..
.
;r '
Then s i n c e a c o n s t a n t times a v a r i a b l e has a s i t s d e r i v a t i v e t h e c o n s t a n t
t
+
times t h e d e r i v a t i v e of t h e v a r i a b l e , and s i n c e 1, J , and k a r e c o n s t a n t s ,
we o b t a i n :
While this was a specific example, it should be fairly easy to see that, in general, the derivat-ive is obtained by differentiating the components. Further details are left to the text and the exercises, but we want to emphasize again the fact that our definition of limit for vectors, being so closely modeled after the scalar situation, guarantees that the differentiation formulas with which we are already familiar in the scalar case are also valid in the vector case. MIT OpenCourseWare
http://ocw.mit.edu
Resource: Calculus Revisited: Multivariable Calculus
Prof. Herbert Gross
The following may not correspond to a particular course on MIT OpenCourseWare, but has been
provided by the author as an individual learning resource.
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.