APPLICATIONS OF LINEAR ALGEBRA TO NON-LINEAR FUNCTIONS From many p o i n t s o f view t h i s c h a p t e r s h o u l d b e t h e l a s t few
You may r e c a l l w e m o t i v a t e d t h a t c h a p t e r
s e c t i o n s o f Chapter 6 .
by l o o k i n g a t t h e s y s t e m o f e q u a t i o n s :
and s t u d y i n g t h e s p e c i a l c a s e i n which f l ,
f u n c t i o n s of t h e n v a r i a b l e s xl,
..., and
..., and xn.
f m were l i n e a r
The r e a s o n f o r emphasizing l i n e a r i t y was based on t h e fundamental
r e s u l t t h a t i f fl,...,
and f w e r e n o t l i n e a r b u t w e r e a t l e a s t
m
continuopsly d i f f e r e n t i a b l e f u n c t i o n s of xlr...,
...,
and xn, t h e n
...,
and Axn would produce i n c r e m e n t s Ayl,
i n c r e m e n t s Ax1,
and Aym which c o u l d b e approximated by t h e d i f f e r e n t i a l s dy
and dym, where
Recall t h a t
xl,
..., and
ll...,
x
r e f e r t o increments measured from
n
all...,
a g i v e n p o i n t ; s a y , xl
xn - a n t s o t h a t each o f t h e c o e f f i c i e n t s , ayi/ a x j l i n ( 2 ) i s a c o n s t a n t , namely
--
*
-
A c t u a l l y , had we b e e n c o n s i s t e n t w i t h o u r n o t a t i o n i n ( 1 ) we would an) r a t h e r than ( 2 ' ) .
have w r i t t e n f i x ( a l ,
The c o n f u s i o n a r i s i n g from t h e u s e o f j m u l t i p l e s u b s c r i p t s e n c o u r a g e d u s t o a d o p t t h e s h n r t ~ rn o f a t i o n u s e d i n ( 2 ) and ( 2 ' ) .
...,
7.1
[where x = (xl,.
..,xn)
and a = (alf..
., an ) 1
In any e v e n t , t h e main p o i n t i s t h a t w h i l e s y s t e m (1) i s n o t n e c e s s a r i l y
, and
l i n e a r , system (2) is. That i s , system ( 2 ) e x p r e s s e s dyl,.
dym a s , l i n e a r combinations of dxl,
and dxn.
..
...,
Thus, t h e s t u d y of l i n e a r a l g e b r a , introduced i n c h a p t e r 6 t o h e l p
us s t u d y system (1) i n t h e c a s e t h a t t h e f u n c t i o n s were l i n e a r , can
now be a p p l i e d t o system (2) provided only t h a t t h e f u n c t i o n s a r e
continuously d i f f e r e n t i a b l e ( a f a r weaker [ i . e . , more g e n e r a l ]
c o n d i t i o n than being l i n e a r ) . I n summary, w h i l e yl,
and ym need
and xn; dy
and dym a r e
n o t b e l i n e a r combinations of xl,
...,
...,
...,
lf...,
...,
and dxn; s o t h a t we may view Ayl,
l i n e a r combinations of dxl,
and Aym a s l i n e a r combinations of Ax1,
Axn i n s u f f i c i e n t l y small
...,
neighborhoods of a p o i n t (al...,a,).
Our main reason f o r beginning a new c h a p t e r a t t h i s time i s s o
t h a t w e may b e t t e r emphasize the i m p l i c a t i o n s of system ( 2 ) .
The Jacobian Matrix
I n terms of s t r u c t u r e , system (1) may b e viewed a s an example of
That i s , system (1) may be i d e n t i f i e d w i t h t h e v e c t o r
function
f (z).
d e f i n e d by
where
yl,
..., and ym a r e a s i n
(1).
[Equation ( 4 ) i s a b b r e v i a t e d a s u s u a l by y = f ( f f ) where
(y,,...,~ m
and
x=
y
=
(X~,--.,X~)I
In Chapter 6 w e introduced such q u e s t i o n s a s whether f was o n t o
and/or 1-1. W e could extend our i n q u i r y s t i l l f u r t h e r by asking
what i t would mean to t a l k about, s a y ,
-k
f (5)
-
lirn
x+a
For example, i n l i n e w i t h o u r p r e v i o u s s t r a t e g i e s , it would make
sense t o say t h a t
f(?I) = L
lim
(where 2
E
-E
E~ b u t L
E"
since
f(5)E E ~ )
x+a
means
Given
E
> 0 t h e r e e x i s t s 6 > 0 such t h a t
W e a l s o might have n o t i c e d t h a t
may i t s e l f b e viewed a s an
Namely, i f w e s u b s t i t u t e t h e v a l u e s o f yl,
m-triple.
t'
..., ahd
Ym a s g i v e n by (1) i n t o ( 4 ) w e o b t a i n
S i n c e each component of t h e m - t r i p l e i n e q u a t i o n ( 5 ) is a s c a l a r
x we
a l r e a d y know how t o compute l i m i t s i n v o l v i n g t h e s e
[By way o f review l i r n
f (f)= L means given E > 0 w e
x-+a
can f i n d 6 > O such t h a t o < 1 1 x---z
1 1 < 6 -+ ( f ( x )
<<EJ.
x&,
function of
components.
- LI
I t would t h e n s e e m n a t u r a l t o u s e ( 5 ) - t o d e f i n e l i r n
f(2);i . e . ,
x+g
lim
f (x)
-
= [lim fl(x)
,..., l i r n
fm(x)l
-
( D e f i n i t i o n ( 6 ) c a n be shown t o b e e q u i v a l e n t t o t h e ~ , 6
definitionb
I
but t h i s i s n o t too important i n t h e present context.)
..
1
L
!
R e t u r n i n g t o ( 5 ) , n o t i c e t h a t we may now w r i t e
f =
-
(71'
(fll...lfm)
with t h e understanding t h a t ( f l ,
...,f m ) means
[fl(x) , . . . , f m ( ~ ) l .
I f we w e r e now t o become i n t e r e s t e d i n t h e c a l c u l u s of f it would
seem n a t u r a l t o extend t h e n o t a t i o n i n ( 7 ) by d e f i n i n p
and
dx =
-
(dxl , . . . , d ~
To d e f i n e f ' (5) w e might then t r y t o copy t h e s t r u c t u r e
used i n t h e c a l c u l u s of a s i n g l e v a r i a b l e .
That i s , w e s h a l l t r y t o r e l a t e
t h e form
9 and g,a s
d e f i n e d by ( 8 ) , i n
and then t o c a p t u r e t h e s t r u c t u r e i n (9), d e f i n e M t o be f'( 5 ) .
Observe. t h a t equation (10) i s a " s t r a n g e animaltt, Neither & nor
I n f a c t , each i s a v e c t o r and they need n o t have
d x i s a number.
t h e same dimension. That i s , 9 i s an m - t r i p l e , dx i s an n - t r i p l e
and m need n o t e q u a l n. With t h i s i n mind, equation (10) should
as
suggest matrix arithmetic.
(For example, viewing d~ and $ b ~
is m by 1 w h i l e dx is n by 1.
column m a t r i c e s , we see t h a t
Hence, by t h e u s u a l p r o p e r t i e s of m a t r i x m u l t i p l i c a t i o n M [ i n
equation (10) 1 must be an m by n matrix. )
-
I f w e re-examine system ( 2 ) i n an e q u i v a l e n t m a t r i x form, we s e e
that
r
Equation (11) now t e l l s us how t o d e f i n e f'(5). Namely w e d e f i n e
f ' (x)
- t o be t h e m by n m a t r i x [ayi/ak. ] whereupon (11) becomes
-
3
3 = g u ( ~ ) d.x
[z]
The m a t r i x
i s s o important t h a t it i s given a s p e c i a l name,
Namely
independently o$ whether we e l e c t t o i n t e r p r e t i t a s
(?I).
I'
i f yl....,
and ym a r e continuously d i f f e r e n t i a b l e f u n c t i o n s of n
independent v a r i a b l e s x ~ , . . . , and xn, we d e f i n e t h e Jacobian Matrix
of yl, ...,y m w i t h r e s p e c t t o x l t . . . , x n
t o be t h e m by n m a t r i x
This m a t r i x i s o f t e n denoted by
The remaining s e c t i o n s of t h i s c h a p t e r d e s c r i b e a p p l i c a t i o n s of t h e
Jacobian.
The I n v e r s e Function Theorem
I n t h e previous u n i t , we d i s c u s s e d informally**how t o i n v e r t system
(1) i n t h e c a s e m = n.
Namely, given t h e system
...,
..
* In many tex
xn ) is reserved to name the
determinant o
Notice that determinants
apply only to square matrices, hence, we prefer our notation and
,xn ) when w e mean the determinant.
w e will write 1 a(yl,. .yn) /a(xl,.
.
I
**That is, we used loose expressions such as "dy* A y" and "sufficiently small" without attempting to show what these meant in rigorous, computational terms. w e formed
from which w e concluded (from our s t u d i e s of l i n e a r a l g e b r a ) t h a t
...,
...
and dxn could be expressed uniquely a s l i n e a r combinations
dxl,
and dyn provided t h a t [ a y i l a k . ] was non-singular
of dyl,
3
(i.e . , [ ayi/axj I-'
e x i s t e d , o r i n t h e language of determinants,
01
d e t [ayi/ax. 1
J
+
.
Restated i n t h e language of v e c t o r c a l c u l u s , from system (1) we
obtained system (2). which i s e q u i v a l e n t t o
9=
-f ' (-x )-d x ,
ayi
where
f'(5) = [-XI,
3
and we concluded t h a t we could s o l v e f o r
that
If'(5)1 * #
0.
dx i n
terms of
3 provided
-
In f a c t , i f If'
- (x)1 # 0 t h e n
This r e s u l t can be proven r i g o r o u s l y and once proven i t goes under t h e name of t h e I n v e r s i o n Theorem o r t h e I n v e r s e Function Theorem. The theorem c o r r o b o r a t e s what we a l r e a d y suspected from our informed treatment. More s p e c i f i c a l l y : The I n v e r s e Function Theorem n
Suppose g:E +E" i s continuously d i f f e r e n t i a b l e * * a t 5 = 5 EE". I
1
*
fl(x)
Keep i n mind that
is, i n the present context, a n n by n
m a t r i x s o it m a k e s s e n s e to t a l k about t h e determinant of
,...,
fl(x).
**In terms o f f = (f
fn), w e d e f i n e f to b e continuously d i f f e r e n t i a b l e ++ eacA of its s c a l e r - f u n c t i o n components fl,.
,
and f is c o n t i n u o u s l y differentiable. n 7.6
..
-.--.
I
I
Then if
f ' (2)#
I
I
-
--.
0 there is a (sufficiently small) neighborhood
.
...
I
119'(b)1
I
=
1
1
~ (a)
' 1
-
where b
- = f (a).
-- -The theorem is given a computational application in Exercise
4.6.1.
Pictorially, the theorem says:
X
-2
-- - - - - -f-
-
-
S = f(R)
%
-
En
E"
Summary
1. f is 1-1 on R, but
= y E S where
E R it is possible that there is
2. If f
However z2 cannot
another vector 5 2 ~ ~ 1such
1
that f
=
f is 1-1 on R).
be in R (since 3. This is why the inverse function theorem is said to involve
a local property. That is, all "bets" about f being 1-1 are
"off' once R gets too large.
-(xl)
A
r8
-;.:'
In other words if S = f(R) then f:R-+S is
(i.e.,
f1
Of
= a exists).
f
is then a
invertible, i.e., f'l: s + R exists. Moreover 3 = l'
'continuously differentiable function of yl,
,yn in S and
'
I
-
h that, when restricted to R, f is 1-1 and onto
lR
I
I
I
I
I
I
I
I
I
I
I
I
I
.a-
xl
New Look at the Chain Rule
Suppose
(zl) -f(z2).
.. .
.
.
and
Then c l e a r l y , systems (12) and (13) 'allow us t o conclude t h a t
zm a r e f u n c t i o n s of xl,
x
zl,
n
..., .
...,
Notice t h a t i f w e e l e c t t o w r i t e (12) and (13) i n such a way
a s t o emphasize t h e i d e a of v e c t o r f u n c t i o n s , we have
,...,
where:=
(fl
f k ) , P = (gl
a r e a s i n (12) and ( 1 3 ) .
I q t e r m s of ( 1 4 )
, we
,-...,g m ) ; and f l ,...,f k ,
see that i f
5
=
q
-f then
-h
: En
gl,--,g,
-t
Em;
i. e. , 2 = h(x) , where h (x) = g(: (5)) , and t h i s i n t u r n seems t o
s u g g e s t t h e c h a i n r o l e . I n f a c t i f w e assume t h a t g and 3 a r e
continuously d i f f e r e n t i a b l e , systems (12) and (13) l e a d t o
-
and
Since (15) and (16) a r e l i n e a r systems, we may apply our knowledge
of m a t r i x a l g e b r a t o conclude t h a t
--
r
and s i n c e m a t r i x m u l t i p l i c a t i o n i s a s s o c i a t i v e t h i s says t h a t
I f w e now i n t r o d u c e t h e Jacobian n o t a t i o n a s w e l l a s t h e n o t a t i o n s
dx and dz i n t o (17) i n t o (17) we o b t a i n
-
-
With t h i s n o t a t i o n , n o t i c e a l s o t h a t (15) and (16) t a k e t h e forms
dz
-
p -
-
dx
To i d e n t i f y t h i s d i s c u s s i o n w i t h t h e d e r i v a t i v e s of v e c t o r f u n c t i o n s
\
n o t i c e t h a t e q u a t i o n (19) s a y s
while equation (19) s a y s
'since
=
h(x) ,
i t a l s o follows t h a t
Comparing (21) and (22) w e s e e t h a t t h e chain r u l e a s it e x i s t s
i n t h e s t u d y of t h e c a l c u l u s of a s i n g l e r e a l v a r i a b l e extends t o
t h e c a l c u l u s of v e c t o r f u n c t i o n s .
C
By way of review, r e c a l l t h a t t h e "old chain r u l e s a i d :
I1f f : E l + E l , g : E' - + E l a r e d i f f e r e n t i a b l e and h = gof, then h
is a l s o d i f f e r e n t i a b l e .
Moreover i f xo E don f , h 1 (xo) = g ' (yo)
) where yo = f ( x o ) .
Now what w e have is:
En +Ek and
q : E~ + E ~a r e continuously d i f f e r e n t i a b l e and
-
= gof then h : En
oreover, i f +
x
+
E dom
Em i s a l s o continuously d i f f e r e n t i a b l e .
-
f and yo = f
, then
-h 1 (s)=
[ q l (yJ I
[go(&) I
In f a c t i f we i d e n t i f y 1 by 1 m a t r i c e s w i t h numbers, it i s e a s i l y
seen t h a t t h e "old" c h a i n r u l e i s a s p e c i a l c a s e of the new
chain r u l e .
I n a d d i t i o n n o t i c e t h a t t h e Jacobian m a t r i x n o t a t i o n
x ) i s an e x t e n s i o n of t h e i d e a of w r i t i n g
a(yl, ...,yk)/ a(xl,
n
Namely, i f we look a t (18) and (23) w e s e e t h a t
f ' (x) a s dy/dx.
...,
,
dz =
-
a(zl,
a(Xl,..
...,zm)
.,xn)
-
dx and
3
so that
=
..
,yk) a s a number
Equation (23) i n d i c a t e s t h a t w e may t r e a t (yl,.
and cancel i t from both numerator and denominator on t h e r i g h t
hand s i d e of (231, s o t h a t t h e f r a c t i o n - l i k e n o t a t i o n f o r t h e
Jacobian i s j u s t i f i e d .
Again, a s reinforcement, n o t i c e i n t h e s p e c i a l c a s e t h a t r = k = m = 1,
e q u a t i o n (23) becomes the f a m i l i a r
r = k = m and
e q u a t i o n ( 2 3) becomes
A s another s p e c i a l c a s e i f
exists) ,
-1
- (assuming
= f
-
f 1
and s i n c e
a(xlI".
,xn)
= I
( s e e E x e r c i s e 4.6.8)
a(xl, ...,x nI
e q u a t i o n (24) r e a f f i r m s t h e f a c t t h a t
Other computations a r e l e f t t o E x e r c i s e 4.6.4,
and t h i s concludes
our i n t r o d u c t o r y remarks concerning t h e Jacobian a s an extension
of t h e c h a i n r u l e .
F u n c t i o n a l Dependence
,
.
.
, - -
.
,.
I n o u r study of l i n e a r systems of e q u a t i o n s , we saw t h a t a c r u c i a l
p o i n t was whether any of t h e e q u a t i o n s was a l i n e a r combination of
the others.
For example, we saw t h a t t h e system
was n o t i n v e r t i b l e , even though it was t h r e e e q u a t i o n s i n t h r e e
unknowns, because t h e t h i r d equation was twice t h e second minus
t h e f i r s t . That i s , y3 + 2y2
yl; o r e q u i v a l e n t l y , yl = 2y2
y3, e t c . , b u t t h e important p o i n t i s t h a t a t l e a s t one of t h e
e q u a t i o n s can be w r i t t e n a s a l i n e a r combination of t h e o t h e r s .
-
-
For t h i s reason we s a y t h a t t h e system of e q u a t i o n s (25) i s l i n e a r l y
dependent. The concept of l i n e a r dependence w i l l occur again i n
o u r c o u r s e i n Block 7 when w e d i s c u s s d i f f e r e n t i a l equations and
i n Block 8 when we t a l k more about l i n e a r a l g e b r a .
For now,
however, a l l we want t o p o i n t o u t i s t h a t t h e q u e s t i o n of i n v e r t i b i l i t y of a system of n e q u a t i o n s i n n unknowns was resolved i n
terms of whether t h e system of e q u a t i o n s was l i n e a r l y dependent.
This i d e a could be extended t o t h e c a s e i n which t h e number of
e q u a t i o n s and t h e number of unknowns w e r e unequal. For example,
i f we were given f o u r l i n e a r e q u a t i o n s i n seven unknowns, we would,
i n g e n e r a l , e x p e c t t o be a b l e t o choose t h r e e of t h e unknowns a t
random and t h i s would have t h e e f f e c t of reducing our system t o
f o u r e q u a t i o n s i n f o u r unknowns, from which we could then determine
t h e v a l u e of t h e o t h e r f o u r unknowns. The s u c c e s s of t h i s procedure,
of course, r e q u i r e d t h a t t h e r e s u l t i n g system of f o u r equations i n
f o u r unknowns was n o t l i n e a r l y dependent.
Now, i f we l e a v e o u t any r e f e r e n c e t o t h e e q u a t i o n s being l i n e a r ,
t h e same type of q u e s t i o n s i s s t i l l suggested. For example,
l i n e a r system of t h r e e e q u a t i o n s
suppose w e a r e given t h e
i n t h r e e unknowns,
then we might be tempted t o ask whether t h e system can be i n v e r t e d .
That i s , does (26) d e f i n e x,y, and z ( a t l e a s t , i m p l i c i t l y ) a s
f u n c t i o n s of u,v, and w? Without t r y i n g t o e s t a b l i s h any a n a l y s i s
of how we might o b t a i n t h e r e s u l t , t h e f a c t i s t h a t from (26) w e
can show t h a t
That i s , w i s dependent on u and v, even though now t h e dependence
i s no longer l i n e a r . What t h i s means, by use of ( 2 7 ) , is t h a t any
f u n c t i o n f ( u , v , w ) i s a c t u a l l y a f u n c t i o n only of t h e two independent
v a r i a b l e s u and v (and n o t i c e t h a t we have n o t proven t h a t u and v
a r e independent, b u t a glance a t ( 2 6 ) should convince you t h a t they
are*)
.
In particular
*But i n t h e e v e n t y o u a r e n o t c o n v i n c e d a more i n d e p t h s t a t e m e n t
w i l l b e made later.
I n any e v e n t , when a c o n d i t i o n such a s (27) h o l d s , we say t h a t t h e
f u n c t i o n s ( v a r i a b l e s ) u , v , and w a r e f u n c t i o n a l l y dependent.
This is
Namely,
every
c
a
s
e
i
n
a g e n e r a l i z a t i o n of l i n e a r dependence.
which we have l i n e a r dependence we a l s o have f u n c t i o n a l dependence,
b u t we may have f u n c t i o n a l dependence without having l i n e a r
dependence.
Indeed, (27) shows us t h a t we have f u n c t i o n a l dependence,
b u t t h e f a c t t h a t u2 a p p e a r s on the r i g h t s i d e of (27) t e l l s us
t h a t t h e dependence i s n o n l i n e a r .
There i s a very c o n c i s e mathematical way of s t a t i n g what we mean
by f u n c t i o n s being f u n c t i o n a l l y dependent.
For t h e sake of conc r e t e n e s s w e w i l l i l l u s t r a t e t h e d e f i n i t i o n i n terms of t h r e e
e q u a t i o n s i n t h r e e unknowns, and then s t a t e t h e more g e n e r a l
definition.
u, v, and w a r e f u n c t i o n s of x, y , and z .
Then u, v , and
a r e s a i d t o be f u n c t i o n a l l y dependent i f and only i f t h e r e e x i s t s
;E , such
= 0 b u t f # 0 , ath~ry=$e
-13
'
2
7and w a r e c a l l e d f u n c t i o n a l l y independent.
f : E~
t h a t f (u,i,w)
r
,
A , \
A complete understanding of t h i s d e f i n i t i o n a g a i n r e q u i r e s t h a t we
know t h e d i f f e r e n c e between an i d e n t i t y and an equation.
For example, i f f = 0 , then f ( y l , y 2 , y 3 ) = 0 f o r every 3 - t r i p l e
What our d e f i n i t i o n s a y s t h a t i f f # 0 b u t f ( u , v , w ) = 0
(yl,y2,y3).
t h e n u, v, and w a r e f u n c t i o n a l l y dependent.
To i l l u s t r a t e t h i s i n terms of t h e s p e c i f i c system ( 2 6 ) , w e see
from (27) t h a t
Using ( 2 7 ' ) a s m o t i v a t i o n , c o n s i d e r t h e f u n c t i o n f J defined by
C l e a r l y f i s n o t t h e zero f u n c t i o n , s i n c e among o t h e r t h i n g s ,
i f we l e t yl = y2 = yg = 1 i n (28) we o b t a i n f ( 1 1 , ) = l2
On t h e o t h e r hand, (28) says t h a t f ( u , v , w ) = u2
= -1 f 0.
and t h i s i s i d e n t i c a l l y zero from ( 2 7 ' )
.
1-v-
-
1
W,
More i n t u i t i v e l y , t h e mathematical d e f i n i t i o n o f f u n c t i o n a l
dependence simply s t a t e s t h a t t h e r e i s a n o n - t r i v i a l * r e l a t i o n s h i p
- .
between u, v, and w t h a t makes t h e s e v a r i a b l e s dependent. "
Our aim i n t h i s s e c t i o n is t o show how f u n c t i o n a l d e p e n d e n c e ' i s
r e l a t e d t o a Jacobian m a t r i x ( o r d e t e r m i n a n t ) . Again, o m i t t i n g
..
proofs, t h e major r e s u l t is:
.
A.
-.
u = u ( x , Y , z ) , v = v ( x , Y , z ) , and w = w(x,y,z) where x , y, and z
independent v a r i a b l e s and u,v,w a r e continuously d i f f e r e n t i a b l e .
u, v, and w a r e f u n c t i o n a l d e p e n d e n t - i n some r e g i o n R i f and
1~ 0 i n t h e r e g i o n R (i.e., i f and only
i s s i n g u l a r f o r a l l ( x , y , z ) ~R)
Again, from an i n t u i t i v e p o i n t of view, t h i s c o n d i t i o n i s t h e only
one t h a t p r e v e n t s u s from s o l v i n g f o r dx, dy, and dz i n terms of
du, dv, and dw once we have t h a t
With r e s p e c t t o ( 2 6 ) , observe t h a t
therefore,
5
3 3
= 32x yz + 3
2 + 32x
~ y z ~
~
3
3
5
-32x y z
32xy3z3
32xy z
5
3 3
+32xy3z3 + 32xy5z
32x yz
32x yz
-
-
-
~
~
-
*By " t r i v i a l " we mean t h a t f o r any t h r e e v a r i a b l e s U , V , and w ;
ou
o v + ow 3 0 .
Non-trivial, therefore, i s reflected i n the
d e f i n i t i o n by t h e r e q u i r e m e n t t h a t f $ 0 .
+
.
which t e l l s us t h a t u, v, and w a r e f u n c t i o n a l l y dependent, even
though i t may n o t r e v e a l t h e s p e c i f i c r e l a t i o n s h i p t h a t e x i s t s
between u, v, and w.
This r e s u l t c a r r i e s over i n t o t h e c a s e i n which we have more
unknowns than equations.
For example suppose t h a t f l ,
f n a r e continuously d i f f e r e n t i a b l e f u n c t i o n s of t h e n
X1t-~tXn
+
k-
..., and
+
k variables
Then t h e system of n-equations
i m p l i c i t l y d e f i n e s any n of t h e unknowns a s d i f f e r e n t i a b l e f u n c t i o n s
of t h e o t h e r k unknowns i n a neighborhood of a p o i n t s a t i s f y i n g
...,
( 2 9 ) i f t h e Jacobian determinant of f l ,
f n with respect t o the
n dependent v a r i a b l e s i s not zero a t t h a t p o i n t .
Again, f u r t h e r d e t a i l s a r e l e f t f o r t h e e x e r c i s e s .
The Jacobian i s t o t h e s t u d y of t h e c a l c u l u s of v e c t o r f u n c t i o n s
a s t h e d e r i v a t i v e i s t o t h e s t u d y of c a l c u l u s of a s i n g l e r e a l
variable.
I n o r d e r t o appreciate t h e Jacobian, we f i r s t introduced l i n e a r
a l g e b r a and a p p l i e d t h i s s t u d y t o systems of e q u a t i o n s involving
total differentials.
I n t h i s c h a p t e r we have t r i e d t o g i v e an i n k l i n g a s t o what r o l e
t h e Jacobian, i n p a r t i c u l a r , and m a t r i x a l g e b r a , i n g e n e r a l ,
play i n t h e development of t h e c a l c u l u s of s e v e r a l r e a l v a r i a b l e s .
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.