i2-chapterii

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CHAPTER II

DIFFERENTATION IN SEVERAL VARIABLES

Department of Foundation Year,

Institute of Technology of Cambodia

20142015

Department of Foundation Year ITC 1 / 62
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Contents

1 Introduction to Functions of Several Variables

2 Limits and Continuity

3 Partial Derivatives

4 Differentials

5 Chain rules for functions of several variables

6 Directional Derivatives and Gradients

7 Tangent planes and normal lines8 Extrema of Functions of Several Variables

9 Lagrange Multipliers

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Contents




4 Differentials



7 Tangent planes and normal lines8 Extrema of Functions of Several Variables


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Introduction to Functions of Several Variables

Definition 1

Let D be a nonempty set and that for each element x in D therecorresponds a unique value y =f(x)in R, then f is called a function

ofx. The setD is the domainoff, andRis the rangeoff.We write

f :D R; xf(x).

In this case x is called independent variable and y is called

dependent variable.

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Note

A real-valued function on subset D ofRn is a function whose rangeis R. That is,

f :D Rn R; (x1, x2, . . . , xn)y =f(x1, x2, . . . , xn)

Special cases for n= 2and n= 3will be mainly concerned since theyhelp to visualise their geometrical meaning.

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Some Operations on Rn

Letx= (x1, x2, . . . , xn), y= (y1, y2, . . . , yn) Rn and R.We defineAddition:

x+y = (x1+y1, x2+y2, . . . , xn+yn)

Scalar multiplication:

x= (x1, x2, . . . , xn)

Inner product:

x y=x, y= xy =x1y1+x2y2+ +xnyn

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In this study we use only Euclidean Norm, that is ifx= (x1, x2, . . . , xn) Rn, then norm ofx is defined by

x= n

i=1

x2i1/2

= x21+x

22+ +x

2n.

Ifx= (x1, x2, . . . , xn), y= (y1, y2, . . . , yn) Rn, then norm of thedifference x and y (or Euclidean distance between x and y)is defined by

x y=

ni=1

(xi yi)2

1/2

.

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Let x0 Rn and let >0. A neighbourhood or-neighbourhood about x0 is denoted and defined by

N(x0) ={x Rn :x x0< }.

A point x0 in R is called an interior point ofR if there exists an-neighbourhood about x0 that lies entirely in R. That is,

x0N(x0) R.

The interiorofR, denoted by

R orint(R), is the set of all interiorpoints ofR.

A point x0 is a boundary pointofR if every neighbourhoodabout x0 contains points inside R and points outside R.

>0 :N(x0) R= and N(x0) Rc =

The boundary ofR, denoted by R or b(R), is the set of all

boundary points ofR.Department of Foundation Year ITC 7 / 62
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Figure: Interior pointFigure: Boundary point

A region R is open if it is a subset of its interior. That is, R

R

A region R is closed if it contains its entire boundary. That is,R R

The closure ofR is denoted by

R and defined by

R=

R R

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Level curve, level surface and level hypersurfce

Definition 2 (Level curve, level surface and level hypersurfce)The set of points(x1, x2, . . . , xn)in R

n where a function ofnindependent variables has a constant value f(x1, x2, . . . , xn) =c iscalled a level hypersurface off.

In particular,the set of points in the plane where a function f(x, y)has aconstant value f(x, y) =c is called a level curve off.

the set of points in the space where a function f(x,y,z)has aconstant value f(x,y,z) =c is called a level surface off.

Note that ifn 4, the set of points satisfying the equationf(x1, x2, . . . , xn) =c is called level hypersurface.

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Example of level curves

Figure: Level curves show the lines ofequal pressure (isobars) measured inmillibars

Figure: Level curves show the lines ofequal temperature (isotherms)measured in degree Fahrenheit.

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Contents




4 Differentials



7

Tangent planes and normal lines8 Extrema of Functions of Several Variables


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Limit of a Function of Several Variables

Definition 3

Let f :D Rn Rm and a

D. Then we say that the limit off(x)equals L as x approaches a, written as

limxa

f(x) =L,

if given any >0, there exists a >0such that

f(x) L< whenever x a< .

Theorem 4

If a limit exists, it is unique.

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Theorem 5

Suppose that limxa

f(x) and limxa

g(x) both exist and thatk is a scalar.

Then

limxa

[f(x) g(x)] = [limxa

f(x)] [limxa

g(x)]

limxa

[kf(x)] =k[limxa

f(x)]

limxa

[f(x)g(x)] = [limxa

f(x)][limxa

g(x)]

limxa

[f(x)/g(x)] = [limxa

f(x)]/[limxa

g(x)] provided limxa

g(x)= 0 and

bothf andg are real-valued functions.

Iff(x) g(x) for allx, then limxaf(x) limxag(x), wherefandg are real-valued functions.

Iff(x) L g(x) for allx and if limxag(x) = 0, thenlimxaf(x) =L.

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Note

To show that the limit does not exist, we need to show that thefunction approaches different values as x approaches a along differentpaths in Rn.

Theorem 6

Letf :D Rn Rm be a vector-value functions,f= (f1, f2, . . . , f m)andL= (L1, L2, . . . , Lm). Then lim

xaf(x) =L if and only if

limxa fi(x) =Li fori= 1, 2, . . . , m.

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Continuity

Definition 7

Let f :D Rn Rm. We say that the function f is continuous at apoint a in D if

limxa

f(x) =f(a).

We say that f is a continuous function on D if it is continuous at every

point in its domain D.

Theorem 8 (Algebraic properties)

Letf, g:D Rn Rm be continuous vector-value function and let

R be a scalar. Thenf+g andf g are continuous.

If bothf andg are real-valued functions and ifg(x)= 0, thenf /gis continuous.


C
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Continuity

Theorem 9

Letf :D Rn

Rm

, wheref= (f1, . . . , f m). Thenf is continuousata D (respectivelyf is continous onD) if and only if its componentfunctionsfi:D R are all continuous ata.


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Contents




4 Differentials



7 Tangent planes and normal lines

8 Extrema of Functions of Several Variables



P ti l d i ti f f ti f l i bl
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Partial derivatives of a function of several variables

Definition 10

Let f :D Rn R. Ify=f(x) =f(x1, x2, . . . , xn), then the firstpartial derivative offwith respect to xi, i {1, 2, . . . , n}, is defined

byfxi(x)or fxi (x)and

fxi(x) =

xif(x) = lim

xi0

f(x1, . . . , xi+ xi, . . . , xn) f(x1, . . . , xn)

xi

provided the limit exists.


P ti l D i ti
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Partial Derivatives

Theorem 11

Letf, g be two scalar-valued functions ofn variables and letx= (x1, . . . , xn). Iffxi(x) andgxi(x) exist, then

(f+g)

xi x) =fxi(x) +gxi(x) where is some constant

(f g)

xi(x) =fxi(x)g(x) +f(x)gxi(x)

(f /g)

xi (x) =

fxi(x)g(x) f(x)gxi(x)

g2(x) ifg(x)= 0.


P ti l D i ti
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Partial Derivatives

Definition 12

A function f :D Rn R is said to be of class Ck if all itspartial derivatives of order k are continuous.

A function f :D Rn

R

is said to be of class C

if f hascontinuous partial derivatives of all orders.

A function f :D Rn Rm is said to be of class Ck if each ofcomponent functions is of class Ck.

A function f :D Rn Rm is said to be of class C if each ofcomponent functions is of class C.


Pa tial D i ati s
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Partial Derivatives

Note that if the function if

f

xi has a partial derivative withrespect to xj, we denote the partial derivative by

xj f

xi = 2f

xjxi=fxixj .

The function obtained by differentiating f successively withrespect to xi1 , xi2 , . . . xir at x is denoted by

k

fxirxir1. . . xi1

=fxi1xi2 ...xir where i1+ +ir =k.

It is called a kth-order partial derivative of f.


Partial Derivatives
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Partial Derivatives

Theorem 13

Letf :D Rn R be aCk function. Then

fxi1xi2 ...xir (x) =fxj1xj2 ...xjr (x)

wherei1+i2+ +ir =j1+j2+ +jr =k.


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Contents




4 Differentials







Differentials
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Differentials

Definition 14Let D Rn be open and f :D R. The function f is said to bedifferentiable at a if there is a mapping La: R

n R (calleddifferential off at a denoted by dfa=La ordf=L for short, that isdf(h) =L(h)), such that

1 La(x+y) =La(x) +La(y)for all x, y Rn and R.

2 f(a+h) =f(a) +La(h) +o(h).

Theorem 15

LetD Rn be open andf :D R be a function. If the functionf isdifferentiable ata D then there is only one mappingLa andf iscontinuous ata.


Theorem 16
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Theorem 16

LetD Rn be open andf :D R. If the functionf is differentiableata then all partial derivatives of functionf exist ata and thedifferential off ata is

df=L(h) = f

x1(a)h1+

f

x2(a)h2+ +

f

xn(a)hn

whereh= (h1, h2, . . . , hn).

Note that if we denote the increments

hi = xi=dxi, i= 1, 2, . . . , n

(called the differential of

the independent variable xi, i= 1, 2, . . . , n,respectively), then

df= f

x1(a)dx1+

f

x2(a)dx2+ +

f

xn(a)dxn


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Differentials

and the incrementf=f(a+ x) f(a)of dependent variable f is

f= f

x1(a)x1+ +

f

xn(a)xn+o (x) .

Ifxi=dxi, i= 1, 2, . . . , n are small enough tending to zero, thenfcan be approximated by df.

Theorem 17

LetD Rn be open andf :D R. If all first-order partial derivativesof functionf exist and are continuous ata, then the functionf isdifferentiable ata.

Theorem 18

If a function is differentiable ata, then it is continuous ata.


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Contents




4 Differentials







Chain rules for functions of several variables
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Theorem 19 (Chain rules for function of several variables)

Lety=f(x1, x2, . . . , xn), wheref is differentiable function ofxi, i= 1, 2, . . . , n .If eachxi, i= 1, 2, . . . , n is a differentiable functionofm variablest1, t2, . . . , tm, theny is a differentiable function oft1, t2, . . . , tm and

ytj

=n

i=1

yxi

xitj

.

forj = 1, 2, . . . , m .In particular, ifxi, i= 1, 2, . . . , n is a function of a single variablet,

then we have dy

dt =

ni=1

y

xi

dxidt

.


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Theorem 20 (Chain rule: Implicit Differentiation)

If the equationF(x1, x2, . . . , xn, y) = 0 definesy implicitly as adifferentiable function ofxi, i= 1, 2, . . . , n , then

y

xi=

Fxi(x1, x2, . . . , xn, y)

Fy(x1, x2, . . . , xn, y), Fy(x1, x2, . . . , xn, y)= 0

fori= 1, 2, . . . , n .


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Contents




4 Differentials







Directional Derivatives
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Definition 21

Let f :D Rn R be a real-valued function and a be a point in Rn.Ifu Rn is any unit vector. Then, the directional derivative offat a in the direction of uis defined by

Duf= limt0

f(a+tu) f(a)

t , t R,

provided that the limit exists.

Theorem 22

Letf :D R anda= (a1, a2, . . . , an) D. Suppose that the first

partial derivatives offexist and continue ata. Then thedirectionalderivative off in the direction of aunit vectoru= (u1, u2, . . . , un)is given by

Duf(a) =n

i=1

uifxi(a).


Gradient
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Definition 23

For a real-valued function f(x1

, x2

, . . . , xn

), the gradient off at apoint a, denoted by f(a), is the vector

f(a) =

f

x1(a),

f

x2(a), . . . ,

f

xn(a)

.

Theorem 24

LetD Rn be open, and supposef :D R is differentiable ata D.Then the directional derivative off ata exists for all directions (unit

vectors) u and, moreover, we have

Duf(a) =f(a) u.


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Theorem 25

Letfbe a continuouslydifferentiable real-valued function,withf=0. Then:

The value off(x) increases the

fastest in the direction off.The maximum value ofDuf isf.

The value off(x) decreases thefastest in the direction of

f. The minimum value ofDuf isf.

Figure: The maximum increase off inthe direction off(x, y)in thexy-plane


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Theorem 26

LetD Rn be open, andf :D R be a function of classC1. Ifa is apoint on the level hypersurfaceS={x D:f(x) =c}, then the vectorf(a) is perpendicular to S.


Differential fo Vector-Valued Functions
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Definition 27Let D Rn be open and f :D Rm be a vector-valued function ofnvariables. Then fis said to be differentiable at a if there is a mappingLa : R

n Rm (called the differential off at a denoted by dfa=Laor df=L for short, that is df(h) =L(h)) such that

1 La(x+y) =La(x) +La(y)for all x, y Rn and R.

2 f(a+h) =f(a) +La(h) +o (h).

Theorem 28

If the functionf :D Rn Rm is differentiable ata then there isonly one mappingLa andf is continuous ata.




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Definition 29

Let f :D Rn Rm be a vector-valued function ofn variables. Letx= (x1, . . . , xn)denote a point ofR

n and f= (f1, . . . , f m). We define

the matrix of partial derivatives of f, denoted Df, to be them n matrix whose(i, j)entry is

fixj

. That is, Df=

fixj

. The

matrix Df(a) =

fixj

(a)

is also called Jacobian matrix off at a.




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Theorem 30

LetD Rn be open andf :D Rm be a vector-valued function of

x= (x1, . . . , xn). Iff= (f1, . . . , f m) is differentiable ata then thefirst-order partial derivatives off exist ata and the differential off ata is

df(dx) =Df(a)(dx) =

f1x1

(a) f1xn (a)

... . . . ...fmx1

(a) fmxn (a)

dx1

...dxn

For short,

df=

f1x1

(a) f1xn (a)...

. . . ...

fmx1

(a) fmxn (a)

dx1...

dxn

.




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Theorem 31Iff, g :D Rm are differentiable ata then

d(f+g)(a) =df(a) +dg(a)

where is a constant.

Theorem 32

LetD Rn be open andf :D Rm be a function. If all first-order

partial derivatives off exist ata and are continuous ata, then thefunctionf is differentiable ata.


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Theorem 33 (Chain rule)LetD1 Rn andD2 Rm be open. Iff :D1 Rm is differentiable ata andg:D2 R

p is differentiable atf(a) D2, thenk=g f isdifferentiable ata and

d(g f)(a) =dg (f(a)) df(a).

IfMgf(a) is the Jacobian matrix ofg f ata, Mg(f(a)) the Jacobianmatrix ofg atf(a), andMf(a) is the Jacobian matrix off ata, then

Mgf(a) =Mg(f(a)) Mf(a).


Hessian Matrix
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Definition 34 (Principal Minor)

Let A= (aij)nn be a square matrix. The determinantAk = det (aij)kk is called the kth principal minor of the n n matrix.

Definition 35 (The Hessian of a Function)

Let D Rn and f :D R be a function having second-order partialderivatives

2fxjxi

. The Hessian offis the matrix whose(i, j)entry is

fxixj .. That is,

Hf(a) = 2f

xjxi

(a)nn

.

We call Hs= det

2fxjxi

ss

, sth principal minor ofHf for

s= 1, 2, . . . , n.


Higher Order Differential


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Definition 36 (Higher Order Differential)

Let D Rn and f :D R be a function. Suppose that the partialderivatives off order k 1exist on D. If each(k 1)th order partialderivative off is differentiable at a D. Let h= (h1, . . . , hn). We callthe kth differential of f at a is the expression

d(k)fa(h) =n

i1=1

n

ik=1

kf

xi1. . . xik(a)hi1. . . hik .


Theorem 37 (Taylors formula)
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LetD Rn be open, a, x D, andf :D R be a function, andsuppose that the partial derivatives off orderk 1 exist onD. If each(k 1)th order partial derivative off is differentiable onD and theline segmentL(a, x) ={(1 t)a+tx, 0 t 1} D, then there is apointc L(a, x) such that

f(x) = f(a) +k1

j=1

1

j!d(j)fa(h) +

1

k!d(k)fc(h)

f(x) = f(a) +dfa(h) +1

2htHf(a)h+

k1j=3

1

j!d(j)fa(h) +Rk

whered(j)fa(h) =n

i1=1 n

ij=1jf

xi1 ...xij(a)hi1. . . hij ,

Rk =d(k)fc(h) =

ni1=1

n

ik=1jf

xi1 ...xik(c)hi1. . . hik and

h= (h1, . . . , hn) =x a.


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4 Differentials







Tangent planes and normal lines
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So far, you have represented surfaces in space primarily by equations of

the formz=f(x, y)

In the development to follow, however, it is convenient to use the moregeneral representation F(x,y,z) = 0. For a surface Sgiven by

z=f(x, y), you can convert to the general form by defining F as

F(x,y,z) =f(x, y) z.

Because f(x, y) z = 0, you can considerSto be the level surface ofF

given byF(x,y,z) = 0.


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Let S be a surface given byF(x,y,z) = 0 and let P(x0, y0, z0)be a point onS. LetCbe a curveSon throughPthat is defined by thevector-valued function

r(t) =x(t)

i+y(t)

j+z(t)

k

Then, for all t

F(x(t), y(t), z(t)) = 0.

IfF is differentiable and x(t), y(t)and z(t)all exist, then we have

F(t) =Fx(x,y,z)x(t)+Fy(x,y,z)y

(t)+Fz(x,y,z)z(t) = 0


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At point P(x0, y0, z0), the equivalent vector form is

F(x0, y0, z0).r

(t0) = 0.

This result means that the gradient at P is orthogonal to the tangentvector of every curve on S throughP. So, all tangent lines on Slie in aplane that is normal to F(x0,0, z0)and contains P, as shown in the

Figure.

Definition 38

Let Fbe differentiable at the point P(x0, y0, z0)on the surface givenbyF(x,y,z) = 0such that F(x0, y0, z0)=0.

The plane through Pthat is normal to F(x0, y0, z0)is called thetangent plane to S at P.

The line through Phaving the direction ofF(x0, y0, z0)is calledthe normal line to S at P.


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Theorem 39 (Equation of Tangent plane)

IfF is differentiable at(x0, y0, z0) then an equation of the tangentplane to the surface given byF(x,y,z) = 0 at(x0, y0, z0) is

Fx(x0, y0, z0)(x x0) + Fy(x0, y0, z0)(y y0) + Fz(x0, y0, z0)(z z0) = 0

Theorem 40

If the surface given by equationz=f(x, y), then an equation oftangent line to the the surface at the point(x0, y0, z0) is

fx(x0, y0)(x x0) +fy(x0, y0)(y y0) (z z0) = 0.


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4 Differentials







Extrema of Functions of Several Variables
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Definition 41

Let D Rn, f :D R be a function and a D.

f(a)is called a local minimum offif there is anr >0such thatf(a) f(x)for all x Br(a) D.

f(a)is called a local maximum off if there is an r >0suchthat f(a) f(x)for all x Br(a) D.

f(a)is called a local extremum off if there f(a)is a localminimum or a local maximum off.

f(a)is called a global minimum off on D iff(a) f(x)for allx D.

f(a)is called a global maximum off on D iff(a) f(x)for allx D.

f(a)is called a global extremum off on D iff(a)is a globalminimum or a global maximum off.


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Theorem 42

IfD Rn is closed and bounded andf :D R is a continuousfunction, thenfmust have both a global maximum and a globalminimum somewhere onD.

Theorem 43

LetD Rn be open andf :D R be a function. If the first-orderpartial derivatives off exist ata D andf(a) is a local extremum of

f, thenf(a) = 0.


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Definition 44Let fbe defined on an open region D containing a. The point a is acritical point off if one of the following is true.

1 fxi(a) = 0, for all i= 1, 2, . . . , n .

2

fxi(a)does not exist for some i {1, 2, . . . , n}.

Definition 45

Let D Rn be open and f :D R be differentiable at a D. Then ais called a saddle point off iff(a) = 0and there is an r0>0 such

that given any0< < r0 there are points x, yB(a)which satisfyf(x)< f(a)< f(y).


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Theorem 46 (A necessary condition for extremum)

Iffhas a local extremum ata on an open regionD, thena is a criticalpoint off.

Note that the converse of the theorem above is not true in general.That is, a critical point does not yield a local extremum.


Definition 47 (Quadratic form)

Let bij R such that bij =bji and h= (h1, . . . , hn) Rn.
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ij ij ji ( 1, , n)

1 A quadratic form in h1, . . . , hn is a function defined by

Q(h1, . . . , hn) =ni=1

nj=1

bijhihj.

This quadratic form can be written in term of matrices as

Q(h) = (h1. . . hn)

b11 . . . b1n...

. . . ...

bn1 . . . bnn

h1...

hn

=htBh

where ht

= ( h1 . . . hn )and B= (bij)nn is a symmetricmatrix.

2 The quadratic form Q (and also symmetric matrix B) is said to bepositive definite ifQ(h)> 0 for all h= 0and negativedefiniteifQ(h)< 0 for all h= 0.


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Theorem 48LetA= (aij)nn be a symmetric matrix. Then the matrixA is positivedefinite if and only if allkth principal minorsAk >0 fork= 1, 2, . . . , n .

Theorem 49

LetA= (aij)nn be a symmetric matrix. ThenA is positive definite ifand only if it can be reduced to upper triangular form using onlyelementary row operationsEi,j() and the diagonal elements of

resulting matrix are greater than zero.


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Theorem 50

LetD Rn be open andf :D R be a function. If all second-order

partial derivatives off exist ata D andd(2)fa(h)> 0 for allh= 0,then there is a >0 such thatd(2)fa(x) x

2 for allx Rn.

Theorem 51 (The Second Derivative Test)

LetDRn

be open containinga andf :D R

satisfyf(a) = 0.Suppose further that all second-order partial derivatives offexist onDand continuous ata.

If the quadratic formQ= d(2)fa(h) is positive definite, thenf(a)is a local minimum off.

If the quadratic formQ= d(2)fa(h) is negative definite, thenf(a)is a local maximum off.

If the quadratic formQ= d(2)fa(h) takes on both positive andnegative values forh Rn, thena is a saddle point off.


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Definition 52

Let D Rn and f :D R be a function.

The set D is said to be convex if every two points x, yD, theline segment L(x, y) D.

The function f is said to be convex on a convex set D if for anyx, yD and for any

t [0, 1], f((1 t)x+ty) (1 t)f(x) +tf(y).The function f is said to be concave on a convex set D if for anyx, yD and for anyt [0, 1], f((1 t)x+ty) (1 t)f(x) +tf(y).

Theorem 53

LetD Rn be a convex set andf :D R be aC2 function. Thefunctionf is a convex function onD if and only if for anyx D, foranyk= 1, 2, . . . , n , thekth principal minor of the HessianHk(x) 0.


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Theorem 54

Iff :D R is a convex function on a convex setD andf(a) is a local

minimum thenf(a) is the global minimum onD.Iff :D R is a concave function on a convex setD andf(a) is alocal maximum thenf(a) is the global maximum onD.


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4 Differentials







Lagrange Multipliers
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Theorem 55

Letf, g1, . . . , gk :D R beC1 functions whereD Rn is open and

k < n. Suppose there is ana D such that

det

g1x1

(a) . . . g1xk (a)...

. . . ...

gkx1 (a) . . .

gkxk (a)

= 0.

Iff(a) is local extremum offsubject to the constraintsgi(x) =ci fori= 1, 2, . . . , k, then there exist scalars1, . . . , k (calledLagrangeMultipliers) such that

f(a) +k

i=1

igi(a) = 0.


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Theorem 56 (Second derivative test for constrained local

extremum)

f, g1, . . . , gk :D R beC2 functions whereD Rn is open andk < n.

Denote(; x) and so-called Lagrange functionL(; x) =f(x) ki i(gi(x) ci). Suppose there is anx D suchthat

det

g1x1

(a) g1xk (a)...

. . . ...

gkx1

(a) . . . gkxk (a)

= 0

wherehij = 2f

xjxi(a)

kb=1b

2gbxjxi

(a). Letdj bejth principal minor

ofHL(; a).


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We calculate the sequence ofn k numbers

(s) : (1)kd2k+1, (1)kd2k+2, . . . , (1)

kdk+n

1 If the sequence in(s)consists entirely of positive numbers, then

f(a)is a local minimum subject to the constraints gix) =ci for alli= 1, 2, . . . , k .

2 If the sequence in(s)begin with a negative number and thereafteralternates in sign, then f(a)is a local maximum subject to the

constraintsgi(x) =ci for all i= 1, 2, . . . , k .3 If neither case 1 nor case 2 holds, then fhas a constrained saddle

point at a.


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1. R. Larson and B. Edwards, Multivariable Calculus, Ninth Edition,Brooks/Cole, Cengage Learning, 2010.

2. S. T. TAN, Multivariable Calculus, Brooks/Cole, CengageLearning, 2010.

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