Local Linear Approximation for

Functions of Several Variables

Functions of One Variable

• When we zoom in on a “sufficiently nice” function of one variable, we see a straight line.

Functions of two Variables

Functions of two Variables

Functions of two Variables

Functions of two Variables

Functions of two Variables

When we zoom in on a “sufficiently nice” function of two variables, we see a plane.

( , ) 5 sin( ) cos( )2

yf x y x y x

(a,b)

Describing the Tangent Plane• To describe a tangent line

we need a single number---the slope.

• What information do we need to describe this plane?

• Besides the point (a,b), we need two numbers: the partials of f in the x- and y-directions.

( , )

( , ) ( , )( , ) ( ) ( ) ( , )a b

f a b f a bT x y x a y b f a b

x y

( , )f

a bx

( , )f

a by

Equation?

Describing the Tangent Plane

We can also write this equation in vector form.

Write x = (x,y), p = (a,b), and

( ) ( ) ( ) ( )T f f p x p x p p

Dot product!

( , ) ( , )( ) ,

f a b f a bf

x y

p

( , )

( , ) ( , )( , ) ( ) ( ) ( , )a b

f a b f a bT x y x a y b f a b

x y

Gradient Vector!

General Linear Approximations

For a function general vector-valued function

: and for a point in

we want a linear function : that satisfies

( ) ( ) ( ) for "close" to .

m n m

m n

p

p

F p

L

F x L x F p x p

Why don’t we just subsume F(p) into Lp?Linear--- in the linear

algebraic sense.

In the expression we can think of As a scalar function on 2: . This function is linear.

( ) ( ) ( )T f f p x p x p

( )f p

( )f x p x

General Linear Approximations

For a function general vector-valued function

: and for a point in

we want a linear function : that satisfies

( ) ( ) ( ) for "close" to .

m n m

m n

p

p

F p

L

F x L x F p x p

In the expression we can think of As a scalar function on 2: . This function is linear.

( ) ( ) ( )f f p x p x p

( )f p

( )f x p x

Note that the expressionLp (x-p) is not a product.

It is the function Lp acting on the vector (x-p).

We must understand Linear Linear Vector Fields:

Linear Transformations from

To understand Differentiability Differentiability

inVector FieldsVector Fields

to m n

Linear Functions

A function L is said to be linear provided that

( ) ( ) ( )

and

( ) ( )

L x y L x L y

L x L x

Note that L(0) = 0, since L(x) = L (x+0) = L(x)+L(0).

For a function L: m → n, these requirements are very prescriptive.

Linear Functions

It is not very difficult to show that if L: m →n is linear, then L is of the form:

1 2 3

11 1 12 2 13 3 1

21 1 22 2 23 3 2

31 1 32 2 33 3 3

1 1 2 2 3 3

( ) ( , , , , )m

m m

m m

m m

n n n nm m

x x x x

a x a x a x a x

a x a x a x a x

a x a x a x a x

a x a x a x a x

L x L

where the aij’s are real numbers for j = 1, 2, . . . m and i =1, 2, . . ., n.

Linear Functions

Or to write this another way. . .1 2 3

11 12 13 1 1

21 22 23 2 2

31 32 33 3 3

1 2 3

( ) ( , , , , )m

m

m

m

n n n nm m

x x x x

a a a a x

a a a a x

a a a a x

a a a a x

L x L

Αx

In other words, every linear function L acts just like left-multiplication by a matrix. Though they are different, we cheerfully confuse the function L with the matrix A that represents it! (We feel free to use the same notation to denote them both except where it is important to distinguish between the function and the matrix.)

11 1 12 2 13 3 1

21 1 22 2 23 3 2

31 1 32 2 33 3 3

1 1 2 2 3 3

( )

n n

n n

n n

m m m mn n

a x a x a x a x

a x a x a x a x

a x a x a x a x

a x a x a x a x

A x

One more idea . . .

Suppose that A = (A1 , A2 , . . ., An) . Then for 1 j n

What is the partial of Aj with respect to xi?

Aj(x) = aj1 x1+aj 2 x2+. . . +aj n xn

11 1 12 2 13 3 1

21 1 22 2 23 3 2

31 1 32 2 33 3 3

1 1 2 2 3 3

( )

n n

n n

n n

m m m mn n

a x a x a x a x

a x a x a x a x

a x a x a x a x

a x a x a x a x

A x

The partials of the Aj’s are the entries in the matrix that represents A!

One more idea . . .

Suppose that A = (A1 , A2 , . . ., An) . Then for 1 j n

Aj(x) = aj1 x1+aj 2 x2+. . . +aj n xn

Local Linear Approximation

( )

where is the matrixm n

pL x Αx

A

11 12 13 1

21 22 23 2

31 32 33 3

1 2 3

m

m

m

n n n nm

a a a a

a a a a

a a a a

a a a a

A

For "close" to we have

( ) ( ) ( ) p

x p

F x L x F p

For we have

( ) ( ) ( ) ( )

where ( ) is the error

committed by

in approximating .

all

p

p

x

F x L x F p E x

E x

L F

Local Linear Approximation

What can we say about the relationship between the matrix DF(p) and the coordinate functions F1, F2, F3, . . ., Fn ?

Quite a lot, actually. . .

Suppose that F: m →n is given by coordinate functionsF=(F1, F2 , . . ., Fn) and all the partial derivatives of F exist at p in m and are continuous at p then . . .

there is a matrix Lp such that F can be approximated locally near p by the affineaffine function

( ) ( ) pL x p F p

Lp will be denoted by DF(p) and will be called the

Derivative of F at p or the Jacobian matrix of F at p.

A Deep Idea to take on Faith

I ask you to believe that for all i and j with 1 i n and 1 j m

( ) ( )j j

i i

L F

x x

p p

This should not be too hard. Why?

Think about tangent lines, think about tangent planes.

Considering now the matrix formulation, what is the partial of Lj with respect to xi?

The Derivative of F at p(sometimes called the Jacobian Matrix of F at p)

1 1 1 1

1 2 3

32 2 2

1 2 3

3 3 3 3

1 2 3

1 2 3

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

n

n

n

m m m m

n

F F F F

x x x x

FF F F

x x x x

D F F F F

x x x x

F F F F

x x x x

F

p p p p

p p p p

F p J pp p p p

p p p p

Notice the two common

nomeclatures for the derivative of a vector –

valued function.

In Summary. . .

For "close" to we have

( ) ( )( ) ( )D x p

F x F p x p F p

For all x, we haveF(x)=DF(p) (x-p)+F(p)+E(x)

where E(x) is the error committed by DF(p) +F(p)

in approximating F(x).

If F is a “reasonably well behaved” vector field around the point p, we can form the Jacobian Matrix DF(p).

How should we think about this function E(x)?

E(x) for One-Variable Functions

But E(x)→0 is not enough, even for functions of one variable!

( , ( ))p f p

What happens to E(x)

as x approaches p?

( , ( ))p f p

E(x) measures the vertical distance between f (x) and Lp(x)

( )E x( )E x

( , ( ))px L x

( , ( ))x f x

( , ( ))px L x

( , ( ))x f x

Definition of the DerivativeA vector-valued function F=(F1, F2 , . . ., Fn) is differentiable at p in m provided that 1)All partial derivatives of the component functions

F1, F2 , . . ., Fn of F exist on an open set containing p, and2)The error E(x) committed by DF(p)(x-p)+F(p) in approximating F(x) satisfies

(Book writes this as .)

( )lim 0

x p

E x

x p

( ) ( )( ) ( )lim 0

D

x p

F x F p x p F p

x p