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Local Linear Approximation for
Functions of Several Variables
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Functions of One Variable
• When we zoom in on a “sufficiently nice” function of one variable, we see a straight line.
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Functions of two Variables
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Functions of two Variables
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Functions of two Variables
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Functions of two Variables
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Functions of two Variables
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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
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(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?
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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!
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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
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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).
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We must understand Linear Linear Vector Fields:
Linear Transformations from
To understand Differentiability Differentiability
inVector FieldsVector Fields
to m n
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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.
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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.
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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.)
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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
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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
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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
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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.
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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?
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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.
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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)?
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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
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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