Local Linear Approximation for Functions of Several Variables

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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.
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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.

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

Gradient Vector!
Dot product!
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General Linear Approximations
Why dont we just subsume F(p) into Lp? Linear---
in the linear algebraic sense.
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General Linear Approximations
Note that the expression Lp (x-p) is not a
product. It is the function Lp acting on the
vector (x-p).
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To understand Differentiability inVector Fields
We must understand Linear Vector Fields Linear
Transformations from
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Linear Functions

• A function L is said to be linear provided that

Note that L(0) 0, since L(x) L (x0)
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

where the aijs 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. . .

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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One more idea . . .
Suppose that A (A1 , A2 , . . ., An) . Then
for 1 ? j ? n
Aj(x) aj1 x1aj 2 x2. . . aj n xn
What is the partial of Aj with respect to xi?
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One more idea . . .
Suppose that A (A1 , A2 , . . ., An) . Then
for 1 ? j ? n
Aj(x) aj1 x1aj 2 x2. . . aj n xn
The partials of the Ajs are the entries in the
matrix that represents A!
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Local Linear Approximation
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Local Linear Approximation
Suppose that F ?m ??n is given by coordinate
functions F(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 affine function
What can we say about the relationship between
the matrix DF(p) and the coordinate functions
F1, F2, F3, . . ., Fn ? Quite a lot, actually.
. .
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
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)
Notice the two common nomeclatures for the
derivative of a vector valued function.
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In Summary. . .
How should we think about this function E(x)?

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

For all x, we have F(x)DF(p) (x-p)F(p)E(x) wher
e E(x) is the error committed by DF(p) F(p) in
approximating F(x).
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E(x) for One-Variable Functions
But E(x)?0 is not enough, even for functions of
one variable!
E(x) measures the vertical distance between f (x)
and Lp(x)
What happens to E(x) as x approaches p?
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Definition of the Derivative