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

• What information do we need to describe this
  plane?
• The point (a,b) and the partials of f in the x-
  and y-directions.

• The equation of the plane is
• We can also write this in vector form.
• If we write and

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General Linear Approximations
In the expression
we can think of the gradient as a
scalar function on ?2 This function is linear.
For a general vector-valued function F ?n ? ?m
and for a point p in ?n , we want to find a
linear function Ap ?n ? ?m such that
Why dont we just subsume F(p) into Ap? Linear---
in the linear algebraic sense.
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Linear Functions

• A function A is said to be linear provided that

Note that A (0) 0, since A(x) A (x0)
A(x)A(0).
For a function A ?n ??m, these requirements are
very prescriptive.
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Linear Functions

• It can be shown that if A ?n ??m is linear, then

In other words, the function A is just
left-multiplication by a matrix. Thus we
cheerfully confuse the function A, with the
matrix that represents it!
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Local Linear Approximation
For all x, we have F(x)Ap(x-p)F(p)E(x) Where
E(x) is the error committed by Lp(x)
Ap(x-p)F(p) in approximating F(x)
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Local Linear Approximation
Fact Suppose that F ?n ??m is given by
coordinate functions F(F1, F2 , . . ., Fm) and
all the partial derivatives of F exist at p ? ?n
, then . . . there is some matrix Ap such that
F can be approximated locally near p by the
affine function
What can we say about the relationship between
the matrix Ap and the coordinate functions F1,
F2, F3, . . ., Fm ? Quite a lot, actually. . .
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We Just Compute
First, I ask you to believe that if L(L1,L2, . .
., Ln) 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)
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Local Linear Approximation
How close does x have to be to p?
So F will be locally linear if F(x)
?Df(p)(x-p)F(p) for all x close to p.
It is easy to see that for all x, we can find
E(x) so that F(x)Df(p)(x-p)F(p)E(x). E(x) is
the error committed by Lp(x) Df(p)(x-p)F(p)
in approximating F(x).
How should we think about the error 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!
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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Differentiability of Vector Fields
A vector-valued function F ?n ? ?m is said to be
differentiable provided that there exists a
function E ?n ? ?m such that for all x,
F(x)Ap(x-p)F(p)E(x) where E(x) satisfies