5.12 The Frechet Derivative
Let U be an open set in Fn, and let f : U → Fm be a function.
Definition 5.12.1 A function g is o
A function f : U → Fm is differentiable at x ∈ U if there exists a linear transformation
This linear transformation L is the definition of Df
. This derivative is often called the
Frechet derivative. .
Usually no harm is occasioned by thinking of this linear transformation as its matrix
taken with respect to the usual basis vectors.
The definition 5.12.14 means that the error,
converges to 0 faster than
. Thus the above definition is equivalent to saying
Now it is clear this is just a generalization of the notion of the derivative of a function of
one variable because in this more specialized situation,
due to the definition which says
For functions of n variables, you can’t define the derivative as the limit of a difference
quotient like you can for a function of one variable because you can’t divide by a vector.
That is why there is a need for a more general definition.
The term o
is notation that is descriptive of the behavior in
and it is only
this behavior that is of interest. Thus, if t
and other similar observations hold. The sloppiness built in to this notation is useful
because it ignores details which are not important. It may help to think of o
as an adjective describing what is left over after approximating
Theorem 5.12.2 The derivative is well defined.
Proof: First note that for a fixed vector, v, o
. Now suppose both
work in the above definition. Then let v
be any vector and let t
be a real scalar which
is chosen small enough that tv
+ x ∈ U
Therefore, subtracting these two yields
. Now let
0 to conclude that
= 0. Since this is true for all
, it follows L2
. This proves the
Lemma 5.12.3 Let f be differentiable at x. Then f is continuous at x and in fact,
there exists K > 0 such that whenever
is small enough,
Proof: From the definition of the derivative, f
be small enough that
1 so that
. Then for such
This proves the lemma with K
Theorem 5.12.4 (The chain rule) Let U and V be open sets, U ⊆ Fn and V ⊆ Fm.
Suppose f : U → V is differentiable at x ∈ U and suppose g : V → Fq is differentiable at
∈ V . Then g ∘ f is differentiable at x and
Proof: This follows from a computation. Let B
and let r
also be small
enough that for
, it follows that f
. Such an r
exists because f
is continuous at x
, the definition of differentiability of g
It remains to show o
By Lemma 5.12.3, with K given there, letting ε > 0, it follows that for
Since ε > 0 is arbitrary, this shows o
By 5.12.17, this shows
which proves the theorem.
The derivative is a linear transformation. What is the matrix of this linear
transformation taken with respect to the usual basis vectors? Let ei denote the vector of
Fn which has a one in the ith entry and zeroes elsewhere. Then the matrix of the linear
transformation is the matrix whose ith column is Df
What is this? Let t ∈ ℝ
is sufficiently small.
Then dividing by t
and taking a limit,
Thus the matrix of Df
with respect to the usual basis vectors is the matrix of the
As mentioned before, there is no harm in referring to this matrix as Df
but it may
also be referred to as
This is summarized in the following theorem.
Theorem 5.12.5 Let f : Fn → Fm and suppose f is differentiable at x. Then
all the partial derivatives
exist and if Jf
is the matrix of the linear
transformation with respect to the standard basis vectors, then the ijth entry is given
by fi,j or
What if all the partial derivatives of f exist? Does it follow that f is differentiable?
Consider the following function.
Then from the definition of partial derivatives,
However f is not even continuous at
which may be seen by considering
the behavior of the function along the line
and along the line x
By Lemma 5.12.3
this implies f
is not differentiable. Therefore, it is necessary
to consider the correct definition of the derivative given above if you want to
get a notion which generalizes the concept of the derivative of a function of
one variable in such a way as to preserve continuity whenever the function is
class=”left” align=”middle”(Fn, Fm)5.13. C1 FUNCTIONS