The Derivative And Integral, One Variable
The following definition is on the derivative and integral of a vector valued function of
Definition 16.0.1 The derivative of a function f′
, is defined as the following
limit whenever the limit exists. If the limit does not exist, then neither does
The function of h on the left is called the difference quotient just as it was for a scalar
valued function. If f
dt exists for each i
dt is defined as the vector
This is what is meant by saying f is Riemann integrable.
Here is a simple proposition which is useful to have.
Proposition 16.0.2 Let a ≤ b, f =
is vector valued and each fi is
Proof: This follows from the Cauchy Schwarz inequality.
As in the case of a scalar valued function, differentiability implies continuity but not
the other way around.
Theorem 16.0.3 If f′
exists, then f is continuous at t.
Proof: Suppose ε > 0 is given and choose δ1 > 0 such that if
then for such h, the triangle inequality implies
letting δ <
it follows if
, this shows that if
which proves f
is continuous at
As in the scalar case, there is a fundamental theorem of calculus.
Theorem 16.0.4 If f ∈ R
and if f is continuous at t ∈
Proof: Say f
. Then it follows
from the fundamental theorem
of calculus for scalar valued functions. Therefore,
Example 16.0.5 Let f
c where c is a constant. Find f′
The difference quotient,
Example 16.0.6 Let f
where a,b are constants. Find f′
From the above discussion this derivative is just the vector valued functions whose
components consist of the derivatives of the components of f. Thus f′