Higher derivatives can be calculated by using the product rule for differentiation
together with the rule for differentiating a composite function.
Integration of Vectors
Let u(s) = u 1 (s)ˆe 1 +u 2 (s)ˆe 2 +u 3 (s)ˆe 3 denote a vector function of arc length, where
the components ui(s), i = 1 , 2 , 3 are continuous functions. The indefinite integral of
u (s) is defined as the indefinite integral of each component of the vector. This is
expressed in the form
∫
u(s)ds =
∫
u 1 (s)ds ˆe 1 +
∫
u 2 (s)ds ˆe 2 +
∫
u 3 (s)ds ˆe 3 +C ,
=U(s) + C .
(6 .89)
where U(s)is a vector such that ddsU =u (s)and C is a vector constant of integration.
The definite integral of u is defined as
∫b
a
u (s)ds =U(s)
b
a
=U(b)−U(a), where d
U(s)
ds =u (s). (6 .90)
The following are some properties associated with the integration of vector func-
tions. These properties are stated without proof.
1. For c a constant vector
∫
c ·u (s)ds =c ·
∫
u (s)ds and
∫
c ×u(s)ds =c ×
∫
u(s)ds
2. For c 1 and c 2 constant vectors, the integral of a sum equals the sum of the
integrals ∫
[c 1 ·u (s) + c 2 ·v (s)] ds =c 1 ·
∫
u (s)ds +c 2 ·
∫
v (s)ds,
3. Integration by parts takes on the form
∫b
a
f(s)u (s)ds =f(s)U(s)
b
a
−
∫b
a
f′(s)U(s)ds, (6 .91)
where f(s)is a scalar function and
dU(s)
ds =u (s).
