where the components ui(t),vi(t)and wi(t), i = 1, 2 , 3 are continuous and differentiable,
the following differentiation rules can be verified using the definition of a derivative
as given by equation (6.49).
The derivative of a sum is the sum of the derivatives and dtd(u +v ) = dudt +dudt
The derivative of a dot product of two vectors is the first vector dotted with the
derivative of the second vector plus the derivative of the first vector dotted with
the second vector and one can write
d
dt
(u ·v ) = u ·dv
dt
+du
dt
·v
The derivative of a cross product of two vectors gives a similar result
d
dt
(u ×v ) = u ×
dv
dt
+
du
dt
×v
The derivative of a scalar function times a vector is similar to the product rule
and one finds
d
dt (f(t)u ) = f(t)
du
dt +
df
dtu
where f =f(t)is a scalar function. In the special case f=cis a constant one
finds dtd(cu ) = cdudt.
If u =u (s)and s=s(t), then the chain rule for differentiating vector functions is
given by
du
dt =
du
ds
ds
dt
The derivative of a triple scalar product is found to be
d
dt
(u ·v ×w) = u ·v ×d w
dt
+u ·dv
dt
×w+du
dt
·v ×w
Each of the above derivative relations can be derived using the definition of a
derivative.
Kinematics of Linear Motion
In the study of dynamics or physics one encounters Newton’s three laws of
motion. These three laws are sometimes expressed in the following form.
1. A body at rest remains at rest and a body in motion remains in motion, unless
acted upon by an external force.
2. The time rate of change of the linear momentum of a body is proportional to
the force acting on the body, with the body moving in the direction of the applied
force.
3. For every action there is an equal and opposite reaction.
