Vector Calculus
Suppose that vector
varies with time, so that
. The time
derivative of the vector is defined
![$\displaystyle \frac{d {\bf a}}{dt} = \lim_{\delta t\rightarrow 0} \left[\frac{{\bf a}(t+\delta t) - {\bf a}(t)}
{\delta t}\right].$](img4681.png) |
(1.63) |
When written out in component form this becomes
 |
(1.64) |
Suppose that
is, in fact, the product of a scalar
and another vector
. What now is the time derivative of
? We have
 |
(1.65) |
which implies that
 |
(1.66) |
Moreover, it is easily demonstrated that
 |
(1.67) |
and
 |
(1.68) |
Hence, it can be seen that the laws of vector differentiation are analogous to those in
conventional calculus.