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Vector Calculus
Suppose that vector
varies with time, so that
. The time
derivative of the vector is defined
![\begin{displaymath}
\frac{d {\bf a}}{dt} = \lim_{\delta t\rightarrow 0} \left[\frac{{\bf a}(t+\delta t) - {\bf a}(t)}
{\delta t}\right].
\end{displaymath}](img3364.png) |
(1320) |
When written out in component form this becomes
 |
(1321) |
Suppose that
is, in fact, the product of a scalar
and another vector
. What now is the time derivative of
? We have
 |
(1322) |
which implies that
 |
(1323) |
Moreover, it is easily demonstrated that
 |
(1324) |
and
 |
(1325) |
Hence, it can be seen that the laws of vector differentiation are analogous to those in
conventional calculus.
Next: Line Integrals
Up: Vector Algebra and Vector
Previous: Vector Triple Product
Richard Fitzpatrick
2011-03-31