Derivative of a vector: After
reviewing the vector algebra, we would now like to introduce
you to the idea of differentiating a vector quantity. Here
we take a vector
as
depending on one parameter, say time t , and evaluate
the derivative
.
This is similar to what we do for a regular function. We
evaluate the vector
at
time (t+ Δt) , subtract
from
it, divide the difference
by Δt and
then take the limit Δ t → 0 .
This is shown in figure 14. Thus
![](1_4_clip_image170.gif)
The derivative is easily understood if we think in terms of its derivatives. If we write a vector as
![](1_4_clip_image176.gif)
then the derivative of the vector is given as
![](1_4_clip_image178.gif)
Notice that only the components are differentiated, because the unit vectors
are fixed in space and therefore do not change with time. Later when we learn about polar coordinates, we will encounter unit vectors which also change with time. In that case when taking derivative of a vector, the components as well as the unit vectors both have to be differentiated.
Using the definition above, it is easy to show that in differentiating the product of two vectors, the usual chain rule can be applied. This gives
![](1_4_clip_image182.gif)
and
![](1_4_clip_image184.gif)