We’ll follow up the previous lesson by deriving an equation which connects the directional derivative and the del operator.
While the definition of the directional derivative may suggest that it is hard to calculuate, it turns out that it can be expressed quite elegantly.
VECTOR INTEGRAL CALCULUS >
DIRECTIONAL DERIVATIVE AND THE DEL OPERATOR
May want to check out:
Similar to:
VIDEO LECTURE
MAIN CONCEPTS
The directional derivative and the del operator is related by the equation  Get the Printer Friendly Version COMMENTS
Feel free to leave any comments on the lesson - your views, improvements, mistakes, clarification of concepts, or vote to have this lesson revised. |
CONCEPT
Previously, we concluded with an expression of
 as
The graphical illustration of
We are concern with the rate of change of
Now, using the partial differentiation version of chain rule, we have
For the rate of change of
Can you spot the grad
a vector and that
also a vector. The dot product of these two vectors gives us the above expression, namely
And that my friends, is how the directional derivative is related to the gradient of Remember that the vector All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
Video courtesy of YouTube.com service. |
can be seen from the previous diagram or replicated below.
as we travel along unit vector 
. So, using process of differentiation, we find the derivative of 
with respect to 
, evaluated at 
. Thus
at point 
, we are left to evaluate these partial derivatives at 
or 
then
hidden somewhere there. First recall that
,
through the del operator. Lovely expression isn’t it?
needs to be a unit vector. This can be easily solved by letting 
if traveling to 
with a vector 
whose magnitude is not unit.
