After talking about the gradient vector field, we did mention that the gradient is related to the directional derivative of
. In this lesson and the next, we shall make the relationship clear. VECTOR INTEGRAL CALCULUS >
DEFINING THE DIRECTIONAL DERIVATIVE
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VIDEO LECTURE
MAIN CONCEPTS
Using vector algebra we can express a general form of in terms of the associated points and the unit vector, that is Get the Printer Friendly Version COMMENTS
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CONCEPT
Let
 have the coordinates  and let  be any point on the line from  in the direction of the vector  . Using normal vector algebra, we can specify the vector
which is parallel and in the same direction as
Hence for some
We define
and after rearranging,
This set of equations giving us the coordinates of
Finally, we can find the derivative of We shall continue this in our next lesson and finally see how the del operator is used in finding the directional derivative. All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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as shown below.
, we write
as 
and then equate the separate unit vectors to give 
is key in finding the gradient of 
. They link a certain point 
and the unit vector in which we are travelling to by giving the coordinates 
in a ‘general form’. Thus, we apply the function 
. 
by differentiating with respect to the constant 
. All the definitions of the directional derivative is taken into account here - The unit vector 
is represented by 
and the point 
is represented by 
. The rate of change is thus to differentiate w.r.t to 
. 
