I have question about presentation of directional derivative.
Sometimes I see directional derivates and gradient to be represented as a function but I think that it is convenient to use "vector presentation" instead.
My question is that is there some cases and/or good reasons why should use functional presentation instead of vector?
To make my point clear I wrote an example what I mean about "presentations".
Simply I mean the presentation of vector, using comma versus using i and j components
There is one example twice, for "function presentation" and for "vector presentation".
Let we have a function: and we calculate gradient and directional derivative of function .
Function presentation:
Gradient (I include all phases here):
Next we calculate directional derivative of function f at this point (2,-1) to the direction of
The vector has magnitude . The unit vector in the direction is thus
Directional derivative is then
and plugging the point (2,-1) we get
Vector presentation
Next we calculate directional derivative of function f at this point (2,-1) to the direction of
Let's plugin the point to the Gradient:
The vector has magnitude . The unit vector in the direction is thus
Directional derivative is then
Calculations above are the same no matter when we plugin the point (2,-1).
I think that representing the gradient as a vector is more clear than function.
Now, I will repeat my question:
Is there some cases and/or good reasons why should use functional presentation (i,j components) instead of vector presentation in points (x,y)?