Does anyone know some links (exclude wiki) that explain the "Directional derivative of a vector field". I've been googling but found none, and i have a task that deals with it but have no literature.
It seems to me as you may have confused two different concepts.
First, you began this thread by asking about directional derivatives. They apply to functionals: functions from a vector space to the reals.
Then you now appear to be asking vector value functions or vector fields.
There is the concept of the divergence in the field. But we don’t find divergence in a particular direction.
Rather divergence is direction at a particular point in the field.
So what exactly are you asking? As put the current question is meaningless.
If $\displaystyle \vec{a}=yz\vec{i}+xz\vec{j}+xy\vec{k}$ is a vector field, do you mean to ask for the divergence at the point $\displaystyle (1,1,1)$
I have the solution of the task, but i don't know what it means. Perhaps it's divergence and i messed things up. Here it is, maybe you can figure out something out of it.
The text goes:
Find the derivate of a vector field $\displaystyle \vec{a}=yz\vec{i}+xz\vec{j}+xy\vec{k}$ in direction of the vector $\displaystyle \vec{p}=(1,1,1)$ in the point $\displaystyle T(1,1,1)$.
Solution:
$\displaystyle \frac{\partial \vec{a}}{\partial \vec{p}}=(\frac{\partial \vec{a_1}}{\partial \vec{p}},\frac{\partial \vec{a_2}}{\partial \vec{p}},\frac{\partial \vec{a_3}}{\partial \vec{p}})$
$\displaystyle \frac{\partial \vec{a_1}}{\partial \vec{p}}=\nabla a_1 \frac{\vec{p}}{p}=(0,z,y)\frac{(1,1,1)}{\sqrt{3}}= (0+z+y)\frac {1}{\sqrt{3}}=\frac{z+y}{\sqrt{3}}$
$\displaystyle \frac{\partial \vec{a_2}}{\partial \vec{p}}=\nabla a_2 \frac{\vec{p}}{p}=\frac{z+x}{\sqrt{3}}$
$\displaystyle \frac{\partial \vec{a_3}}{\partial \vec{p}}=\nabla a_3 \frac{\vec{p}}{p}=\frac{y+x}{\sqrt{3}}$
$\displaystyle |\frac{\partial \vec{a}}{\partial \vec{p}}|=|\frac{2}{\sqrt{3}},\frac{2}{\sqrt{3}},\ frac{2}{\sqrt{3}}|$
Ah! If I'm right it's something akin to what I have come to call a "vector gradient." (With all due apologies, I don't know what the actual name of it is.)
When the gradient of a scalar function $\displaystyle \vec{\nabla}s$ is taken it returns a vector. But you can define a version of this that acts on a vector as operating on each component of the vector individually. $\displaystyle \vec{\nabla} \vec{v} = \left < \vec{\nabla}v_x , \vec{\nabla}v_y , \vec{\nabla}v_z \right >$.
I've seen it mentioned in my Electrodynamics text, but have never actually used the thing.
-Dan