A directional, partial derivative of a scalar product?

[particlezoo]

Let's say I have two vector fields a(x,y,z) and b(x,y,z).

Let's say I have a scalar field f equal to a•b.

I want to find a clean-looking, simple way to express the directional derivative of this dot product along a, considering only changes in b.

Ideally, I would like to be able to express this without invoking unit vectors and without || 's, while still using vector notation.

 

[particlezoo]

$\vec a \cdot (\nabla \vec b)$

If vectors in a had x-components only, and vectors in b had y-components only, wouldn't $\vec a \cdot (\nabla \vec b)$ still return values with y-components only provided that b varies with x, as opposed to zero, which a•b would be equal to everywhere?

Edit: Maybe I am conflating $\vec a \cdot (\nabla \vec b)$ with $(\vec a \cdot \nabla) \vec b$

Edit2: Also, It seems like you gave the gradient of the dot product of a•b, considering only changes in b. I was wondering about the directional derivative of a•b along a, considering only changes in b.

 

[Henryk]

My mistake