Function of angle between vector field and scalar function

[Kavorka]

I was curious, if you were given a vector field F(x,y,z) = <F_x(x,y,z), F_y(x,y,z), F_z(x,y,z)>, and then some scalar function f(x,y,z), how would you define a function θ(x,y,z) of the angle θ between the scalar function and the vector field at any given point. I know how I would find this at a single point, by finding the equation of the line tangent to f(x,y,z) at that point via gradient and then finding its corresponding vector starting at the origin, and then taking the dot product of this vector with F(x,y,z) at that point, but I'm having difficulty generalizing it across all (x,y,z). Am I correct in saying:

cosθ = <F_x(x,y,z), F_y(x,y,z), F_z(x,y,z)> ⋅ <∂f/∂x, ∂f/∂y, ∂f/∂z> =
(∂f/∂x)F_x + (∂f/∂y)F_y + (∂f/∂z)F_z

 

[Ssnow]

The idea is correct but there are a few observations.
1) In this way you find the angle between the vector field and the tangent space of $f(x,y,z)$.
2)You must normalize vectors in order to have a number between $0$ and $1$.
3)the partial derivatives of $f$ must be taken at the point $(x,y,z)$.