Yes, all of the above are defined whenever they occur and denote multivariable functions and vector fields which are twice continuously differentiable on a common domain.
What is vector field?
A vector field is the assignment of the a vector to each point inside a subset of space in the fields of vector calculus and physics. For example, a vector field inside the plane could be visualised as a group of arrows, each attached to the a point in the plane and each with a specific magnitude and direction. Vector fields are frequently used to simulate various physical phenomena, such as the strength and motion of a force as it shifts through one point to the next or the speed and trajectory of a fluid moving through space.
The first expression, V . (F . Vf), denotes a multivariable function which is the dot product of a vector V and the composition of a function F with a vector field Vf.
The second expression, V . (V x Vf), denotes a multivariable function which is the dot product of a vector V and the cross product of a vector V and a vector field Vf.
The third expression, V x (V . F), denotes a vector field which is the cross product of a vector V and the dot product of a vector V and a function F.
The fourth expression, V x (V x f), denotes a vector field which is the cross product of a vector V and the cross product of a vector V and a function f.
The fifth expression, V x (V x F), denotes a vector field which is the cross product of a vector V and the cross product of a vector V and a function F.
The sixth expression, V . (V x F), denotes a multivariable function which is the dot product of a vector V and the cross product of a vector V and a function F.
To learn more about vector field
brainly.com/question/24332269
#SPJ4
