V = V_xi + V_yj + V_zk

Explanation:
To understand why the correct answer is option 'A', let's break down the problem step by step.

1. Curl of a Vector Function:
The curl of a vector function V, denoted as curl(V), is a vector that represents the rotation or circulation of the vector field. Mathematically, it is defined as the cross product of the del operator (∇) and the vector V.

2. Scalar Function:
A scalar function f is a function that takes a vector as an input and returns a scalar value. It can be thought of as a scalar field.

3. Vector Function:
A vector function fV is a function that takes a vector as an input and returns another vector as an output. It can be thought of as a vector field.

4. The Cross Product of a Scalar and a Vector:
The cross product of a scalar function f and a vector V, denoted as (grad f) x (V), is a vector that represents the cross product of the gradient of f and V. The gradient of f (∇f) is a vector that points in the direction of the steepest increase of f.

5. The Cross Product of Two Vectors:
The cross product of two vectors A and B, denoted as A x B, is a vector that is perpendicular to both A and B and has a magnitude equal to the product of the magnitudes of A and B multiplied by the sine of the angle between them.

6. The Relationship:
Now, let's consider the expression curl(fV). According to the properties of the curl operator, we can expand it as follows:
curl(fV) = curl(f * V)
= (∇ x (f * V))

7. Applying the Product Rule:
Using the product rule of vector calculus, we can expand (∇ x (f * V)) as follows:
(∇ x (f * V)) = (∇ f) x V + f * (curl(V))

8. Comparing the Result:
Comparing the expanded expression (∇ f) x V + f * (curl(V)) with the given options:
a) (grad f) x (V) (f curl V)
b) 0
c) f curl (V)
d) (grad f) x (V)

We can see that option 'A' matches the expanded expression (∇ f) x V + f * (curl(V)). Therefore, the correct answer is option 'A'.

Summary:
In summary, the curl of a vector function is given by (∇ x (f * V)), and using the product rule of vector calculus, we can expand it as (∇ f) x V + f * (curl(V)). Comparing this expansion with the given options, we find that option 'A' is the correct answer.