Velocity Field:
The given velocity field V=k(yi xk) represents the velocity of fluid in three-dimensional space. Here, k is a constant, and yi and xk represent the unit vectors in the y and x directions, respectively.

Definition of Vorticity:
Vorticity is a measure of the local rotation of a fluid element within a flow. It is defined as the curl of the velocity field. Mathematically, vorticity (ω) can be written as:

ω = ∇ x V

where ∇ is the del operator and x denotes the cross product.

Calculating Vorticity:
In order to calculate the vorticity (ω) for the given velocity field V=k(yi xk), we need to take the curl of V. Let's break it down step by step:

1. Write the velocity field in terms of its components:
V = k(yi xk) = k(yk - xj)

2. Compute the curl of V using the del operator:
∇ x V = ∇ x (k(yk - xj))

3. Apply the curl operator to each component:
∇ x V = ∇ x (kyk) - ∇ x (xj)

4. Use the properties of the cross product and the del operator:
∇ x V = (∇k) x yk + k∇ x yk - (∇x)xj - x∇xj

5. Simplify the expressions:
∇ x V = 0 + k(∇ x yk) - 0 - 0

6. Evaluate the curl of yk:
∇ x yk = (∂/∂x, ∂/∂y, ∂/∂z) x (0, 1, 0)
= (∂/∂x)(0) - (∂/∂y)(0) + (∂/∂z)(1)
= (0, 0, 1)

7. Substitute the result back into the vorticity equation:
∇ x V = k(0, 0, 1) = kzi

Final Result:
After performing the calculations, we find that the vorticity (ω) for the given velocity field V=k(yi xk) is ωz=k. In other words, the vorticity vector points in the z-direction with a magnitude equal to the constant k.

Summary:
The vorticity ωz for the velocity field V=k(yi xk) is found to be ωz=k. This implies that the fluid elements within the flow rotate around the z-axis with a rotation rate determined by the constant k.