Linear Algebra & Vector Calculus

Consider the vector field V⃗ consisting of the velocities of points on a thin horizontal disc of radius R = 2 m, moving anticlockwise with uniform angular speed \omega = 2 rad/sec about an axis passing through its center. If V = |V⃗ | , then which of the following options is(are) CORRECT ? (In the options, r̂ and 𝜃̂ are unit vectors corresponding to the plane polar coordinates r and 𝜃). You may use the fact that in cylindrical coordinates (s, \phi, z) (s is the distance from the z-axis), the gradient, divergence, curl and Laplacian operators are: ∇⃗⃗ f = 𝜕f 𝜕s ŝ + 1 s 𝜕f 𝜕\phi \phî + 𝜕f 𝜕z ẑ ; ∇⃗⃗ . A = 1 s 𝜕 𝜕s (sAs) + 1 s 𝜕A\phi 𝜕\phi + 𝜕Az 𝜕z ; ∇⃗⃗ × A = ( 1 s 𝜕Az 𝜕\phi - 𝜕A\phi 𝜕z ) ŝ + ( 𝜕As 𝜕z - 𝜕Az 𝜕s ) \phî + 1 s ( 𝜕 𝜕s (sA\phi) - 𝜕As 𝜕\phi ) ẑ ; ∇⃗⃗ 2f = 1 s 𝜕 𝜕s (s 𝜕f 𝜕s) + 1 s2 𝜕2f 𝜕\phi2 + 𝜕2f 𝜕z2 .

Options:

• (A) ∇⃗⃗ V = 2r̂
• (B) ∇⃗⃗ . V⃗ = 2
• (C) ∇⃗⃗ × V⃗ = 4ẑ, where ẑ is a unit vector perpendicular to the (r, 𝜃) plane
• (D) ∇⃗⃗ 2V = 4 3 at r = 1.5 m