Solved Examples/Numericals - Divergence of a Vector Field.
Q.1 Determine the divergence of the following vector fields and evaluate them at the specified points: a) A = yz a x + 4xy a y + y a z at point (1, -2, 3) Ans: Given A = yz a x + 4xy a y + y a z at point (1, -2, 3) Divergence of A is given as: ∇ . A = 0 + 4x + 0 = 4x At Point (1, -2, 3) Hence, ∇ . A = 4 b) A = ρzsinφ a ρ + 3ρz 2 cosφ a φ at (5, π / 2, 1) Ans: Divergence of a vector field A in cylindrical co-ordinate system is given as: => ∇ . A = (1/ρ) 2ρz sinφ - (1/ρ) 3ρz 2 sinφ = 2zsinφ - 3z 2 sinφ = (2 - 3z) zsinφ At point (5, π / 2, 1) ∇ . A = (2 – 3) (1) = -1 c) A = 2r cosθ cosφ a r + r 1/2 a φ at (1, π / 6, π / 3) Ans: Divergence of a vector field A in Spherical co-ordinate system is given as: => ∇ . A = (1/r 2 ) 6r 2 cosθ cosφ = 6 cosθ cosφ At point (1, π/6, π/3) ∇ . A= 6 cos π/6 cos π/3 = 2.598. Q. 2 Determine the flux of D = ρ 2 cos 2 ...