Question

ar,a7, that V2u V.Vu 6.4. Verify directly from the gradient operator that V uK +uyy-see Definition 6.5

Answer 1

We have to verify from the fradient operator del = [delta_x_1 delta_y] T that del^2 4 = del. Del 4 = 4_xx + 4_yy R. H. S, del. Del 4 = (delta/delta x I + delta/delta y i). (delta 4/delta x I + delta 4 delta y I) = delta/delta x + delta/delta y + delta/delta y (delta 4/delta y) = delta 24 delta_x2 + delta 24 delta x 2 = 4_xx + 4_yy = (delta 2/delta x2 + delta 24/delta y 2) = 4_xx + 4_yy del. Del 4 = del^24 = 4_xx + 4_yy