. = Prove the following identities using simplified index notation: a) V.(u y)= u Vy + y (V.u) b) V.(V!)= y.(V.?") + :Vy (where " denotes transpose) c) V(u y) - u x(Vxv) - yx(Vxu )= (u.V)y + (v.V)u D 1 d) V(kv) vx(Vxv) Dt at D where V: V is known as the substantial derivative operator. Dt at + E Hint: for part (d), it will be easier to work from the right side toward the left. . = Prove the following identities using simplified index notation: a) V.(u y)= u Vy + y (V.u) b) V.(V!)= y.(V.?") + :Vy (where " denotes transpose) c) V(u y) - u x(Vxv) - yx(Vxu )= (u.V)y + (v.V)u D 1 d) V(kv) vx(Vxv) Dt at D where V: V is known as the substantial derivative operator. Dt at + E Hint: for part (d), it will be easier to work from the right side toward the left