Apply the operator nabla to products of smooth scalar

Question

Problem 1- Apply the operator ∇ to products of smooth scalar fields Φ, Ψ, vector fields u, v, and tensor fields A, B. Establish the identities

div(Φu) = Φdivu + (u, gradΦ)

grad(u, v) = (grad^Tu)v + (grad^Tv)u

div(A^Tu) = (divA,u) + A : gradu

Problem 2- Compute the derivate DG(A) for each of the following functions

G(A) = (trA)A

G(A) = ABA

G(A) = A^TA

G(A) = (u, Au)A

Answer 1

As the specified assignment that is about to operators find out the derivations Problem 1- concern the operator ? to products of smooth scalar fields F, ?, vector fields u, v, and tensor fields A, B. set up the identities div(Fu) = Fdivu + (u, gradF) grad(u, v) = (gradTu)v + (gradTv)u div(ATu) = (divA,u) + A : gradu Problem 2- Calculate the derivate DG(A) for each of the subsequent functions