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**Example text**

7. Using a 2 × 2 tensor A, write out the differential of ln(A). 1 INTRODUCTION TO VARIATIONAL METHODS Let u(x) be a vector-valued function of position vector x, and consider a vectorvalued function F(u(x), u′(x),x), in which u′(x) = ∂u/∂x. Furthermore, let v(x) be a function such that v(x) = 0 when u(x) = 0 and v′(x) = 0 when u′(x) = 0, but which is otherwise arbitrary. The differential d F measures how much F changes if x changes. The variation δ F measures how much F changes if u and u′ change at fixed x.

27) Observe from the following results that ∇ 2 A = ∇(∇ ⋅ A T ) − ∇ × [∇ × A T ]T . 28) An integral theorem for the Laplacian of a tensor is now found as ∫ ∇ AdV = ∫ (n∇ )AdS − ∫ n × [∇ × A ] dS. 30) in which I1 = tr(A) I2 = 1 2 [tr (A) − tr(A 2 )] 2 I3 = det(A). 31) Here, tr (A) = δijaij denotes the trace of A. 33) = I [A − I1A + I2 I] 2 The trace of any n × n symmetric tensor B is invariant under orthogonal transformations (rotations), such as tr(B′) = tr(B), since a ′pqδ pq = q pr qqs arsδ pq = ars q pr qqs = arsδ rs .

13) th The j variation of a vector-valued quantity F is defined by d jΦ δ jF = e j j . 14) It follows that δ u = 0 and δ u′ = 0. By restricting F to a scalar-valued function F and x to reduce to x, we obtain 2 δ 2 F = {δ u T δu δ u ′ T}H , δ u ′ 2 ∂ T ∂ F ∂u ∂u H= T ∂ ∂ ∂u ∂u F ′ and H is known as the Hessian matrix. 16) in which V again denotes the volume of a domain and S denotes its surface area. In addition, h is a prescribed (known) function on S.