The first term on the right hand side needs to be broken into a symmetric part and a skew part as follows: ∫ ∫ V ( ∂ ∂ u i ∂ ∂ x j σ σ i j − − u i f i ) d V = ∫ ∫ V ( 1 2 [ ( ∂ ∂ u i ∂ ∂ x j + ∂ ∂ u j ∂ ∂ x i ) + ( ∂ ∂ u i ∂ ∂ x j − − ∂ ∂ u j ∂ ∂ x i ) ] σ σ i j − − u i f i ) d V = ∫ ∫ V ( [ ϵ ϵ i j + 1 2 ( ∂ ∂ u i ∂ ∂ x j − − ∂ ∂ u j ∂ ∂ x i ) ] σ σ i j − − u i f i ) d V = ∫ ∫ V ( ϵ ϵ i j σ σ i j − − u i f i ) d V = ∫ ∫ V ( ϵ ϵ : σ σ − − u ⋅ ⋅ f ) d V where ϵ ϵ is the strain that is consistent with the specified displacement field. The 2nd to last equality comes from the fact that the stress matrix is symmetric and that the product of a skew matrix and a symmetric matrix is zero.