ere B is the total derivative. The Ths can be expanded as D (pdv)v+ but di (pdv)=0 from conservation of mass, so the principle reads fdv+tdS Now, using what we've learned about the tractions and their relation to the stress tensor. Padv=/fdv+/n ads This is the linear momentum balance equation in integral form. We can replace the surface integral with a volume integral with the aid of the diver- gence theorem and then(18)becomes f-V·adv=0 Since this principle applies to an arbitrary volume of material, the integrand must vanish f = 0 at This is the linear momentum balance equation in differential form. In com- 可x+f=p at Angular momentum balance and the symmetry of the stress tensor The principle of conservation of angular momentum states that the rate of change of angular momentum is equal to the sum of the moment of all the xternal forces acting on the body Dt/ ex xvdv x×fdV+)x×tdS� � � � � � � � � � � � where D is the total derivative. The lhs can be expanded as: Dt D D ∂v ρvdV = Dt (ρdV )v + ρ dV Dt V V V ∂t D but Dt(ρdV ) = 0 from conservation of mass, so the principle reads: ∂v ρ dV = fdV + tdS (17) V ∂t V S Now, using what we’ve learned about the tractions and their relation to the stress tensor: � V ρ ∂v ∂t dV = � V fdV + � S n · σdS (18) This is the linear momentum balance equation in integral form. We can replace the surface integral with a volume integral with the aid of the diver￾gence theorem: � � n · σdS = � · σdV S V and then (18) becomes: ∂v ρ − f − � · σ dV = 0 V ∂t Since this principle applies to an arbitrary volume of material, the integrand must vanish: ∂v ρ − f − � · σ = 0 (19) ∂t This is the linear momentum balance equation in differential form. In com￾ponents: ∂vi σji,j + fi = ρ ∂t Angular momentum balance and the symmetry of the stress tensor The principle of conservation of angular momentum states that the rate of change of angular momentum is equal to the sum of the moment of all the external forces acting on the body: D ρx × vdV = x × fdV + x × tdS (20) Dt V V S 7