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Coulomb's law of kinetic friction requires that the friction force be aligned with the relative slip velocity between the two surfaces. Despite the name, Coulomb's law is a constitutive relation, and hence must obey the Principle of Material-Frame Indifference; this requires a frame-invariant relative velocity. As points in stick do not experience relative motion, the development of velocities below is only concerned with opposing contact points in slip.
As parametric coordinates of integration points on the primary surface represent material points, the velocity of these points is evaluated from the material time derivative in the material frame, In contrast, different material points may convect through the intersection point , and so the velocity at the intersection point on is evaluated from the material time derivative in the spatial frame, where represents the velocity of the intersection point on , while are the contravariant components of the convective velocity of material passing through the intersection point . The quantity represents the relative slip velocity between the material on and that on . Importantly, by definition is evaluated while holding constant. From these relations, a more practical formulation of the slip velocity can be achieved . Taking the material time derivative of eq.(184.108.40.206-1) and recalling the contact persistency condition  produces Substituting Eqs.(220.127.116.11-1)-(18.104.22.168-2) into this expression yields the desired frame-invariant measure of relative velocity between and , where is evaluated from eq.(22.214.171.124-2) as Here, is the material time derivative of in the material frame, evaluated from eq.(7.2.1-3) as and is the tangential plane projection tensor, A unit vector in the slip direction can then be found by projecting the relative velocity onto the tangent plane of , yielding