ABSTRACT
The word \tensor" has a physically motivated root: namely the word \tension". In Continuum mechanics, it is customary to classify the forces acting upon dierential volume elements dV . One class corresponds to body forces b, dened as external forces per unit dierential mass dM = dV , where is the mass density of the continuous media, so that the corresponding external forces per unit dierential volume is b. Typical examples are gravitational forces, with b = gk, and centrifugal forces, due to rotation. The other class corresponds to the surface forces acting upon the surface
bounding dV . In order to isolate the dierential volume element dV from the rest of the
continuous media, it is necessary to consider the bounding surface as a mathematically ideal surface. This gives rise to two surfaces, which are congruent to the bounding surface. One of them can be thought of as made of molecules within the volume element dV , and the other can be considered as made of molecules lying inside the set-complement. These two sets of molecules act upon each other, thus exerting force throughout the separating surface, according to Newton's Law of Action and Reaction. Therefore, in the case of an elastic medium being stretched, there will be one force per unit dierential surface area d, (td) as the traction force exerted from the complement set to dV , and a contraction force td exerted by the boundary molecules from dV toward the complement set. Such vector eld t is generally designated as the tension. In cases involving nonviscous uids, the surface forces do not act as trac-
tions. As it is familiar, the liquid pressure forces act upon surfaces (such as a diver's eardrum) in a compressive way, and, in a direction normal to the surface, so that in case of td = (pn)d, where n is a unit normal outwardly directed with respect to dV . From this simple denition, it might appear that the scalar-valued pressure
eld p might depend not only on the space location for d, but also on the direction of the normal n to dV . Pascal's principle states that the pressure at a point within an inviscid uid,
i.e., a liquid free from tangential surfaces, is the same in all directions. We will prove this in detail, since the geometry and symbols needed to dene the general Cartesian tensor are already present in this very simple and familiar
situation. Suppose that the pressure force pn = pnn is dierent for n = i; j; k, so
that we are dealing with four scalar values pn, px, py, pz at the same point p = (x; y; z) (See Figure B.1). Now consider the dierential volume dV for a tetrahedron PABC, where
A = (x+x; y; z), B = (x; y +y; z), and C = (x; y; z +z). Let dx, dy, dz, and dn denote the areas of the boundary triangular
faces PBC, PCA, PAB, and ABC, respectively (see Figure B.1.)
Figure B.1: Four scalar values for pressure acting at the same point p = (x; y; z).
