Coordinate subscript convention

The stress existing at any point in a material may always be resolved into components acting on the faces of a diifferential element in three arbitrary coordinate directions (Fig. 17.1). The stress components acting on the faces of the element are of two types normal stresses (forces acting normal to the surface per unit surface area) and shear stresses (forces acting parallel to the surface per unit surface area). The first subscript conventionally identifies the direction perpendicular to the surface in question, and the second identifies the direction of the force itself. In general, there are three normal stresses, Th, T22, and T33, and six shear stresses, ti2, Ti3, T21, 1 23, T32. These nine quantities,... 

It is convention to use Greek letters for a subscript that runs over all four coordinates, and Roman letters for subscripts only over the space components. 

Placement of indices as superscripts or subscripts follows the conventions of tensor analysis. Contravariant variables, which transform like coordinates, are indexed by superscripts, and coavariant quantities, which transform like derivatives, are indexed by subscripts. Cartesian and generalized velocities and 2 thus contravariant, while Cartesian and generalized forces, which transform like derivatives of a scalar potential energy, are covariant. 

The mass balance of the bimodal particle is composed of two equations. One describes the mass balance inside the crystal, while the other describes the mass balance in the pellet, and the two are coupled through the boundary of the zeolite crystal. The coordinate framework for the zeolite pellet is shown as in Figure 10.4-2, with r being the radial distance for the pellet and r being the radial distance for the crystal. We have used the notation convention that the subscript p is for the adsorbed phase. Here the crystal is acting as the adsorbed phase. 

Here, the subscript h indexes different helices in the unit cell. In this convention, the fractional coordinates for each unique helix in the unit cell are specified with the helix axis directed along the z-coordinate and passing through the origin. Each helix may have a distinct setting angle, o)/, and h and k may take fractional values to indicate helices at nonprimitive lattice positions. The lattice index / is replaced by the helix repeat index, n. 

The scalar product k.n equals the sum of kjU, in an arbitrary Cartesian coordinate system for which the components of k are kj and those of n are rij It is a common convention to write k.n= k,Tij, meaning that if an index appears twice in a product, then the sum is performed implicitly with respect to that index, here subscript j. Then the vector h can be written alternatively by expressing the components h of h... 

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