The mass/energy conservation laws are derived for two commonly used co-ordinate systems—the Cartesian co-ordinate system in Section 7.1 and the spherical co-ordinate system in Section 7.2. For unidirectional transport, we have seen that the conservation equation has different forms in different co-ordinate systems. Here, conservation equations are first derived using shell balance in three dimensions for the Cartesian and spherical co-ordinate systems. The conservation equations have a common form when expressed in terms of vector differential operators, the gradient, divergence, and Laplacian operators; the expressions for these operators are different in different co-ordinate systems. The conservation equation derived using shell balance is used to identify the differential operators in the the Cartesian and spherical co-ordinate system, and the procedure for deriving these in a general orthogonal co-ordinate system is explained.

Since the conservation equation is universal when expressed using vector differential operators, it is not necessary to go through the shell balance procedure for each individual problem; it is sufficient to substitute the appropriate vector differential operators in the conservation equation expressed in vector form. It is important to note that the derivation here is restricted to orthogonal co-ordinate systems, where the three co-ordinate directions are perpendicular to each other at all locations.

The discussion in Section 7.1 and 7.2 is restricted to mass/energy transfer. The constitutive relation (Newton's law) for momentum transfer for general three-dimensional flows is more complicated than that for mass/heat transfer. Mass and heat are scalars, and the flux of mass/heat is a vector along the direction of decreasing concentration/temperature. Since momentum is a vector, the flux of momentum has two directions associated with it: the direction of the momentum vector and the direction in which the momentum is transported. Due to this, the stress or momentum flux is a ‘second order tensor’ with two physical directions—the direction of momentum and the orientation of the perpendicular to the surface across which momentum is transported.