Cartesian Tensors

Abstract

In this text we shall deal, almost exclusively, with Cartesian coordinates. They are certainly the most ‘familiar’ coordinates and, mathematically, the easiest to deal with. This is not to imply that they are always the most convenient to use. In physical situations which display certain symmetries then it may be appropriate to suspend the use of Cartesian coordinates and to use a coordinate system more attuned to the natural symmetry inherent in the problem. However, there is often a hefty price to be paid when using ‘curvilinear’ coordinates — particularly when dealing with vectors. The base vectors $\hat i,\hat j,\hat k$ in Cartesians are constant in both magnitude and direction. This is not the case for curvilinear coordinates; at least one of the base vectors ${{\hat e}_1},{{\hat e}_2},{{\hat e}_3}$ has a direction which is position dependent. This considerably complicates the treatment of vector differentiation.