ABSTRACT
In the preceding chapter, an orthogonal Cartesian coordinate system was assumed. Here, we consider an “oblique coordinate system” that is not necessarily orthogonal. If the axes are not orthogonal, the basis vectors are not orthogonal, either. Then, vector components are defined in two different ways: a vector can be expressed as a linear combination of the basis vectors, or it can be expressed as a linear combination of another set of vectors, called “reciprocal basis vectors,” that are orthogonal to the basis vectors. The inner product, the vector product, and the scalar triple product have different expressions depending on which convention we use. It is shown, however, that these different expressions can be transformed to each other by means of the “metric tensor” that specifies the inner products among the basis vectors. If we use another oblique coordinate system, the same vector has a different expression, but the “coordinate transformation” can be described in a systematic way. For simplicity, the Cartesian coordinate system is used throughout the subsequent chapters, so the readers who want to know the idea of geometric algebra quickly can skip this chapter in the first reading. This chapter requires some knowledge of linear algebra.
