Vector Algebra Review

The purpose of this chapter is to review some of the salient operations involving scalar and vector fields and to broaden these concepts to dyadics and tensors in general. Here we briefly discuss variant and invariant scalars, the concept of scalar and vector fields, and the utility of phasor forms of these quantities. Classical arithmetic vector operations of addition, subtraction, and dot and cross products are discussed along with physical applications of these. The direct vector-vector product is mentioned in Section 2.4.3 as having a dyadic resultant; however, the details of this process are left to later chapters. The basic building blocks of open and closed line and surface integrals of vector fields are discussed. These are essential for both the definitions of vector differential operators, covered in Chapter 4, and the integral forms that shape the basis of divergence, Stokes', and Green's theorems covered in Chapter 5. Other highly useful applications of dot- and cross-product operations conclude the sections of this chapter. These are vector field direction lines and equivalue surfaces of scalar fields.