Mathematical Physics Vol 1

2. Vector analysis

2.1 Vector analysis 2.1.1 Vector function

Let T be a set of real (complex) numbers (scalars), and V a vector set.

Definition If to each number t ∈ T , according to a certain rule, corresponds a specific value of a vector v ∈ V , then v is called a vector function of a scalar argument t and shortly denoted by v = v ( t ) . (2.1) The vector function can also be defined in another way: a single-valued mapping of a set of real (complex) numbers T to a vector set V , according to a certain rule v = v ( t ) . (2.2) is called a vector function v ( t ) of one scalar argument t . The set of real (complex) numbers (scalars) T on which the function is defined is called the domain of the function v ( t ). As, according to 1.38, v = [ v 1 , v 2 , v 3 ] (in 3–D), the single-valued mapping of set T on set V consequently amounts to a mapping of the first set to the second set by means of three scalar functions, which represent the projections of the vector function v ( t ) on the coordinate axes v ( t )=[ v 1 ( t ) , v 2 ( t ) , v 3 ( t )] . (2.3) On basis of the aforementioned, the analysis of vector functions of one scalar argument amount to the analysis of three (in 3–D) scalar functions–projections of vector v ( t ) on the axes of the corresponding coordinate system.