Mathematical Physics Vol 1

Chapter 2. Vector analysis

40

2.1.2 Hodograph of a vector function

Definition Hodograph of a vector function v = v ( t ) is the geometric location of the end points of the vector v for all possible values of t , where all these vectors start at a single fixed point, e.g. O , which is called the pole of the hodograph .

v 1

If the vector function represents a position vector (in 3–D), then the hodograph of this vector function represents a spatial curve (3–D). The vector equation of this curve is denoted by r = r ( t ) . (2.4) To this function three scalar (parametric) equations corre spond, which represent the equation of the curve in space, in parametric form

v i

v n

O

x = x ( t ) , y = y ( t ) , z = z ( t ) ,

(2.5)

Figure 2.1: Hodograph of a vector func tion.

2.1.3 Limit processes. Continuity Basic concepts in vector analysis, such as convergence and continuity can be introduced as follows.

Definition It is said that a series of vectors a n , n = 1 , 2 ,..., converges if there exists a vector a such that lim n → ∞ | a n − a | = 0 . (2.6) The vector a is called the limit vector of the series a n and denoted by

a n = a .

lim n → ∞

(2.7)

If a coordinate system is introduced, then it is said that the series of vectors a n converges to a iff the three series (in 3-D) of vector components converge to the respective components of vector a . Definition It is said that the vector function v ( t ) has vector ℓ as its limit value when the argument t tends to t 0 , if for a given number ε > 0 a number δ = δ ( ε ) > 0 can be determined, such that for each t , forwhich | t − t 0 | < δ , (2.8)