Mathematical Physics Vol 1

2.1 Vector analysis

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the following relation holds

| v ( t ) − ℓ | < ε .

(2.9)

This can be symbolically denoted by

lim t → t 0 v ( t )= ℓ or v ( t ) → ℓ when t → t 0 .

(2.10)

Definition A vector function v ( t ) is continuous inpoint t = t 0 , if 1 ◦ it is defined in point t = t 0 , 2 ◦ has a limit value when t → t 0 , and 3 ◦ if

v ( t )= v ( t 0 ) .

lim t → t 0

(2.11)

Let v ( t ) and v ( t + ∆ t ) be the values of the vector function v for argument values t and t + ∆ t ( ∆ t is the increment of the argument). The difference v ( t + ∆ t ) − v ( t )= ∆ v ( t ) (2.12) is called the geometric increment of the function v ( t ). Let us represent this on the hodograph of the function v ( t ), Fig. 2.2. The geometric increment is represented by the vector −→ AB . There is also another way of defining continuity. Definition A vector function v ( t ) is continuous for a given value of the argument t , if its geometric increment ∆ v tends to zero, when the increment of the argument tends to zero, i.e. when lim ∆ t → 0 | ∆ v ( t ) | = 0 . (2.13)

As v ( t )= [ v 1 ( t ) , v 2 ( t ) , v 3 ( t )] , the necessary and sufficient condition for the continuity of the function v ( t ) in point t is that the projections v i ( t ) of this vector are continuous functions in this point, i.e., that:

∆ v i ( t )= 0 ⇒ lim ∆ t → 0

[ v i ( t + ∆ t ) − v i ( t )]= 0 , i = 1 , 2 , 3 .

lim ∆ t → 0

(2.14)

From here, the continuity of the function v ( t ) also follows, i.e. | ∆ v ( t ) | = q ( ∆ v 1 ( t )) 2 +( ∆ v 2 ( t )) 2 +( ∆ v 3 ( t ))

2 → 0 .

(2.15)

2.1.4 Derivative of a vector function of one scalar variable Observe now some (arbitrary) value of the scalar t and the corresponding value of the vector v ( t ). This vector is represented in Fig. 2.2 by the oriented segment −→ OA .