Mathematical Physics Vol 1

Chapter 2. Vector analysis

42

d v

Let us now increase the value of the scalar t by the value of ∆ t . The vector that corresponds to value t + ∆ t of the scalar is denoted by v ( t + ∆ t ). In Fig 2.2 this vector is represented by the oriented segment −→ OB . The change of the vector v ( t ), which corresponds to the increment of the scalar t by ∆ t , is given by the difference ∆ v = v ( t + ∆ t ) − v ( t ) . (2.16) It can be observed in Fig. 2.2 that this difference, the geo metric increment, is represented by the oriented segment −→ AB (= ∆ v ).

d t

A

∆ v

v ( t )

∆ v

B

∆ t

v ( t + ∆ t )

O

Figure 2.2: The increment of the vector function.

Consider now the vector ∆ v ∆ t , which represents the mean change in the value of v with respect to the parameter t . The vector defined in this way has the same direction as the vector ∆ v ( ∆ t is a scalar) if ∆ t > 0, and opposite direction if ∆ t < 0.

Definition The value defined by the relation:

∆ v ∆ t

lim ∆ t → 0 (2.17) is called the derivative of the vector v (ordinary derivative, as opposed to directional derivative, which will be defined later) with respect to the scalar t , if such a limit exists. This value will be denoted shortly by v ′ .

The derivative will be symbolically denoted by d v d t

, and thus

∆ v ∆ t

d v d t

= lim ∆ t → 0

(2.18)

.

f ( x )

Definition It is said that a vector function is differentiable inpoint t , if the first derivative in that point exists, i.e. if the following limit value exists

∆ v ∆ t

v ′ = lim ∆ t → 0

(2.19)

.

x

O

A differentiable function is also contin uous. The reverse is not true (see Fig. 4.30).

Figure 2.3: An example of a continuous non differentiable function.