Mathematical Physics Vol 1

2.1 Vector analysis

43

2.1.5 Properties of the derivative

Some properties of the derivative follow d d t

d a d t

d b d t

( a + b )=

(2.20)

+

,

d a d t

d d t

d m d t

( m a )= m

a , m = m ( t ) ,

(2.21)

+

d b d t

d a d t ·

d d t

( a · b )= a ·

b ,

(2.22)

+

d b d t

d a d t ×

d d t

( a × b )= a ×

b .

(2.23)

+

2.1.6 Differential of the vector function Suppose that the geometric increment of the vector function v ( t ) can be represented as ∆ v = v ( t + ∆ t ) − v ( t )= L ( t ) ∆ t + ε ( ∆ t ) , (2.24) where ε ( ∆ t ) , when ∆ t → 0, is a higher order vector infinitesimal with respect to ∆ t , i.e. lim ∆ t → 0 ε ( ∆ t ) ∆ t = 0 . (2.25) Definition Differential of the vector function v ( t ) is the linear part of the increment of the argument L ( t ) ∆ t in the geometric increment of the function. This is symbolically denoted by d v = L ( t ) ∆ t . (2.26) For sufficiently small values of the increment of the variable ∆ t = d t , the geometric increment of the function v ( t ), can be approximated by its differential, i.e. v ( t + ∆ t ) − v ( t ) ≈ L ( t ) ∆ t , (2.27) that is v ( t + ∆ t ) − v ( t ) ≈ d v . (2.28) It follows from (2.24) that v ( t + ∆ t ) − v ( t ) ∆ t = L ( t )+ ε ( ∆ t ) ∆ t , (2.29) namely, according to (2.25) and (2.18), we obtain

v ( t + ∆ t ) − v ( t ) ∆ t

d v d t

= v ′ .

= L ( t )=

lim ∆ t → 0

(2.30)

Now (2.26) becomes

d v d t ·

d v = (2.31) To obtain the last relation (2.31) we used the definition of the derivation (2.17) and the assumption that ∆ t = d t . Thus, the differential d v is a vector whose direction is tangent to the hodograph. ∆ t = v ′ · d t .