Mathematical Physics Vol 1

2.1 Vector analysis

45

Definition The function v = v ( T ) is differentiable in point A , if its increment can be represented in the form ∆ v =[ p 1 ( T )( t 1 − a 1 )+ ··· + p n ( T )( t n − a n )]+ ω ( T ) · ρ ( T , A ) , (2.41) where ρ ( T , A ) is the distance between points T and A (in Euclidian space) ρ = q ( t 1 − a 1 ) 2 +( t 2 − a 2 ) 2 + ··· +( t n − a n ) 2 , (2.42) and ω ( T ) is a continuous function in point A , inwhich

ω ( T )= ω ( A )= 0 .

lim T → A

(2.43)

Definition Differential of the vector function v ( T ) , in point A , is the linear part with respect to the increment of variables ∆ t i = t i − a i , in the expression for the increment of the function ∆ v , i.e. d v ( T , A )= p 1 ( t 1 − a 1 )+ ··· + p n ( t n − a n ) . (2.44)

If all variables except the k -th, namely

( a 1 = t 1 , ··· , a k

t k

a k + 1 = t k + 1 , ··· , a n = t n ) ,

− 1 =

− 1 ,

are fixed, then the distance ρ becomes

ρ ( T , A )= q ( t k − a k )

2 = | t

k − a k | ,

(2.45)

and the increment of the function takes the form ∆ v = v ( a 1 , ··· , a k − 1 ,

t k , a k + 1 , ··· , a n ) − v ( a 1 , a 2 ··· , a n )=

(2.46)

p k ( t k − a k )+ ω ( T ) ·| t k − a k | .

If the symbol t k − a k = ∆ t k is introduced, it follows from the last equation that ∆ v ∆ t k = p k ± ω ( T ) .

(2.47)

and consequently

∆ v ∆ t k

ω ( T ) ,

= p k ± lim T → A

lim ∆ t k → 0

(2.48)

or

∂ v ∂ t k T = A

p k =

(2.49)

.

The expression for the differential (2.44) can now take one of the following forms

∂ v ∂ t 1

∂ v ∂ t n

d v ( T , A )=

( t 1 − a 1 )+ ··· +

( t n − a n )

(2.50)