Mathematical Physics Vol 1

Chapter 2. Vector analysis

44

2.1.7 Higher order derivatives and differentials Since the derivative of a vector function is also a vector function (of the same variable), a derivative of that function can subsequently be determined, and thus a second derivative obtained d d t ( v ′ )= d 2 v d t 2 = v ′′ . (2.32) In the same way, higher derivatives can also be obtained. Since by definition v ( 0 ) = v , the n –th derivative is d d t d n − 1 v d t n − 1 = d n v d t n = v ( n ) , (2.33) where this relation is true for each n ∈ N . The n –th differential of the vector function is, by analogy with (2.31), a product of the n –th derivative and the n –th degree of the differential d t , i.e. d n v = v ( n ) d t n . (2.34) 2.1.8 Partial derivative of a vector function of several independent variables Consider a vector function v , which depends of n scalar variables t i , i = 1 , 2 , ··· , n , i.e. v = v ( t 1 , t 2 , ··· , t n )= v ( T ) . (2.35) T can be perceived as a point in the n –dimensional space, with coordinates t i ( i = 1 , 2 , ··· , n ). Let us now find the increment of this function, if only one variable, say k , changes, and the other variables remain "frozen", i.e. they are held constant. The increment of the function is v ( t 1 , ··· , t k + ∆ t k , ··· , t n ) − v ( t 1 , ··· , t k , ··· , t n ) . (2.36) The limit value

Definition

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

∂ v ∂ t k

lim ∆ t k → 0

(2.37)

is called the partial derivative of the vector function v , with respect to the variable t k , it such a limit value exists.

As this value is also a vector function, of the same variables, partial derivatives of higher order can also be defined, as for example ∂ 2 v ∂ t 2 i , ∂ 2 v ∂ t i ∂ t j , ∂ 3 v ∂ t 3 i , ∂ 3 v ∂ t 2 i ∂ t j , ∂ 3 v ∂ t i ∂ t j ∂ t k , ··· (2.38) 2.1.9 Differential of a vector function of n scalar variables Observe the vector function v = v ( T ) , (2.39) its point T ( t 1 , t 2 , ··· , t n ) and some other point A ( a 1 , a 2 , ··· , a n ) . Points A and T belong to the n –dimensional Euclidian space E n . The increment of the function v is ∆ v = v ( T ) − v ( A )= v ( t 1 , t 2 , ··· , t n ) − v ( a 1 , a 2 , ··· , a n ) . (2.40)