Mathematical Physics Vol 1

2.2 Integration

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2.2 Integration

2.2.1 Indefinite integral of a vector function In the previous section, vector function differentiation was defined. Namely, for a vector function a its derivative can be found following the definition (2.17). However, is often necessary to solve the reverse task, namely to find a vector function if its derivative is given. Let a ( t ) be a continuous vector function of the scalar argument t .

Definition The primitive function of the function a ( t ) is the function b ( t ) the derivative of which d b d t = a . (2.54)

However, as the derivative of a constant vector is equal to zero, i.e. d c d t = 0 ,

(2.55)

then if b ( t ) is a primitive function of a continuous function a ( t ) , there exists an unlimited set of primitive functions, each of them differing from b ( t ) by a vector constant c , i.e. d ( b + c ) d t = a . (2.56) Definition The indefinite integral of a vector function a is the set of all its primitive functions, denoted by Z a d t = b + c . (2.57) the indefinite integral of a vector function amounts to a sum of indefinite integrals of scalar functions Z a d t = Z a x d t i + Z a y d t j + Z a z d t k . (2.59) 2.2.2 Definite integral Let a be a bounded function of the parameter t , on the interval ( t A , t B ) . Let that interval be divided into a finite number of parts ∆ t i = t i − t i − 1 (2.60) by the points t A = t 0 < t 1 < ··· < t n = t B . (2.61) As a can be represented by the relation a = a x i + a y j + a z k , (2.58)