Mathematical Physics Vol 1

4.6 Examples

169

parametric form x = t 2 , y = 2 t , z = t 3 . Time changes in the interval t ∈ [ 0 , 1 ] . Calculate the following line integrals a) R c φ d r , b) R c F × d r .

Solution a) Let us express φ by means of parameter t :

φ = 2 xyz 2 = 2 ( t 2 )( 2 t )( t 3 ) 2 = 4 t 9 .

Given that

r = x i + y j + z k , it follows that d r , expressed by means of t is d r = 2 t i + 2 j + 3 t 2 k d t . It further follows Z c φ d r = 1 Z 0 4 t 9 ( 2 t i + 2 j + 3 t 2 k ) d t =

1 Z 0

1 Z 0

1 Z 0

8 11

4 5

8 t 10 d t + j

8 t 9 d t + k

12 t 11 d t =

= i

i +

j + k .

b) Analogously to the result under a)

F = xy i − z j + x 2 k ⇒ F = 2 t 3 i − t 3 j + t 4 k .

Then F × d r =( 2 t 3 i − t 3 j + t 4 k ) × ( 2 t i + 2 j + 3 t 2 k ) d t =

i k 2 t 3 − t 3 t 4 2 t 2 3 t 2 j

=[( − 3 t 5 − 2 t 4 ) i +( 2 t 5 − 6 t 5 ) j +( 4 t 3 + 2 t 4 ) k ] d t

=

1 Z 0

1 Z 0

1 Z 0

Z c

( − 3 t 5 − 2 t 4 ) d t + j

( 2 t 5 − 6 t 5 ) d t + k

( 4 t 3 + 2 t 4 ) d t =

F × d r = i

9 10

2 3

7 5

i −

j +

k .

= −