Chap. 9. Vector Integral Calculus. Integral Theorems

Chap. 9. Vector Integral Calculus. Integral Theorems

- Green, Gauss, Stokes theorems … 9.1. Line Integrals

- Line integral (or curve integral):

Definition and Evaluation of Line Integrals

(curve C = path of integration: r(t) = x(t)i +y(t)j + z(t)k)

(every path of integration of a line integral  piecewise smooth) For closed curve:

ex.) Work done by a force F in a displacement along C



a^bf(x)dx dt dt

r )) d t ( r ( F r

d ) r ( F

b a

C





^{⋅ = ⋅}

( )

( )







+ +

=

+ +

=

⋅

b

a 1 2 3

C 1 2 3

C

dt ' z F ' y F ' x F

dz F dy F dx F r

d ) r (

F



C ^



C

Ex. 1) Evaluation of a line integral in the plane

Ex. 2) Line integral in space

- Choice of representation ? - Choice of path ?

Ex. 3) Dependence of a line integral on path (from A(0,0,0) to B(1,1,1))

k 3 j t cos i

t sin )

t ( ' r , k t 3 j t sin i

t cos )

t ( r , k y j x i z ) r (

F = + + = + + = − + +







^{⋅ = ⋅ =} 0 ^{/2 −}

2 b 2

a

C dt (sin t cos tsint)dt

dt r )) d t ( r ( F r

d ) r (

F ^π





^{⋅ =} 0^{2π − + +}

2

CF(r) dr ( 3tsint cos t 3sint)dt

j t cos i

t sin )

t ( ' r , j t sin t cos i

t sin ))

t ( r ( F

j t sin i

t cos )

t ( r , j xy i

y )

r ( F

+

−

=

−

−

=

+

=

−

−

=

) 1 t 0 ( k t j t i t ) t ( r ), 1 t 0 ( k t j t i t ) t ( r , k z x j xy i

z 5 ) r (

F = + +

² ₁

= + + ≤ ≤

₂

= + +

²

≤ ≤

12 dt 28

) t 2 t

t 5 ( r

d ) r ( F )

b (

12 dt 37

) t t t 5 ( r

d ) r ( F )

a (

1 0

5 2

2 C

1 0

3 2 C

2 1

= +

+

=

⋅

= +

+

=

⋅









Line integral depends on

- F

- Endpoints A, B of the path - path

- Conditions for guaranteeing independence of the path: see Section 9.2 Motivation of the Line Integral: Work Done by a Force

Sum of these n works: , n∞:  line integral Ex. 5) Work done equals the gain in kinetic energy

Other Forms of Line Integrals

Special cases of line integral when F=F₁i or F₂j or F₃k 

Ex. 6)

[

m 1 ^m

]

^{m m m}

m

m

F ( r ( t )) r ( t ) r ( t ) F ( r ( t )) r ' ( t ) t

W = ⋅ − ≈ ⋅ Δ

Δ

₊

1 n 0

n

W W

W = Δ +  + Δ

₋

b t

a t b 2

a b

a

b a b

a C

2 v dt m

2 ' v m v

dt v ' v m W

) t ( ' v m ) t ( '' r m F

dt ) t ( v )) t ( r ( F dt dt

r )) d t ( r ( F r

d F W

=

=

=



 



 ⋅

=

⋅

=

=

=

⋅

=

⋅

=

⋅

=

















C 3

C 2

C

F

1

dx , F dy , F dz

( ^{F F i f F x '} )

dt )) t ( r ( f dt

) r (

f

^b _{1 1}

a

C

=  =  =



General Properties of the Line Integral

Theorem 1: Direction-preserving transformations of parameter

Any representations of C that give the same positive direction on C also yield the same value of the line integral.

Proof:

( )

r d F r

d F r

d F

r d G r

d F r

d G F

r d F k r d F k

2

1 C

C C

C C

C

C C

⋅ +

⋅

=

⋅

⋅ +

⋅

=

⋅ +

⋅

=

⋅

















r d ) r ( F dt dt

r )) d t ( r ( F

dt dt dt dt

r )) d t ( ( r ( F dt dt

r ) d t ( r ( F r

d ) r ( F

C b

a

b a

*

* b *

a

*

*

*

* *

* C

*

*

*

*

*

⋅

=

⋅

=

⋅

 =

 



 



 ⋅

=

⋅









 ^φ