Upload
rai-university
View
326
Download
7
Embed Size (px)
Citation preview
Unit-5 VECTOR INTEGRATION
Unit-V: VECTOR INTEGRATION
Sr.
No.
Name of the Topic Page
No.1 Line Integral 2
2 Surface integral 5
3 Volume Integral 6
4 Green’s theorem (without proof) 8
5 Stoke’s theorem (without proof) 10
6 Gauss’s theorem of divergence (without proof) 13
7 Reference book 16
RAI UNIVERSITY, AHMEDABAD 1
Unit-5 VECTOR INTEGRATION
Vector integration
1.1 LINE INTEGRAL:
Line integral ¿∫c(F . drds )ds=∫c F .dr
Note:1) Work: If F represents the variable force acting on a particle along arc AB,
then the total work done ¿∫A
B
F .dr
2) Circulation: If V represents the velocity of a liquid then ∮cV .dr is called the
circulation of V round the closed curve c .If the circulation of V round every closed curve is zero then V is said to be irrotational there.
3) When the path of integration is a closed curve then notation of integration is ∮ in place of∫ .
Note: If ∫A
B
F .dr is to be proved to be independent of path, then F=∇∅
here F is called Conservative (irrotational) vector field and ∅ is called the Scalar potential. And ∇× F=∇×∇∅=0
Example 1: Evaluate ∫cF .dr where F=x2 i+xy j and C is the boundary of the square
in the plane z=0 and bounded by the linesx=0 , y=0 , x=a∧¿
y=a.
Solution: ∫cF .dr=∫
OAF .dr+∫
ABF .dr+∫
BCF .dr+∫
COF .dr
Here r=x i+ y j , dr=dx i+dy j , F=x2 i+xy j
RAI UNIVERSITY, AHMEDABAD 2
Unit-5 VECTOR INTEGRATION
F .dr=x2dx+xydy _______ (i)
On OA , y=0 ∴F .dr=x2dx (From (i))
∫OAF .dr=∫
0
a
x2dx=[ x3
3 ]0
3
=a3
3 _______ (ii)
On AB , x=a ∴dx=0
∴F .dr=ay dy (From (i))
∫ABF .dr=∫
0
a
ay dy=a[ y2
2 ]0
a
=a3
2 _______ (iii)
On BC , y=a ∴dy=0
∴F .dr=x2dx (From (i))
∫BCF .dr=∫
a
0
x2dx=[ x3
3 ]a
0
=−a3
3 _______ (iv)
On CO, x=0
∴F .dr=0 (From (i))
∫CO
F .dr=0 _______ (v)
On adding (ii), (iii), (iv) and (v), we get
∫CF .dr=a3
3+ a3
2−a3
3+0=a3
2 ________ Ans.
Example 2: A vector field is given by
F=(2 y+3 ) i+( xz ) j+( yz−x) k . Evaluate ∫CF .dr along the path c is
x=2 t , y=t , z=t3 ¿ t=0¿ t=1.
Solution:
RAI UNIVERSITY, AHMEDABAD 3
Unit-5 VECTOR INTEGRATION
∫CF .dr=∫
C(2 y+3 ) dx+ (xz ) dy+( yz−x )dz
[ since x=2 t y=t z=t3
∴ dxdt
=2 dydt
=1 dzdt
=3 t 2] ¿∫
0
1
(2 t+3 ) (2dt )+ (2t ) ( t3 )dt+(t 4−2 t )(3 t 2dt)
¿∫0
1
( 4 t+6+2t 4+3 t 6−6 t 3 )dt
¿ [4 t2
2 +6 t+ 25 t
5+37 t7−6
4 t 4]0
1
¿ [2 t 2+6 t+ 25t 5+ 3
7t7−3
2t 4]0
1
¿2+6+ 25+3
7−3
2
¿7.32857 _________ Ans.
Example 3: Suppose F ( x , y , z )=x3 i+ y j+z k is the force field. Find the work done by F along the line from the (1, 2, 3) to (3, 5, 7).
Solution: Work done ¿∫cF .dr
¿ ∫(1,2,3)
(3,5,7)
(x3 i+ y j+z k ) . d ( i dx+ j dy+k dz )
¿ ∫(1,2,3)
(3,5,7)
(x3dx+ ydy+ zdz )
¿∫1
3
x3dx+∫2
5
y dy+∫3
7
z dz
¿ [ x4
4 ]1
3
+[ y2
2 ]2
5
+[ z2
2 ]3
7
RAI UNIVERSITY, AHMEDABAD 4
Unit-5 VECTOR INTEGRATION
¿ [ 814
−14 ]+[25
2− 4
2 ]+[492
−92 ]
¿ 804
+ 212
+ 402
¿ 2024
¿50.5 units _______ Ans.
1.2Exercise:
1) If a force F=2 x2 y i+3 xy j displaces a particle in the xy-plane from (0, 0) to (1, 4) along a curvey=4 x2. Find the work done.
2) If A=(3 x2+6 y ) i−14 yz j+20 x z2 k, evaluate the line integral ∮ A dr from (0, 0, 0) to (1, 1, 1) along the curveC.
3) Show that the integral ∫(1,2)
(3,4)
( x y2+ y3 )dx+(x2 y+3 x y2)dy is independent of
the path joining the points (1, 2) and (3, 4). Hence, evaluate the integral.
2.1 SURFACE INTEGRAL:
Let F be a vector function and S be the given surface.Surface integral of a vector function F over the surface S is defined as the integral of the components of F along the normal to the surface.
Component of F along the normal¿F . nWhere n = unit normal vector to an element ds and
n= grad f|grad f|
ds=dx dy(n . k )
RAI UNIVERSITY, AHMEDABAD 5
Unit-5 VECTOR INTEGRATION
Surface integral of F over S
¿∑ F . n ¿∬S
(F .n )ds
Note:
1) Flux ¿∬S
(F .n )ds where, F represents the velocity of a liquid.
If∬S
(F . n )ds=0, then F is said to be a Solenoidal vector point function.
3.1 VOLUME INTEGRAL:
Let F be a vector point function and volume V enclosed by a closed surface.
The volume integral ¿∭V
Fdv
Example 1: Evaluate ∬S
( yz i+zx j+xy k ) . ds where S the surface of the sphere
is x2+ y2+z2=a2 in the first octant.
Solution: Here, ∅=x2+ y2+ z2−a2
Vector normal to the surface ¿∇∅
¿ i ∂∅∂x + j ∂∅∂ y
+k ∂∅∂z
¿( i ∂∂x + j ∂∂ y
+k ∂∂ z ) (x2+ y2+ z2−a2 )
RAI UNIVERSITY, AHMEDABAD 6
Unit-5 VECTOR INTEGRATION
¿2 x i+2 y j+2 z k
n= ∇∅|∇∅|=
2x i+2 y j+2 z k√4 x2+4 y2+4 z2
¿ x i+ y j+ z k√ x2+ y2+z2
¿ x i+ y j+ z ka
[∵ x2+ y2+ z2=a2 ]
Here, F= yz i+zx j+xy k
F . n=( yz i+zx j+xy k ) .( x i+ y j+z ka )=3 xyza
Now, ∬SF . n ds=∬
S
( F .n ) dx dy|k . n|
¿∫0a
∫0
√a2−x2
3xyz dx dy
a( za )
¿3∫0
a
∫0
√a2−x2
xy dy dx
¿3∫0
a
x ( y2
2 )0
√a2− x2
dx
¿ 32∫0
a
x (a2−x2)dx
¿ 32 ( a2 x2
2− x4
4 )0
a
¿ 32 ( a4
2−a4
4 ) ¿ 3a4
8 ________ Ans.
RAI UNIVERSITY, AHMEDABAD 7
Unit-5 VECTOR INTEGRATION
Example 2: IfF=2 z i−x j+ y k, evaluate ∭V
F dv where, v is the region bounded by
the surfacesx=0 , y=0 , x=2 , y=4 , z=x2 , z=2.
Solution: ∭V
F dv=∭ (2 z i−x j+ y k )dx dydz
¿∫0
2
dx∫0
4
dy∫x2
2
(2 z i−x j+ y k ) dz
¿∫0
2
dx∫0
4
dy [ z2 i−xz j+ yz k ]x2
2
¿∫0
2
dx∫0
4
dy [ 4 i−2 x j+2 y k−x4 i+x3 j−x2 y k ]
¿∫0
2
dx [4 y i−2 xy j+ y2 k−x4 y i+x3 y j− x2 y2
2 k ]0
4
¿∫0
2
(16 i−8 x j+16 k−4 x4 i+4 x3 j−8x2 k )dx
¿ [16 x i−4 x2 j+16 x k−4 x5
5 i+x4 j−8 x3
3 k ]0
2
¿32 i−16 j+32 k−1285
i+16 j−643k
¿ 32 i5
+ 32 k3
¿ 3215
(3 i+5 k ) _________ Ans.
3.2 Exercise:
1) Evaluate∬S
(F . n ) ds, where, F=18 z i−12 j+3 y k and S is the surface of the
plane 2 x+3 y+6 z=12 in the first octant.
2) IfF=(2 x2−3 z) i−2 xy j−4 x k, then evaluate∭V∇ F dv, where V is bounded
by the plane x=0 , y=0 , z=0 and2 x+2 y+z=4.
RAI UNIVERSITY, AHMEDABAD 8
Unit-5 VECTOR INTEGRATION
4.1 GREEN’S THEOREM: (Without proof)
If ∅ (x , y ) ,Ψ ( x , y ) ,
∂ϕ∂ y
∧∂Ψ
∂ x be continuous functions over a region R bounded
by simple closed curve C in x− y plane, then
∮C
(ϕdx+Ψ dy )=∬R
( ∂Ψ∂ x − ∂ϕ∂ y )dxdy
Note: Green’s theorem in vector form
∫cF .dr=∬
R(∇×F ) . k dR
Where, F=∅ i+Ψ j , r=x i+ y j , k is a unit vector along z-axis and dR=dx dy.
Example 1: Using green’s theorem, evaluate∫c(x2 y dx+x2dy ), where c is the
boundary described counter clockwise of the triangle with vertices(0,0 ) , (1,0 ) ,(1,1).Solution: By green’s theorem, we have
∮C
(ϕdx+Ψ dy )=∬R
( ∂Ψ∂ x − ∂ϕ∂ y )dxdy
∫c
(x2 y dx+x2dy )=∬R
(2x−x2 )dx dy
¿∫0
1
(2 x−x2 )dx∫0
x
dy
¿∫0
1
(2 x−x2 )dx [ y ]0x¿¿
¿∫0
1
(2 x2−x3)dx
¿( 2 x3
3− x4
4 )0
1
¿( 23−1
4 ) ¿ 5
12 _______ Ans.
RAI UNIVERSITY, AHMEDABAD 9
Unit-5 VECTOR INTEGRATION
Example 2: Use green’s theorem to evaluate
∫c
(x2+xy )dx+(x2+ y2)dy, where c is the square formed by the lines
y=±1 , x=±1.
Solution: By green’s theorem, we have
∮C
(ϕdx+Ψ dy )=∬R
( ∂Ψ∂ x −∂ϕ∂ y )dxdy
¿∫−1
1
∫−1
1
[ ∂∂x
(x2+ y2 )− ∂∂ y
( x2+xy )]dxdy¿∫
−1
1
∫−1
1
(2x−x )dxdy
¿∫−1
1
∫−1
1
x dxdy
¿∫−1
1
x dx∫−1
1
dy
¿∫−1
1
x dx ( y)−1
1¿ ¿
¿∫−1
1
x dx (1+1)
¿∫−1
1
2 xdx
¿(x2)−1
1¿¿
¿1−1
¿0 ________ Ans.
RAI UNIVERSITY, AHMEDABAD 10
Unit-5 VECTOR INTEGRATION
4.2 Exercise:
1) Apply Green’s theorem to evaluate
∫C
[ (2x2− y2 )dx+(x2+ y2)dy ], where C is the boundary of the area enclosed by
the x-axis and the upper half of circlex2+ y2=a2.
2) A vector field F is given byF=sin y i+x (1+cos y) j.
Evaluate the line integral ∫CF .dr where C is the circular path given by
x2+ y2=a2.
5.1 STOKE’S THEOREM: (Relation between Line integral and Surface integral) (Without Proof)
Surface integral of the component of curl F along the normal to the surfaceS, taken over the surface S bounded by curve C is equal to the line integral of the vector point function F taken along the closed curveC.
Mathematically
∮F .dr=∬Scurl F .n ds
Where n=cos∝ i+cos β j+cos γ k
is a unit external normal to any surface ds.
OR
The circulation of vector F around a closed curve C is equal to the flux of the curve of the vector through the surface S bounded by the curveC.
∮F .dr=∬ScurlF .n ds=∬
Scurl F . d S
RAI UNIVERSITY, AHMEDABAD 11
Unit-5 VECTOR INTEGRATION
Example 1: Apply Stoke’s theorem to find the value of
∫c( y dx+ zdy+x dz)
Where c is the curve of intersection of x2+ y2+z2=a2 andx+z=a.
Solution: ∫c( y dx+ zdy+x dz)
¿∫c
( y i+z j+x k ) .(i dx+ j dy+ k dz)
¿∫c
( y i+z j+x k ) . d r
¿∬Scurl ( y i+z j+x k ). n ds (By Stoke’s theorem)
¿∬S
(i ∂∂ x + j ∂∂ y
+ k ∂∂ z )× ( y i+z j+x k ) . n ds
¿∬S– (i+ j+ k) . n ds _______ (i)
Where S is the circle formed by the integration of x2+ y2+z2=a2 and
x+z=a.
n=∇∅
|∇∅|
¿(i ∂∂ x
+ j ∂∂ y
+ k ∂∂ z )(x+z−a)
|∇∅|
¿ i+ k√1+1
RAI UNIVERSITY, AHMEDABAD 12
Unit-5 VECTOR INTEGRATION
¿ i√2
+ k√2
Putting the value of n in (i), we have
¿∬S– (i+ j+ k) .( i
√2+ k
√2 )ds
¿∬S
−¿( 1√2
+ 1√2 )ds ¿ [Use r2=R2−p2=a2−
a2
2=a2
2 ] ¿− 2
√2∬Sds=−2
√2π ( a
√2 )2
=−π a2
√2 ______ Ans.
Example 2: Evaluate ∮CF .dr by stoke’s theorem, where
F= y2 i+x2 j−(x+z ) k and C is the boundary of triangle with vertices at (0,0,0 ) ,(1,0,0) and (1,1,0).
Solution: We have, curl F=∇× F
¿| i j k∂∂ x
∂∂ y
∂∂z
y2 x2 −(x+z )|
¿0. i+ j+2(x− y ) k
We observe that z co-ordinate of each vertex of the triangle is zero.
Therefore, the triangle lies in the xy-plane.
∴ n=k
∴ curlF .n=[ j+2 (x− y) k ] . k=2 ( x− y ) .
In the figure, only xy-plane is considered.
The equation of the line OB is y=x
By Stoke’s theorem, we have
RAI UNIVERSITY, AHMEDABAD 13
Unit-5 VECTOR INTEGRATION
∮CF .dr=∬
S(curl F . n)ds
¿ ∫x=0
1
∫y=0
x
2 ( x− y )dxdy
¿2∫0
1 [ x2− x2
2 ]dx ¿2∫
0
1 x2
2dx
¿∫0
1
x2dx
¿ [ x3
3 ]0
1
¿ 13 ________ Ans.
5.2 Exercise:
1) Use the Stoke’s theorem to evaluate ∫C
[ ( x+2 y ) dx+( x−z )dy+( y−z )dz ]
where C is the boundary of the triangle with vertices (2,0,0 ) , (0,3,0 )∧(0,0,6) oriented in the anti-clockwise direction.
2) Apply Stoke’s theorem to calculate ∫c
4 y dx+2 z dy+6 y dz
Where c is the curve of intersection of x2+ y2+z2=6 z and z=x+3
3) Use the Stoke’s theorem to evaluate∫Cy2dx+xy dy+ xzdz, where C is
the bounding curve of the hemispherex2+ y2+z2=1 , z≥0, oriented in the positive direction.
6.1 GAUSS’S THEOREM OF DIVERGENCE: (Without Proof)
RAI UNIVERSITY, AHMEDABAD 14
Unit-5 VECTOR INTEGRATION
The surface integral of the normal component of a vector function F taken around a closed surface S is equal to the integral of the divergence of F taken over the volume V enclosed by the surfaceS.Mathematically
∬SF . n ds=∭
V¿ F dv
Example 1: Evaluate ∬SF . n ds where F=4 xz i− y2 j+ yz k and S is the surface
of the cube bounded byx=0 , x=1 , y=0 , y=1 , z=0 , z=1.Solution: By Gauss’s divergence theorem,
∬SF . n ds=∭
V
(∇ . F )dv
¿∭v
(i ∂∂ x
+ j ∂∂ y
+ k ∂∂ z ) . (4 xz i− y2 j+ yz k )dv
¿∭v
[ ∂∂ x
(4 xz )+ ∂∂ y
(− y2)+ ∂∂z
( yz )]dxdy dz ¿∭
v(4 z−2 y+ y )dx dy dz
¿∭v
(4 z− y )dxdy dz
¿∫0
1
∫0
1
( 4 z2
2− yz )
0
1¿ dx dy¿
¿∫0
1
∫0
1
(2 z2− yz )0
1¿ dxdy ¿
¿∫0
1
∫0
1
(2− y )dx dy
¿∫0
1
(2 y− y2
2 )0
1
dx
¿ 32∫0
1
dx
¿ 32
[ x ]01¿¿
¿ 32(1)
RAI UNIVERSITY, AHMEDABAD 15
Unit-5 VECTOR INTEGRATION
¿ 32 ________ Ans.
Example 2: Evaluate surface integral ∬ F . n ds , where F=(x2+ y2+ z2 ) ( i+ j+ k ) , S is the surface of the tetrahedron x=0 , y=0 , z=0 , x+ y+z=2 and n is the unit normal in the outward direction to the closed surfaceS.
Solution: By gauss’s divergence theorem,
∬SF . n ds=∭
V¿ F . dv
Where S is the surface of tetrahedron x=0 , y=0 , z=0 , x+ y+z=2
¿∭V
(i ∂∂ x
+ j ∂∂ y
+ k ∂∂ z ) . (x2+ y2+z2 ) ( i+ j+k )dv
¿∭V
(2 x+2 y+2 z)dv
¿2∭V
(x+ y+z )dx dydz
¿2∫0
2
dx∫0
2− x
dy ∫0
2− x− y
( x+ y+z )dz
¿2∫0
2
dx∫0
2− x
dy (xz+ yz+ z2
2 )0
2− x− y¿ ¿
¿2∫0
2
dx∫0
2− x
dy [2x−x2−xy+2 y−xy− y2+ (2−x− y )2
2 ] ¿2∫
0
2
dx [2xy−x2 y−x y2+ y2− y3
3−
(2−x− y)3
6 ]0
2− x¿
¿
¿2∫0
2
dx [2x (2−x )−x2 (2−x )−x (2−x )2+(2−x )2−(2−x )3
3+(2−x)3
6 ] ¿2∫
0
2 [4 x−2x2−2 x2+x3−4 x+4 x2− x3+(2−x )2−(2−x )3
3+(2−x)3
6 ]
RAI UNIVERSITY, AHMEDABAD 16
Unit-5 VECTOR INTEGRATION
¿2[2x2− 4 x3
3+ x4
4−2x2+ 4 x3
3− x4
4−
(2−x)3
3+(2−x )4
12−
(2−x)4
24 ]0
2
¿2[−(2−x)3
3+(2−x)4
12−
(2−x )4
24 ]0
2
¿2[ 83−16
12+ 16
24 ]¿4 ________ Ans.
6.2 Exercise:
1) Evaluate ∬SF . n ds where S is the surface of the sphere x2+ y2+z2=16
andF=3 x i+4 y j+5 z k.
2) Find∬SF . n ds, where F=(2 x+3 z ) i−( xz+ y ) j+( y2+2 z) k and S is the
surface of the sphere having centre (3,-1, 2) and radius 3.
3) Use divergence theorem to evaluate∬SA . ds, where A=x3 i+ y3 j+ z3 k
and S is the surface of the spherex2+ y2+z2=a2.
4) Use divergence theorem to show that∬S∇ (x2+ y2+z2 ). ds=6V , where S
is any closed surface enclosing volumeV .
7.1 REFERECE BOOKS:
1) Introduction to Engineering MathematicsBy H. K. DASS. & Dr. RAMA VERMA S. CHAND
2) Higher Engineering MathematicsBy B.V. RAMANAMc Graw Hill Education
3) Higher Engineering MathematicsBy Dr. B.S. GREWAL
RAI UNIVERSITY, AHMEDABAD 17
Unit-5 VECTOR INTEGRATION
KHANNA PUBLISHERS4) http://mecmath.net/calc3book.pdf
RAI UNIVERSITY, AHMEDABAD 18