BLACKSHADE

http://blackshadejbrec.blogspot.com UNIT-V VECTOR CALCULUS SYLLABUS Vector Calculus: Scalar point function and vector point function, Gradient- Divergence- Curl and their related properties, - Laplacian operator, Line integral – work done – Surface integrals -Volume integral. Green’s Theorem,Stoke’s theorem and Gauss’s Divergence Theorems (Statement & their Verification). Solenoidal and irrotational vectors, Finding Potential function.

SYNOPSIS VECTOR: A quantity which is completely specified by its magnitude as well as direction is called a ‘Vector”. Ex: - Velocity, Force.

FORMULAE: ___

____

____

1. Position Vector = AB  OB  OA 



2. If a  a1i  a2 j  a3 k then a  a12  a22  a32  



3. If a is any Vector then unit vector a 

a 

a 







 



 

4. Dot product a . b  a b cos  where    a .b  i.i  j. j  k .k  1 and i. j  j.k  k .i  0 

5. Cross product









a b  a b sin  n i  i  j  j  k  k  0, i  j  k , j  k  i, k  i  j j  i   k , k  j  i , i  k   j





a  a1i  a2 j  a3 k and b  b1i  b2 j  b3k 6. If

then

 



i

j

k

a . b  a1b1  a2 b2  a3b3

a b  a1 b1

a2 b2

a3 b3



Scalar Point Function: Let ‘S’ be a domain in space. If to each point PES there corresponds a unique real number (Scalar)   P  , then  is called a scalar point function and S is called ‘Scalar Field’.

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http://blackshadejbrec.blogspot.com Vector Point Function:



Let ‘S’ be a domain in space. If to each point PES, there corresponds a unique vector f  p  , then f is called a vector point function and S is called vector field.

Vector Differential Operator: The Vector differential Operator  (read as Del) is defined as

i

     j  k  i Where I, j, k are unit vectors along x, y, z axes. x y z x

Gradient of a Scalar Point Function: Let  (x, y, z) be a scalar point function defining in a scalar field, then the vector i

   j k x y z

is called gradient of  or . If is denoted by grad 

 grnd    i

or 

    j k  i x y z x

NOTE: 1.  at any point is a vector normal to the surface  (x,y,z)= c through that point. Where C is a constant

   

2. If  is the angle b/w two surfaces  and  then cos  

3. The directional derivative of a scalar point function  at a point P(x,y,z) is the direction of a unit vector 





e is grand  , e or  e 4. Max . Value of directional derivative of  = grand   5. Unit normal to the surface  (x,y,z) is defined as

 

Divergence of A Vector Point Function: 

Let

f  f1i  f 2 j  f 3k then 



div f   f  



f be any vector point function then





f x

f y

i.  j. k.



f z is called





divergence of vector f . 



It is denoted by div f or  f 











 div f   f  i. xf  j. yf  k . zf   i. xf

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BLACKSHADE

http://blackshadejbrec.blogspot.com 

If f  f1i  f 2 j  f3 k 

then





 f  i x  j y  k z .  f1i  f2 j  f3 k 



Div f =



f1 x



f 2 y



f3 z



Solenoidal Vector: -



If div f = 0 then the vector f is called solenoid vector.

Laplacian Operator: -   2  Let  be a scalar point function with variables x,y,z. Then grad  =  =



 i x

(vector)

Now div (grad  ) = . 

             i   i  i    i.   i  x  x  x  x  x    2  2  2  2   (i, i ) 2  2  2  2   2 x x y z 2  div (grad  ) =  

2 2 2   is called Laplacian operator. x 2 y 2 z 2 Curl of a Vector Point Function: 2 The operator  









 f f f Let f be any vector point function then i  is called curl of a vector f ,  j k x y z 







f f f f  curl f    f  i   j k   ii  x y z x 





If f  f1i  f 2 j  f3 k then





Curl f    f 

i

j

k

 x f1

 y f2

 z f3 

Irrotational Vector : - (or) Conservative; 







If curl f  0 then the vector f is called 



irrotational vector. NOTE: - If Curl f  0 the is a scalar potential function of such that f   0

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http://blackshadejbrec.blogspot.com OPERATORS 1. Vector differential Operator 

i

    j k x y z 

2. Scalar differential Operator a .           a    a .i    a .i    a .k    x   y   z 

3. Vector differential operator a      a    a i    a   x 

     j    a k   y   z 4. Scalar differential operator       i.  j.  k. x y z 5. Vector differential operator x    x  i   j   k  x y z 2 6. Laplacian operator  2 2 2 2  2  2  2 x y z NOTE : 1. If  2  0 then  is said to satisfy Laplacian equation and  is called Harmonic function.             a  b c    a . c  b  a . b  c                      3.  a b   c   a . c  b   b . c  a       Vector Identities

2.





If  is differentiable scalar function and a . b are two differentiable vector function.then.       a     a     a  . a    a       = grad  a   div a (or)       1. div 



 



 





 



 

 



 

2. Curl   a   grand  a   curl a ; x   a       a      a 

 



   

   

 













3. grand  a . b    b .  a  a .  b b curl a a curl b

 



 





 



 

(2008)



 



(2008)

 



 



 

4. div  a b   b .curl a  a .curl b(or )  a b   b    a   a    b 

(2007)

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http://blackshadejbrec.blogspot.com Proof: - div

         a b        b a   a b     a b    i.  a b    i   x x  x                     a  b  a  a            i  b   i a  i b  i  a  a b    b a   x   x   x   x                             i   a  b    i   b  . a     a  b     b  a  b .curl a  a curl b  x   x                     5. curl  a b   a div b  b div a   b .  a   a .  b                         x  a b   a  . b   b  . a    b .  a   a    b           

6. Curl grad   0 

NOTE: - Curl grad   0 

7. div curl f  0

(2004,2007) 

NOTE: -  div curl f  0 

 Curl f is always solenoidal.  



 

 





 

8.      a     . a    2 a

Vector Integration Open Curve and Closed Curve: Let C be a curve with initial point A and terminal point B. If the initial point and terminal point and terminal point of a curve are coincide (i.e. A=B) then the curve C is called closed curve. If the initial point and terminal point of a curve are not coincide (i.e A  B) then the curve C is called Open curve. Simple Curve: - A closed curve which does not intersect anywhere is called simple curve Circulation: - An integral along the simple curve is called circulation. It is denoted by fc. 



Vector Integration: - If f (t )and g (t ) are two vector point function of a scalar variable ‘t’ such that

  d    g ( t )  f ( t ), then g ( t ) is called an integral of f (t ) w.r.t ‘t’ and is written as   dt  



 f (t )dt 

 (t )





g  c, where c constant vector this is called indefinite integral.

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http://blackshadejbrec.blogspot.com 

The definite integral of f (t ) b/w the limits t=a and t = b written as b 

 f (t ) dt  g (b)  g (a) a

b 



If f (t )  f1i  f 2 j  f 2 k then

 a

b

b

b

f (t ) dt  i  f1dt  j  frdt  k  f3 dt a

a

a

Line Integral Any integral which can be evaluated along a curve is called line integral. 







 F .d r where r  xi  yj  zk

Line integral of F along curve c is denoted by

c





If F  F1i  F2 j  F3 k

then



 F .d r  F dx  F dy  F dz 1

c

2

3

c

B 



Work done by Forve vector F displacement from A to B is given by the line integral



 F .d r A

SURFACE INTEGRALS Any integral which can be evaluated over a surface is called surface 



Surface integral of F over the surface S is denoted by

integral .



 F . n ds s



Where n is unit normal vector to the surface S.

Volume Integrals Any integral which can be evaluated over a volume is called volume integral

Flux: - An integration along the closed surface is called flux. 



NOTE: - If F represents the velocity of a fluid particle then the total outward flux of F across a 

closed surface s is the surface integral







 F .d s When d s  n ds c



1. If R1 is projection of S on XY-Plane then



c

2. If Rr is projection of S on Y 2 -plane then

R1







dxdy n .k



 F . n ds   F . n s

R2



3. If R3 is projection of S on Zx-Planw then

 

 F . n .ds   F n







 F . n ds   F . n s

R3

dydz n .i

dxdz n. j

Vector Integral Theorems Green’s Theorem:If R is a closed region in xy plane bounded by a simple closed curve C and M,N are continuous and differentiable scalar functions of xy in R, then

 N

 Mdx  Ndy    x  c

R

M y

 dxdy 

.

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http://blackshadejbrec.blogspot.com Gauss Divergence Theorem: 

Let S be a closed surface enclosing a volume ‘V’. If F is a continuous differentiable vector point function, then 







 div F dv   F . n ds v

Where n is the outward unit normal vector drawn to the surface S.

s

Cartesian Form: 



Let F  F1i  F2 j  F3 k & n  cos  i  cos  j  cos rk then

 

F1 x



 Fy2  Fz3 dxdydz .

r

  F cos   F 1



cos   F3 cos r ds   F1dydz  F2 dzdx  F3dxdy

2

s

s

dxdy  ds   cos ds  dxdy n.k Stoke’s Theorem: 

Let s be a surface bounded by a closed, non-intersecting curve ‘c’. If F is any continuous differentiable vector point function, them 







 F . dr   curl F . n ds c

s



Where n is the outward unit normal vector drawn to the surface S.

Short Answer Questions 1. Prove that (rn)= nrn-2 r . 2. Show that [f(r)] =

f i (r ) r where r = xi  yj  zk . r

3. Find the directional derivative of f = xy+yz+zx in the direction of vector i  2 j  2k at the point (1,2,0). 4. Find the greatest value of the directional derivative of the function f = x2 yz3 at (2,1,-1). 5. Find a unit normal vector to the given surface x2 y+2xz = 4 at the point (2,-2,3). 6. Evaluate the angle between the normal to the surface xy= z2 at the points (4,1,2) and (3,3,3). 7. If a is constant vector then prove that grad ( a . r )= a 8. If  = yzi  zxj  xyk , find . 9. If f = xy 2 i  2 x 2 yzj  3 yz 2 k find div f at(1, -1, 1). 10. Find div f when grad(x3+y3+z3-3xyz) 410

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http://blackshadejbrec.blogspot.com 11. If f = ( x  3 y )i  ( y  2 z ) j  ( x  pz )k is solenoidal, find P. r 12. Evaluate .  3  where r  xi  yj  zk and r  r . r 

13. Find div r where r = xi  yj  zk 14. If f = xy 2 i  2 x 2 yz j  3 yz 2 k find curl f at the point (1,-1,1). 15. Find curl f where f = grad(x3+y3+z3-3xyz) 16. Prove that curl r = 0 17. Prove that f = ( y  z )i  ( z  x) j  ( x  y ) k is irrotational. 18. If f = x 2 yi  2 zx j  2 yz k find (i) curl f (ii) curl curl f . 19. If  = yzi  zxj  xyk , find . 20. Prove that curl grad  = 0. 21. Prove that div curl f  0 22. Prove that (f xg)is solenoidal 23. State the Greens theorem 24. State the Gauss Divergence theorem 25. State the Stokes Theorem

Long Answer Questions 1. If a=x+y+z, b= x2+y2+z2 , c = xy+yz+zx, then prove that [grad a, grad b, grad c] = 0. 2. Find the directional derivative of the function xy2+yz2+zx2 along the tangent to the curve x =t, y = t2, z = t3 at the point (1,1,1). 3. Find the directional derivative of the function f = x2-y2+2z2 at the point P =(1,2,3) in the direction of the line PQ where Q = (5,0,4). 4. Find the directional derivative of xyz2+xz at (1, 1 ,1) in a direction of the normal to the surface 3xy2+y= z at (0,1,1). 5. Find the values of a and b so that the surfaces ax2-byz = (a+2)x and 4x2 y+z3= 4 may intersect orthogonally at the point (1, -1,2). (or) 411

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http://blackshadejbrec.blogspot.com Find the constants a and b so that surface ax2-byz=(a+2)x will orthogonal to 4x2 y+z3=4 at the point (1,-1,2). 6.Find the angle of intersection of the spheres x2+y2+z2 =29 and x2+y2+z2 +4x-6y-8z-47 =0 at the point (4,-3,2). 7. Find the angle between the surfaces x2+y2+z2 =9, and z = x2+y2- 3 at point (2,-1,2). 8.Prove that if r is the position vector of an point in space, then rn r is Irrotational. (or) Show that curl

9.Show that the vector ( x 2  yz )i  ( y 2  zx ) j  ( z 2  xy ) k is irrotational and find its scalar potential 10. Find constants a,b and c if the vector f = ( 2 x  3 y  az )i  (bx  2 y  3 z ) j  ( 2 x  cy  3 z ) k is Irrotational 11.If f(r) is differentiable, show that curl { r f(r)} = 0 where r = xi  yj  zk . 12.Find constants a,b,c so that the vector A = ( x  2 y  az )i  (bx  3 y  z ) j  ( 4 x  cy  2 z ) k is Irrotational. Also find  such that A = .

13.If  is a constant vector, evaluate curl V where V = x r . 14.If f = ex+y+z (i  j  k ) find curl f . 15. Prove that curl ( a  r )=2 a where a is a constant vector. 16.Prove that div.(grad rm)= m(m+1)rm-2 (or) 2(rm) = m(m+1)rm-2 (or) 2(rn) = n(n+1)rn-2 17. Show that 2[f(r)]=

d 2 f 2 df 2   f 11 (r )  f 1 ( r ) where r = r . 2 r dr r dr

18.If  satisfies Laplacian equation, show that  is both solenoidal and irrotational. 19.Show that (i) ( a .)= a . (ii) ( a .) r = a . (iii) ( f x). r =0

(iv). ( f x)x r =  2 f

20. If f= (x2+y2+z2)-n then find div grad f and determine n if div grad f= 0. 21 If a is a differentiable function and  is a differentiable scalar function, then prove that div( a )= (grad ). a + div a or .( a )= (). a +(. a ) 22. Prove that curl ( a )= (grad )x a + curl a

412

BLACKSHADE

http://blackshadejbrec.blogspot.com 23.Prove that grad ( a . b )= (b .) a  ( a .)b  b  curl a  a  curl b 24.Prove that div ( a  b ) = b . curl a  a . curl b 25.Prove that curl ( a  b )  a div b  bdiv a  (b . )a  (a .)b 26.If f and g are two scalar point functions, prove that div(f g)= f 2g+f. g 27. Prove that x(x a )= (. a )-2 a .  r 2 28. Prove that  .   3 r .  r r

29.Find (Ax), if A = yz2 i - 3xz2 j +2xyz k and  = xyz. 









30.If F (x2-27) i -6yz j +8xz2 k , evaluate

d r from the point (0,0,0) to the point (1,1,1)

along the Straight line from (0,0,0) to (1,0,0), (1,0,0) to (1,1,0) and (1,1,0) to (1,1,1). 







31.If F =(5xy-6x2) i +(2y-4x) j , evaluate  F . d r along the curve C in xy-plane y=x3 from C

(1,1) to (2,8). 

32. Find the work done by the force F = z i + x j + y k , when it moves a particle along the arc 



of the curve r = cost i + sint j -t k from t = 0 to t = 2 





33.If F =3xy i-5z j +10x k evaluate



2

2

3

 F .d r along the curve x=t +1,y=2 t , z = t

from t = 1 to

C

t= 2. 





34. If F =y i+z j +x k , find the circulation of F round the curve c where c is the circle x2 +y2 =1, z=0. 

35. If   x 2 yz 3 , evaluate d r along the curve x= t, y =2t, z=3t from t = 0 to t=1. c 







36. Find the work done by the force F  x 2  yz i  ( y 2  zx ) j  ( z 2  xy ) k in taking particle from (1,1,1) to (3,-5,7). 

37. Find the work done by the force F   2 y  3 i  ( zx) j  ( yz  x)k when it moves a particle from the point (0,0,0) to (2,1,1) along the curve x = 2t2, y = t, z=t3 38. Evaluate  F.ndS where F = zi + xj  3y2zk and S is the surface x2 + y2 = 16 included in the first octant between z = 0 and z = 5.

413

BLACKSHADE

http://blackshadejbrec.blogspot.com 39.If F = zi + xj  3y2zk, evaluate

 F.ndS where S is the surface of the cube bounded by

S

x = 0, x = a, y = 0, y= a, z = 0, z = a. 40. Verify Gauss Divergence theorem for

taken over the

surface of the cube bounded by the planes x = y = z = a and coordinate planes. 41.Apply divergence theorem to evaluate

  ( x  z)dydz  ( y  z)dzdx  ( x  y)dxdy

S is the

s

surface of the sphere x2+y2+z2=4 42.Use divergence theorem to evaluate

  F .d S where

F =x3i+y3 j+z3k and S is the surface of

s

the sphere x2+y2+z2 = r2 43. Use divergence theorem to evaluate

and S is

where

the surface bounded by the region x2+y2=4, z=0 and z=3. 44.Verify divergence theorem for

over the surface S of the solid cut off

by the plane x+y+z=a in the first octant. 45.Verify divergence theorem for 2x2 y i -y2 j +4xz2 k taken over the region of first octant of the cylinder y2+z2=9 and x=2. (or) Evaluate

2

2

  F .ndS , where F =2x y i -y

j +4xz2 k and S is the closed surface of the region in

s

the first octant bounded by the cylinder y2+z2 = 9 and the planes x=0, x=2, y=0, z=0 46. Use Divergence theorem to evaluate   xi  y j  z 2 k .nds. Where S is the surface



2

2



2

bounded by the cone x +y =z in the plane z = 4. S is 47.Use Gauss Divergence theorem to evaluate 2 2 2 the closed surface bounded by the xy-plane and the upper half of the sphere x +y +z =a2 above this plane. taken over the cube bounded 48. Verify Gauss divergence theorem for

by x = 0, x = a, y= 0, y = a, z = 0, z = a. 49. Find by

.

+

where

=2

+ 4xz and S is the region in the first octant bounded

=9 and x=0,x=2.

50. Find

Where S is the region bounded by

+

=4, z=0

and z=3. 51. Verify divergence theorem for F=6z + (2x+y)

the surface of the cylinder

+

-x , taken over the region bounded by

=9 included in z=0, z=8, x=0 and y=0. [JNTU 2007 S(Set No.2)]

414

BLACKSHADE

http://blackshadejbrec.blogspot.com where C is the

52.Verify Green’s theorem in plane for

region bounded by y=

.

and y=

53. Evaluate by Green’s theorem

where C is the triangle

enclosed by the lines y=0, x= ,

54.Evaluate by Green’s theorem for

where C is the

,

rectangle with vertices

55.A Vector field is given by F  (sin y)i  x(1  cos y ) j Evaluate the line integral over the circular path

, z=0

+

(i) Directly (ii) By using Green’s theorem

and hence

56. Show that area bounded by a simple closed curve C is given by

find the area of 57.Verify Green’s theorem for

where C is bounded by

y=x and y=

58.Verify Green’s theorem for

where c is the region

bounded by x=0, y=0 and x+y=1. 59.Apply Green’s theorem to evaluate the boundary of the area enclosed by the x-axis and upper half of the circle

60. Verify Stokes theorem for the sphere

over the upper half surface of bounded by the projection of the xy-plane.

61. Apply Stokes theorem, to evaluate

 ( ydx  zdy  xdz ) where c is the curve of c

intersection of the sphere 62.Apply the Stoke’s theorem and show that

and x+z=a.

is any vector and

S= 63.Evaluate by Stokes theorem

where C is the

boundary of the triangle with vertices (0,0,0), (1,0,0) and (1,1,0). 64.Use Stoke’s theorem to evaluate

over the surface of the paraboloid

z  x 2  y 2  1, z  0 where

65.Verify Stoke’s theorem for the lines x= 66.Verify Stoke’s theorem for

taken round the rectangle bounded by where S is the surface of

the cube x =0, y=0, z=0, x=2, y=2,z=2 above the xy plane.

415

BLACKSHADE

http://blackshadejbrec.blogspot.com and surface is the part of the sphere

67.Verify the Stoke’s theorem for





68.Verify Stoke’s theorem for F  x2  y 2 i  2 xy j over the box bounded by the planes x=0,x=a,y=0,y=b. 69.Verify Stoke’s theorem for =

x=

– 2xy taken round the rectangle bounded by

, y=0,y=a.

70.Using Stroke’s theorem evaluate the integral

where

=2

-(2x+z

+3

C is the boundary of the triangle whose vertices are (0,0,0),(2,0,0),(2,2,0).

Objective Questions 1. If ∅ =

+

+

a). 6 + 6 + 6 2. If ∅ =

+

+

̅+

̅+

then

a)0 ̅+

̅+

a). 3 ̅

∅) = +

+

[

c).

+

[ +

b).0

[ c). 3

6.If ̅ =

b) ̅+

̅+

̅=

[ c). 3 ̅

c) -2

c)

]

d) 3 [

]

[

]

d)-

then ∇ ̅ = b) ̅

]

d) none

is a constant vector then curl( ̅ ) =

a)2

]

d).none

̅=

then

]

d) none

∅=

then

b).0

4. If ̅ =

̅

−3

( c).

b). 6 + 6 + 6

3. If ̅ =

a)

then

b). 0

a). 0

5.If

−3

̅

d) ̅

416

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