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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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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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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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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 xg)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
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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.
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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)]
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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.
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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) ̅
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