Unit 5
Vector Integral Calculus and Applications
Q(1) Evaluate 
where F= cos y.i-x siny j and C is the curve y=
in the xy plane from (1,0) to (0,1)
Ans-
The curve y=
i.e x2+y2 =1. Is a circle with centre at the origin and radius unity.
= 
= 
=
=-1
Q(2). Evaluate 
where 
= (2xy +z2) I +x2j +3xz2 k along the curve x=t, y=t2, z= t3 from (0,0,0) to (1,1,1).
Ans-
F x dr = 
Put x=t, y=t2, z= t3
Dx=dt ,dy=2tdt, dz=3t2dt.
F x dr = 
=(3t4-6t8) dti – ( 6t5+3t8 -3t7) dt j +( 4t4+2t7-t2)dt k
=
t4-6t3)dti –(6t5+3t8-3t7)dt j+(4t4 + 2t7 – t2)dt k
=
=
+
Que(3) Prove that ͞͞͞F = [y2 cos x +z3] i+(2y sin x – 4) j +(3xz2 + 2) k is a conservative field. Find (i) scalar potential for͞͞͞F (ii) the work done in moving an object in this field from (0, 1, -1) to (
/ 2,-1, 2)
Ans-
(a) The fleld is conservative if cur͞͞͞͞͞͞F = 0.
Now, curl͞͞͞F = 
; Cur 
= (0-0) – (3z2 – 3z2) j + (2y cos x- 2y cos x) k = 0
; F is conservative.
(b) Since F is conservative there exists a scalar potential ȸ such that
F = ȸ
(y2 cos x=z3) i + (2y sin x-4) j + (3xz2 + 2) k = 
i + 
j + 
k
= y2 cos x + z3, 
= 2y sin x – 4, 
= 3xz2 + 2
Now, 
= 
dx + 
dy + 
dz
= (y2 cos x + z3) dx +(2y sin x – 4)dy + (3xz2 + 2)dz
= (y2 cos x dx + 2y sin x dy) +(z3dx +3xz2dz) +(- 4 dy) + (2 dz)
=d(y2 sin x + z3x – 4y -2z)
ȸ = y2 sin x +z3x – 4y -2z
(c) now, work done =
.d ͞r
= 
dx + (2y sin x – 4) dy + ( 3xz2 + 2) dz
= 
(y2 sin x + z3x – 4y + 2z) (as shown above)
= [ y2 sin x + z3x – 4y + 2z ]( 
/2, -1, 2)
= [ 1 +8 
+ 4 + 4 ] – { - 4 – 2} =4
+ 15
Q(4) Evaluate 
where 
=yz i+zx j+xy k and C is the position of the curve.
= (a cost)i+(b sint)j+ct k , from y=0 to t=π/4.
Ans-
= (a cost)i+(b sint)j+ct k
The parametric eqn. of the curve are x= a cost, y=b sint, z=ct (i)
=
Putting values of x,y,z from (i),
dx=-a sint
dy=b cost
dz=c dt
=
=
=
= 
Q(5) Find the circulation of 
around the curve C where 
=yi+zj+xk and C is circle 
.
Ans-
Parametric eqn of circle are:
x=a cos
y=a sin
z=0
=xi+yj+zk = a cos
i + b cos
+ 0 k
d
=(-a sin
i + a cos
j)d
Circulation =
=
+zj+xk). d
=
-a sin
i + a cos
j)d
=
= 
Q(6) Verify green’s theorem for 
and C is the triangle having verticles A (0,2 ) , B (2,0 ) , C (4,2 ).
Ans-
By green theorem.
Here ,
(a) Along AB , since the equation of AB is
Putting 
Along BC , since the equation of BC , 
.
Along CA , since the equation of CA, is y = 2 , dy = 0.
(b) 
.
From (1) and (2) , the theorem is verified .
Q(7) Evaluate by Green ‘s theorem 
= - xy (xi –yj) and c is r= a (1+ cos 
)
Ans-
By Green’s Theorem , 
) dx dy
Now, 
.dṝ = 
2yi + xy2 j ) . (d xi + dy j) = 
2y dx + xy2dy )
By comparison p= - x2y, Q = xy2
2, 
= - x2
2 + x2) dx dy
To evaluate the integral , we put x = r cos 
, y = r sin 
for the cardioid r = a ( 1 + cos 
), we take the integral from 
2 .rdr d
= 2 
3dr d
= 
]a ( 1 + cos 
) dθ = 
(1+ cos θ)4dθ
=8a4
= 
a4
Q(8) Verify Green’s Theorem in the plane for 
where C is the closed curve of region bounded by y=x and y=
.
Ans-
Soln. By Green’s Theorem,
A(1,1)
y=x
y=
B
(a) Along 
, y=
and dy=2x dx and x varies from 0 to 1.
Along 
y=x and dy=dx and x varies from 1 to 0.
=
=
= 1
= L.H.S.
RHS=
Q(9) Use divergence theorem to show that 
where Sisanyclosed surface enclosing a volume V.
Ans-
By divergence theorem
Here ,
= 6V
Q(10 )Show that 
Ans-
By divergence theorem, 
..…(1)
Comparing this with the given problem let
Hence, by (1)
………….(2)
Now ,
Hence, from (2),We get,
Q(11) Show that
Ans-
We have Gauss Divergence Theorem
By data, F=
=(n+3)
Q(12) Prove that 
=
Ans-
By Gauss Divergence Theorem,
=
=
=
.[
= 
Q(13) Use stoke’s theorem to evaluate 
Ans-
We have by stoke’s theorem 
Now ,
Q(14):- velocity distribution for a fluid flow is given by
find the equation of
Stream line passing through the point (1,1,2)
Ans:-
the diffential equation of stream line are given by
Putting for u,v,w we get
These are two independent equations, consider
Integrating 
-------1
As stream the passes through (1,1,2)
Put x=1 , y=1 in eq. 1
Consider now, 
Integrating gives , 
Putting 
we get 
Equation 2 &4 together represent curve of intersection of surface
Which represents the stream line
Q(15):- if the velocity of an incompressible fluid at (x,y,z) is given by
Where 
Then determine the stream line at motion
Ans:-
equation of stream lines are given by
Similarly
Integrating , 
Eq 1 & 2 together represents stream line
Que(16)given the velocity fields
What is the acceleration of a particle at (3,0,2) at time t=1
Ans :- we have u=
, V = 
, W =25
The component is given by
Substituting x=3, y=0 & z=2 &t=1
Q(17):- a liquid is in equilibrium under the action of field 
per unit mass given by
Find the pressure at any point on the field
Ans:- let v be the force potential
From the Bernoullis eq
Which gives pressure at any point
