VECTOR CALCULUS AND ITS APPLICATIONS

V. LAKSHMANA GOMATHI NAYAGAM

ASSOCIATE PROFESSOR
DEPARTMENT OF MATHEMATICS
NATIONAL INSTITUTE OF TECHNOLOGY
TIRUCHIRAPPALLI.
E-MAIL: velulakshmanan@nitt.edu
 SCALAR AND VECTOR POINT FUNCTIONS
SCALAR POINT FUNCTION:
If to each point 𝑃 𝑅 of a region 𝑬 in space there corresponds a definite scalar denoted by
𝑓 𝑅 , then 𝑓 𝑅 is called a scalar point function in 𝑬. The region 𝑬 so defined is called a
scalar field.
EXAMPLES:
The temperature at any instant, density of a body and potential due to gravitational matter
are all examples of scalar point functions.
VECTOR POINT FUNCTION:
If to each point 𝑃 𝑅 of a region 𝑬 in space there corresponds a definite vector denoted by
𝐹 𝑅 , then it is called the vector point function in 𝑬. The region 𝑬 so defined is called a
vector field.
The velocity of a moving fluid at any instant, the gravitational intensity of force are examples
of vector point functions.
 VECTOR OPERATOR DEL.
! ! !
The operator on the right side of the equation 𝑑𝐹 = 𝑑𝑥 + 𝑑𝑦 + 𝑑𝑧 𝐹 is in
!" !# !$
! ! !
the form of a scalar product of 𝐼 !"+ 𝐽 !# + 𝐾 !$ and 𝐼𝑑𝑥 + 𝐽𝑑𝑦 + 𝐾𝑑𝑧.
! ! ! ! !
If ∇ be defined by the equation ∇= 𝐼 !" + 𝐽 !# + 𝐾 !$ then 𝑑𝐹 = -!" 𝑑𝑥 + !# 𝑑𝑦 +
!
!$
𝑑𝑧 .may be written as 𝑑𝐹 = ∇ / 𝑑𝑅 𝐹 for when 𝑅 = 𝑥𝐼 + 𝑦𝐽 + 𝑧𝐾, 𝑑𝑅 = 𝐼𝑑𝑥 +
𝐽𝑑𝑦 + 𝐾𝑑𝑧.
DEL APPLIED TO SCALAR POINT FUNCTIONS-GRADIENT:
The vector function ∇𝑓 is defined as the gradient of the scalar point function 𝑓 and is
written as 𝒈𝒓𝒂𝒅 𝒇.
𝝏𝒇 𝝏𝒇 𝝏𝒇
Thus 𝒈𝒓𝒂𝒅 𝒇 = 𝜵𝒇 = 𝑰 + 𝑱 + 𝑲 .
𝝏𝒙 𝝏𝒚 𝝏𝒛
Thus 𝑔𝑟𝑎𝑑 𝑓 is a vector normal to the surface 𝑓 = constant and has a magnitude
equal to the rate of change of 𝑓 along this normal.
 DIRECTIONAL DERIVATIVE
The directional derivative of 𝑓 in the direction of 𝑁 ! is the resolved part of ∇𝑓
in the direction 𝑁′.
Since∇𝑓. 𝑁 ! = ∇𝑓 cos 𝛼 ≤ ∇𝑓
Thus ∇𝑓 gives the maximum rate of change of 𝑓.
Problems:
1) Prove that ∇𝑟 " = 𝑛𝑟 "#$ 𝑅, where 𝑅 = 𝑥𝐼 + 𝑦𝐽 + 𝑧𝐾.
2) Find the values of constants 𝑎, 𝑏, 𝑐 so that the directional derivative of 𝑝 =
𝑎𝑥𝑦 $ + 𝑏𝑦𝑧 + 𝑐𝑧 $ 𝑥 % at (1,2, −1) has a maximum magnitude 64 in the
direction parallel to the 𝑧 −axis.
 Problems:
3) Find the directional derivation of 𝜙 = 𝑥 & + 𝑦 & + 𝑧 & at the point
𝐴 1, −2,1 in the direction 𝐴𝐵 where 𝐵 is (2, 6, −1). Also find the maximum
directional derivative of 𝜙 at (1, −2, 1).
4) In what direction from 3,1, −2 is the directional derivative of 𝜙 =
𝑥 $ 𝑦 $ 𝑧 & maximum? Find also the magnitude of this maximum.
5) Calculate the angle between the normals to the surfaces 𝑥𝑦 = 𝑧 $ at the
points (4,1,2) and (3,3, −3).
6) Find the values of 𝑎 and 𝑏 so that the surface 5𝑥 $ − 2𝑦𝑧 − 9𝑧 = 0 may
cut the surface 𝑎𝑥 $ + 𝑏𝑦 % = 4 orthogonally at (1, −1,2).
 Divergence
The divergence of a continuously differentiable vector point function 𝐹 is
denoted by 𝑑𝑖𝑣 𝐹 and is defined by the equation
𝜕𝐹 𝜕𝐹 𝜕𝐹
𝑑𝑖𝑣 𝐹 = ∇. 𝐹 = 𝐼. + 𝐽. + 𝐾.
𝜕𝑥 𝜕𝑦 𝜕𝑧
If 𝐹 = 𝑓𝐼 + 𝜙𝐽 + 𝜓𝐾 then
𝜕 𝜕 𝜕 𝜕𝑓 𝜕𝜙 𝜕𝜓
𝑑𝑖𝑣 𝐹 = ∇. 𝐹 = 𝐼 +𝐽 +𝐾 . 𝑓𝐼 + 𝜙𝐽 + 𝜓𝐾 = + +
𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑥 𝜕𝑦 𝜕𝑧
 Curl
The curl of a continuously differentiable vector point function 𝐹 is defined by the
!* !* !*
equation 𝑐𝑢𝑟𝑙 𝐹 = ∇×𝐹 = 𝐼× !" + J× !# + 𝐾× !$
! ! !
If 𝐹 = 𝑓𝐼 + 𝜙𝐽 + 𝜓𝐾 then 𝑐𝑢𝑟𝑙 𝐹 = ∇×𝐹 = (𝐼 !" + J !# + !$
)×(𝑓𝐼 + 𝜙𝐽 + 𝜓𝐾)
𝐼 𝐽 𝐾
! ! ! !+ !, !- !+ !, !-
= !" !# !$
=𝐼 !#
− !$
+J !$
− !"
+ 𝐾( !" − !#
).
𝑓 𝜙 𝜓
 Irrotational and solenoidal
Let 𝑉 = 𝑣' 𝐼 + 𝑣( 𝐽 + 𝑣) 𝐾.
𝑑𝑖𝑣 𝑉 gives the rate at which fluid is originating at a point per unit volume.
The 𝑐𝑢𝑟𝑙 of any vector point function gives the measure of the angular velocity
at any point of the vector field.
Irrotational:
Any motion in which the curl of the velocity vector is zero is said to be
irrotational, otherwise rotational.
Solenoidal:
If the flux entering any element of space is the same as that leaving it, i.e.,
𝑑𝑖𝑣 𝑉 = 0 everywhere then such a point function is called a solenoidal vector
function.
 DEL APPLIED TWICE TO POINT FUNCTIONS
$ *!+ *!+ *!+
1) 𝑑𝑖𝑣 𝑔𝑟𝑎𝑑 𝑓 = ∇ 𝑓 = *' ! + *( ! + *) !
2) 𝑐𝑢𝑟𝑙 𝑔𝑟𝑎𝑑 𝑓 = ∇×∇𝑓 = 0
3) 𝑑𝑖𝑣 𝑐𝑢𝑟𝑙 𝐹 = ∇. ∇×𝐹 = 0
4) 𝑐𝑢𝑟𝑙 𝑐𝑢𝑟𝑙 𝐹 = 𝑔𝑟𝑎𝑑 𝑑𝑖𝑣 𝐹 − ∇$ 𝐹, i.e.,∇× ∇×𝐹 = ∇ ∇. 𝐹 − ∇$ 𝐹
5) 𝑔𝑟𝑎𝑑 𝑑𝑖𝑣 𝐹 = 𝑐𝑢𝑟𝑙 𝑐𝑢𝑟𝑙 𝐹 + ∇$ 𝐹, i.e., ∇ ∇. 𝐹 = ∇× ∇×𝐹 + ∇$ 𝐹
 DEL APPLIED TO PRODUCTS OF POINT FUNCTIONS

1) 𝑔𝑟𝑎𝑑 𝑓𝑔 = 𝑓 𝑔𝑟𝑎𝑑 𝑔 + 𝑔 (𝑔𝑟𝑎𝑑 𝑓) i.e., ∇ 𝑓𝑔 = 𝑓∇𝑔 + 𝑔∇𝑓.

2) 𝑑𝑖𝑣 𝑓𝐺 = 𝑔𝑟𝑎𝑑 𝑓 . 𝐺 + 𝑓(𝑑𝑖𝑣 𝐺) i.e., ∇ 𝑓𝐺 = ∇𝑓. 𝐺 + 𝑓∇. 𝐺
3) 𝑐𝑢𝑟𝑙 𝑓𝐺 = 𝑔𝑟𝑎𝑑 𝑓 ×𝐺 + 𝑓(𝑐𝑢𝑟𝑙 𝐺) i.e., ∇× 𝑓𝐺 = ∇𝑓×𝐺 + 𝑓∇×𝐺
4) 𝑔𝑟𝑎𝑑 𝐹. 𝐺 = 𝐹. ∇ 𝐺 + 𝐺. ∇ 𝐹 + 𝐹×𝑐𝑢𝑟𝑙 𝐺 + 𝐺×𝑐𝑢𝑟𝑙 𝐹
i.e., ∇ 𝐹. 𝐺 = 𝐹. ∇ 𝐺 + 𝐺. ∇ 𝐹 + 𝐹× ∇×𝐺 + 𝐺×( ∇×𝐹
5) 𝑑𝑖𝑣 𝐹×𝐺 = 𝐺. 𝑐𝑢𝑟𝑙 𝐹 − 𝐹. 𝑐𝑢𝑟𝑙 𝐺
i.e., ∇. 𝐹×𝐺 = 𝐺. ∇×𝐹 − 𝐹. (∇×𝐺)
6) 𝑐𝑢𝑟𝑙 𝐹×𝐺 = 𝐹 𝑑𝑖𝑣 𝐺 − 𝐺 𝑑𝑖𝑣 𝐹 + 𝐺. ∇ 𝐹 − 𝐹. ∇ 𝐺
i.e., ∇× 𝐹×𝐺 = 𝐹 ∇. 𝐺 − 𝐺 ∇. 𝐹 + 𝐺. ∇ 𝐹 − 𝐹. ∇ 𝐺
 PROBLEMS:
1) Show that ∇$ 𝑟 " = 𝑛 𝑛 + 1 𝑟 "#$
2) Evaluate 𝑑𝑖𝑣 𝐹 and 𝑐𝑢𝑟𝑙 𝐹 at the point (1,2,3) given 𝐹 = 𝑥 $ 𝑦𝑧𝐼 + 𝑥𝑦 $ 𝑧𝐽 +
𝑥𝑦𝑧 $ 𝐾.
$
3) If 𝐹 = 𝑥 + 𝑦 + 1 𝐼 + 𝐽 − 𝑥 + 𝑦 𝐾, show that ∇. 𝑉 = and
' ! ,( ! ,) !
∇×∇= 0.
4) Find the value of 𝑎 if the vector 𝑎𝑥 $ 𝑦 + 𝑦𝑧 𝐼 + 𝑥𝑦 $ − 𝑥𝑧 $ 𝐽 + (2𝑥𝑦𝑧 −
2𝑥 $ 𝑦 $ )𝐾 has zero divergence. Find the 𝑐𝑢𝑟𝑙 of the above vector which has
zero divergence.
5) If 𝐴 and 𝐵 are irrotational, prove that 𝐴×𝐵 is solenoidal.
 PROBLEMS:
- $/ /
6) If 𝑅 = 𝑥𝐼 + 𝑦𝐽 + 𝑧𝐾 and 𝑟 ≠ 0, show that (i) ∇/(. !)=− . " ; ∇. .!
= 1/𝑟 $
(ii)𝑑𝑖𝑣 𝑟 " 𝑅 = 𝑛 + 3 𝑟 " ; 𝑐𝑢𝑟𝑙 𝑟 " 𝑅 = 0
/ $/
(iii) 𝑔𝑟𝑎𝑑 𝑑𝑖𝑣 =− .
. .#
7) If 𝐴 is a constant vector and 𝑅 = 𝑥𝐼 + 𝑦𝐽 + 𝑧𝐾, prove that
(i) 𝑔𝑟𝑎𝑑 𝐴. 𝑅 = 𝐴
(ii) 𝑑𝑖𝑣 𝐴×𝑅 = 0
(iii)𝑐𝑢𝑟𝑙 𝐴×𝑅 = 2𝐴
(iv)𝑐𝑢𝑟𝑙 𝐴. 𝑅 𝑅 = 𝐴×𝑅
8) For a solenoidal vector 𝐹, show that 𝑐𝑢𝑟𝑙 𝑐𝑢𝑟𝑙 𝑐𝑢𝑟𝑙 𝑐𝑢𝑟𝑙 𝐹 = ∇& 𝐹.
9) Find the directional derivative of ∇. (∇𝜙) at the point (1, −2,1) in the
direction of the normal to the surface 𝑥𝑦 $ 𝑧 = 3𝑥 + 𝑧 $ , where 𝜙 = 2𝑥 % 𝑦 $ 𝑧 & .
 LINE INTEGRAL
If 𝐹 𝑅 = 𝐼𝑓 𝑥, 𝑦, 𝑧 + 𝐽𝜙 𝑥, 𝑦, 𝑧 + 𝐾𝜓(𝑥, 𝑦, 𝑧)
and 𝑑𝑅 = 𝐼𝑑𝑥 + 𝐽𝑑𝑦 + 𝐾𝑑𝑧
Then ∫0 𝐹(𝑅) . 𝑑𝑅 = ∫0 (𝑓𝑑𝑥 + 𝜙𝑑𝑦 + 𝜓𝑑𝑧).
Circulation: If 𝐹 represents the velocity of a fluid particle then the line integral
∫0 𝐹. 𝑑𝑅 is called the circulation of 𝐹 around the curve. When the circulation of
𝐹 around every closed curve in a region 𝐸 vanishes, 𝐹 is said to be irrotational in
𝐸.
Work: If 𝐹 represents the force acting on a particle moving along an arc 𝐴𝐵 then
the work done during the small displacement 𝛿𝑅 = 𝐹. 𝛿𝑅.
Thus the total work done by 𝐹 during the displacement from 𝐴 to 𝐵 is given by
2
the line integral ∫1 𝐹. 𝑑𝑅.
 Problems:
1) If 𝐹 = 3𝑥𝑦𝐼 − 𝑦 $ 𝐽, evaluate ∫ 𝐹. 𝑑𝑅, where 𝐶 is the curve in the 𝑥𝑦-plane
𝑦 = 2𝑥 $ from (0,0) to (1,2).
2) Find the work done in moving a particle in the force field 𝐹 = 3𝑥 $ 𝐼 +
2𝑥𝑧 − 𝑦 𝐽 + 𝑧𝐾, along (a) the straight line from (0,0,0) to (2,1,3).
(b) The curve defined by 𝑥 $ = 4𝑦, 3𝑥 % = 8𝑧 from 𝑥 = 0 to 𝑥 = 2.
3) Compute the line integral ∫0 (𝑦 $ 𝑑𝑥 − 𝑥 $ 𝑑𝑦) about the triangle whose
vertices are 1,0 , (0,1) and −1,0 .
4) If 𝐴 = 3𝑥 $ + 6𝑦 𝐼 − 14𝑦𝑧𝐽 + 20𝑥𝑧 $ 𝐾, evaluate ∫ 𝐴. 𝑑𝑅 from (0,0,0) to
(1,1,1) along the path 𝑥 = 𝑡, 𝑦 = 𝑡 $ , 𝑧 = 𝑡 % .
 Problems:
5) Find the total work done by the force 𝐹 = 3𝑥𝑦𝐼 − 𝑦𝐽 + 2𝑧𝑥𝐾 in moving a
particle around the circle 𝑥 $ + 𝑦 $ = 4.
6) Find the total work done in moving a particle in a force field given by 𝐹 =
3𝑥𝑦𝐼 − 5𝑧𝐽 + 10𝑥𝐾 along the curve 𝑥 = 𝑡 $ + 1, 𝑦 = 2𝑡 $ , 𝑧 = 𝑡 % from 𝑡 = 1 to
𝑡 = 2.
7) If 𝐹 = 2𝑦𝐼 − 𝑧𝐽 + 𝑥𝐾, evaluate ∫0 𝐹×𝑑𝑅 along the curve 𝑥 = cos 𝑡 , 𝑦 =
sin 𝑡, 𝑧 = 2 cos 𝑡 from 𝑡 = 0 to 𝑡 = 𝜋/2.
 SURFACE INTEGRAL
The surface integral ∫3 𝐹. 𝑑𝑆 or ∫3 𝐹. 𝑁𝑑𝑠 where 𝑁 is a unit outward
normal at 𝑃 to 𝑆.
Flux across a surface:
If 𝐹 represent the velocity of a fluid particle then the total outward flux of 𝐹
across a closed surface 𝑆 is the surface integral ∫3 𝐹. 𝑑𝑆.
 Problems:
1) Evaluate ∫3 𝐹. 𝑁𝑑𝑠 where 𝐹 = 2𝑥 $ 𝑦𝐼 − 𝑦 $ 𝐽 + 4𝑥𝑧 $ 𝐾 and 𝑆 is the
closed surface of the region in the first octant bounded by the cylinder
𝑦 $ + 𝑧 $ = 9 and the planes 𝑥 = 0, 𝑥 = 2, 𝑦 = 0 and 𝑧 = 0.
2) Evaluate ∫3 𝐹. 𝑑𝑆 where 𝐹 = 𝑥𝐼 + 𝑧 $ − 𝑧𝑥 𝐽 − 𝑥𝑦𝐾 and 𝑆 is the
triangular surface with vertices (2,0,0), (0,2,0) and (0,0,4).
3) If 𝐹 = 2𝑦𝐼 − 3𝐽 + 𝑥 $ 𝐾 and 𝑆 is the surface of the parabolic cylinder
𝑦 $ = 8𝑥 in the first octant bounded by the planes 𝑦 = 4 and 𝑧 = 6,
show that ∫3 𝐹. 𝑁 𝑑𝑠 = 132.
 GREEN’S THEOREM IN THE PLANE
If 𝜙 𝑥, 𝑦 , 𝜓 𝑥, 𝑦 , 𝜙( and 𝜓' be continuous in a region 𝐸 of the 𝑥𝑦-plane
bounded by a closed curve 𝐶, then
𝜕𝜓 𝜕𝜙
e 𝜙𝑑𝑥 + 𝜓𝑑𝑦 = f ( − ) 𝑑𝑥𝑑𝑦
0 4 𝜕𝑥 𝜕𝑦
Problem:
1) Verify Green’s theorem for ∫0 [ 𝑥𝑦 + 𝑦 $ 𝑑𝑥 + 𝑥 $ 𝑑𝑦], where 𝐶 is bounded
by 𝑦 = 𝑥 and 𝑦 = 𝑥 $ .
 STOKE’S THEOREM
If 𝑆 be an open surface bounded by a closed curve 𝐶 and 𝐹 = 𝑓- 𝐼 + 𝑓$ 𝐽 + 𝑓% 𝐾
be any continuously differentiable vector point function, then
∫0 𝐹. 𝑑𝑅 = ∫3 𝑐𝑢𝑟𝑙 𝐹. 𝑁 𝑑𝑠 where 𝑁 = cos 𝛼𝐼 + cos 𝛽𝐽 + cos 𝛾𝐾 is a unit
external normal at any point of 𝑆.
 Problems:
1) Verify Stoke’s theorem for 𝐹 = 𝑥 $ + 𝑦 $ 𝐼 − 2𝑥𝑦𝐽 taken around the
rectangle bounded by the lines 𝑥 = ±𝑎, 𝑦 = 0, 𝑦 = 𝑏.
2) Using Stoke’s theorem evaluate ∫0 [ 𝑥 + 𝑦 𝑑𝑥 + 2𝑥 − 𝑧 𝑑𝑦 + 𝑦 + 𝑧 𝑑𝑧]
where 𝐶 is the boundary of the triangle with vertices 2,0,0 , (0,3,0) and
(0,0,6).
3) Verify Stoke’s theorem for 𝐹 = 𝑦 − 𝑧 + 2 𝐼 + 𝑦𝑧 + 4 𝐽 − 𝑥𝑧𝐾 where 𝑆 is
the surface of the cube 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 = 2, 𝑦 = 2, 𝑧 = 2 above the 𝑥𝑦-
plane.
4) If 𝑆 be the surface of the sphere 𝑥 $ + 𝑦 $ + 𝑧 $ = 1, prove that
∫3 𝑐𝑢𝑟𝑙 𝐹. 𝑑𝑆 = 0.
5) Use Stoke’s theorem to evaluate ∇×𝐹 . 𝑁 𝑑𝑠 where 𝐹 = 𝑦𝐼 + 𝑥 − 2𝑥𝑧 𝐽 −
𝑥𝑦𝐾 and 𝑆 is the surface of the sphere 𝑥 $ + 𝑦 $ + 𝑧 $ = 𝑎$ above the 𝑥𝑦-plane.
 GAUSS DIVERGENCE THEOREM
If 𝐹 is a continuously differentiable vector function in the region 𝐸
bounded by the closed surface 𝑆, then ∫3 𝐹. 𝑁𝑑𝑠 = ∫4 𝑑𝑖𝑣 𝐹𝑑𝑣 where
𝑁 is the unit external normal vector.
Problems:
1) Verify Divergence theorem for 𝐹 = 𝑥 $ − 𝑦𝑧 𝐼 + 𝑦 $ − 𝑧𝑥 𝐽 +
𝑧 $ − 𝑥𝑦 𝐾 taken over the rectangular parallelopiped 0 ≤ 𝑥 ≤ 𝑎, 0 ≤
𝑦 ≤ 𝑏, 0 ≤ 𝑧 ≤ 𝑐.
2) Evaluate ∫3 𝐹. 𝑑𝑠 where 𝐹 = 4𝑥𝐼 − 2𝑦 $ 𝐽 + 𝑧 $ 𝐾 and 𝑆 is the surface
bounding the region 𝑥 $ + 𝑦 $ = 4, 𝑧 = 0 and 𝑧 = 3.
 Problems:
3) Using divergence theorem, prove that
(i) ∫3 𝑅. 𝑑𝑆 = 3𝑉
(ii) ∫3 ∇𝑟 $ . 𝑑𝑆 = 6𝑉
Where 𝑆 is any closed surface enclosing a volume 𝑉 and 𝑟 $ = 𝑥 $ + 𝑦 $ + 𝑧 $ .
4) Using divergence theorem, evaluate ∫3 𝑅. 𝑁𝑑𝑠 where 𝑆 is the surface of the
sphere 𝑥 $ + 𝑦 $ + 𝑧 $ = 9.
5) Evaluate ∬(𝑥𝑑𝑦𝑑𝑧 + 𝑦𝑑𝑧𝑑𝑥 + 𝑧𝑑𝑥𝑑𝑦) over the surface of a sphere of
radius 𝑎.
6) Evaluate ∫3 𝐹. 𝑑𝑆 where 𝐹 = 𝑦 $ 𝑧 $ 𝐼 + 𝑧 $ 𝑥 $ 𝐽 + 𝑥 $ 𝑦 $ 𝐾 and 𝑆 is the upper
part of the sphere 𝑥 $ + 𝑦 $ + 𝑧 $ = 𝑎$ above 𝑋𝑂𝑌 plane.