educational document

1. CURL OF A VECTOR
AND STOKES'S THEOREM

2. The curl of A is an axial (or rotational) vector whose
magnitude is the maximum circulation of A per unit area as
the area tends to zero and whose direction is the normal
direction of the area when the area is oriented so as to make
the circulation maximum.
CURL

4. --{1}
CURL OPERATOR

5. where the area ∆S is bounded by the curve L and an is the
unit vector normal to the surface ∆S and is determined using
the right-hand rule.
To obtain an expression for ▼ X A from the definition in
Eqn{1}, consider the differential area in the yz-plane as in
Figure 1. The line integral in Eqn{1} is obtained as
---{2}
Figure 1. Contour used in evaluating
the x-component of ▼ X A at point
P(x0, y0, z0).A
line
integral is
an
integral w
here
the
function
is
evaluated
along
a
predeterm
ined
curve.

6. Expanding the field components in a Taylor series expansion
about the center point P (xo, yo, zo) as in Eqn{3} and
evaluating Eqn{2}. On side ab, dl = dyay and z = zo – dz/2, so
---{3}

7. On side bc, dl = dz az and y = yo + dy/2, so
On side cd, dl = dy ay and z = z0 + dz/2, so
----{4}
----{6}
----{5}

8. On side da, dl = dz az and y = yo - dy/2, so
----{7}
Substituting Eqn{4} till Eqn{7} into Eqn{2} and ∆S = dy dz,
----{8}
----{9}

9. Similarly y- and x-components of the curl of A can be found
----{11}
----{10}

10. The definition of ▼ X A in Eqn{2} is independent of the
coordinate system. In Cartesian coordinates the curl of A can
be found using
----{12}
----{13}

11. By transforming Eqn{13} using point and vector transformation
techniques, curl of A in cylindrical coordinates can be obtained
----{14}
----{15}

12. In spherical coordinates
----{16}
----{17}

13. The curl of a vector field is another vector field
The curl of a scalar field V, ▼ X V, makes no sense
▼ X(A + B) = ▼ X A + ▼ X B
▼ X (A X B) = A(▼ • B) - B(▼ • A) + (B • ▼) A - (A • ▼)B
▼ X (VA) = V ▼ X A + ▼ V X A
Divergence of the curl of a vector field vanishes, that is,
▼ • (▼ X A) = 0
curl of the gradient of a scalar field vanishes, that is,
▼ X ▼ V = 0
PROPERTIES OF THE CURL

14. (a) (b)
Figure 2. Illustration of a curl: (a) curl at P points out
of the page; (b) curl at P is zero.

15. Physical significance of the curl of a vector field is evident
in Eqn{1}
Curl provides maximum value of the circulation of the field
per unit area (or circulation density) and indicates the
direction along which this maximum value occurs.
Curl of a vector field A at a point P may be regarded as a
measure of the circulation or how much the field curls
around P. For example, Figure 2(a) shows that the curl of a
vector field around P is directed out of the page. Figure 2(b)
shows a vector field with zero curl.

16. A liquid is swirling around in a cylindrical container of
radius 2, so that its motion is described by the velocity field
as shown in Figure 3.
Find
where S is the upper surface
of the cylindrical container.
EXAMPLE 1
Figure 3

17. The curl of F is given by
Letting N = k, we have
SOLUTION

19. EXAMPLE 2
Determine curl of the following vector fields:

20. SOLUTION

22. EXAMPLE 3
If A = ρ cos φ aρ + sin φ aφ, evaluate ∫ A • dl around the path
shown in Figure 4. Confirm this using Stokes's theorem.
Figure 4. For Example 3

23. path L has been divided into segments ab, be, cd, and da as in
Figure 3. Along ab, ρ = 2 and dl = ρ dφ aφ. Hence,
Along bc, φ = 30° and dl = dρ aρ. Hence,

24. Along cd, ρ = 5 and dl = ρ dφ aφ. Hence,
Along da, φ = 60° and dl = dρ aρ. Hence,

25. Putting all these together results in

26. Using Stokes's theorem (because L is a closed path)
dS = ρdφdρaz and

28. EXAMPLE 4
For a vector field A, show explicitly that ▼ • ▼ X A = 0; that is,
the divergence of the curl of any vector field is zero.
SOLUTION:
Assuming that A is in Cartesian coordinates.

30. LINE INTEGRAL
• Line integral: Integral of the tangential component of vector
field A along curve L.
– 2 vectors are involve inside the integral - Result from line integral is a scalar
Line integral Definite integral
Diagram
Maths description
Result Area under the curve
A measure of the total
effect of a given field
along a given path
Information
required
1. Vector field
expression A
2. Path expression
1. Function f(x)
2. Integral limits
Integral limits depends on path

31. SURFACE & VOLUME INTEGRAL
• Surface integral: Integral of the normal
component of vector field A along curve L.
– Two vectors involve inside the integral
– Result of surface integral is a scalar
• Volume integral: Integral of a function f i.e. inside
a given volume V.
– Two scalars involve inside the integral
– Result of volume integral is a scalar

32. SURFACE & VOLUME INTEGRAL
Surface integral Volume integral
Diagram
Maths description
Result
A measure of the total
effect of a scalar function
i.e. temperature, inside a
given volume
A measure of the total
flux from vector field
passing through a given
surface
Information
required
1. Vector field
expression A
2. Surface expression
1. Scalar Function ρv
2. Volume expression
Integral limits depends on surface Integral limits depends on volume

33. Figure 3. Determining the sense of
dl and dS involved in Stokes's
theorem.

34. From the definition of the curl of A in Eqn{1}
----{18}
STOKES'S THEOREM
Circulation of a vector field A around a (closed) path L is
equal to the surface integral of the curl of A over the open
surface S bounded by L (Refer Figure 3) provided that A
and ▼ X A are continuous on V.

35. STOKE’S THEOREM : ELABORATED
Theorem:
–The line integral of field A at the boundary of a
closed surface S is the same as the total rotation of
field A at the surface. i.e. Transformation of surface
integral involving curlA to line integral of A
–Equation:
Physical meaning: The total effect of field A along a
closed path is equivalent to summing all the rotational
component of the field inside the surface of
which the path enclose.

36. PROOF OF STOKES’S THEOREM
The surface S is sub-divided into a
large number of cells. If the kth cell
has surface area ∆Sk and is
bounded by path Lk.
.
. . k
k
L
kL L
k k k
A dl
A dl A dl S
S
= = ∆
∆
∫
∑ ∑∫ ∫
Ñ
Ñ Ñ

37. PROOF OF STOKES’S THEOREM
As shown in Figure, there is cancellation on every interior path,
so the sum of the line integrals around Lk's is the same as the line
integral around the bounding curve L.
Therefore, taking the limit of the right-hand side of above Eqn as
∆Sk  0 and incorporating Eqn leads to:A∇×

38. ( ). .
L S
A dl A dS= ∇×∫ ∫Ñ
The direction of dl and dS in eqn must be chosen using the
right-hand rule or right-handed screw rule. Using the right-hand
rule, if the fingers point in the direction of dl, the thumb will
indicate the direction of dS . Whereas the divergence theorem
relates a surface integral to a volume integral,
Stokes's theorem relates a line integral (circulation) to a surface
integral.

39. EXAMPLE 5
Calculate using Stoke’s theorem the circulation of
A = ρ cos Φ aρ + z sin Φ az around the edge L of the wedge
defined by 0 ≤ ρ ≤ 2, 0 ≤ Φ ≤ 60o, z= 0.
( )
( ) ( )
22
60
0
0
. .
cos
sin
. sin cos
2
1
2 1 1
2
o
L S
z
z
S
A dl A dS
z
A a a
dS d d a
A dS d d
ρ
φ
φ
ρ
ρ φ ρ
ρ
ρ φ φ ρ φ
= ∇×
∇× = +
=
∇× = = −
 
= × − + = ÷
 
∫ ∫
∫ ∫∫
Ñ

40. DIFFERENTIAL OPERATORS
Operator grad div curl Laplacian
is
a
vector
a
scalar
a
vector
a scalar
(resp. a vector)
concerns
a
scalar
field
a
vector
field
a
vector
field
a scalar field
(resp. a vector
field)
Definition ∇φ ∇ ×v
ur
∇ × v
ur 2 2
( v)∇ φ ∇
ur
resp.

41. SUMMARY : GRAD, DIV & CURL
Gradient Divergence Curl
Scalar f(x,y) Vector A Vector A
Expression
(Cartesian)
Expression
(Cylindrical)
Expression
(Spherical)
Result Vector Scalar Vector

42. GRAD, DIV & CURL
Gradient Divergence Curl
Physical
meaning
A vector that gives
direction of the
maximum rate of
change of a
quantity i.e. temp
i.e. Flux out
< flux in
i.e. Flux out
> flux in
Incompressible
Flux out = flux in

43. SUMMARY
1. The differential displacements in the Cartesian, cylindrical,
and spherical systems are respectively
dl = dx ax + dy ay + dz az
dl = dρ aρ + ρ dφ aφ + dz az
dl = dr ar + r dθ aθ + r sin θ dφ aφ
Note: dl is always taken to be in the positive direction; the
direction of the displacement is taken care of by the limits of
integration.
2. The differential normal areas in the three systems are
respectively
dS = dy dz ax
dx dz ay
dx dy az

44. dS = ρ dφ dz aρ
dρ dz aφ
ρ dρ dφ az
dS = r2
sin θ dθ dφ ar
r sin θ dr dφ aθ
r dr dθ aφ
Note : dS can be in the positive or negative direction depending
on the surface under consideration.
3. The differential volumes in the three systems are
dv = dx dy dz
dv = ρ dρ dφ dz
dv = r2
sin θ dr dθ dφ

45. 4. The line integral of vector A along a path L is given by
∫L A • dl. If the path is closed, the line integral becomes the
circulation of A around L; that is, ∫L A • dl.
5. The flux or surface integral of a vector A across a surface S is
defined as ∫s A • dS. When the surface S is closed, the surface
integral becomes the net outward flux of A across S; that is,
∫ A • dS.
6. The volume integral of a scalar ρv over a volume v is defined
as ∫v ρv dv.
7. Vector differentiation is performed using the vector differential
operator ▼. The gradient of a scalar field V is denoted by ▼V,
the divergence of a vector field A by ▼ • A, the curl of A by
▼ X A.

46. 8. The divergence theorem, ∫L A • dS = ∫v ▼ • A dv, relates a
surface integral over a closed surface to a volume integral.
9. Stokes's theorem, ∫L A • dl = ∫s(▼ X A) • dS, relates a line
integral over a closed path to a surface integral.

47. NEXT TIME (INSHAALLAH)
(a)Laplacian of a Scalar
(b) Classification of Vector Fields
(c) Examples and practice problems