CURL OF A VERTOR FIELD
The net outward flux of a vector field A through a surface
bounding a volume indicates the presence of a source. This source
may be called a flow source and the divergence of A is a measure
of the strength of the flow (scalar) source.
There is another kind of source called vortex (vector) source which
causes a circulation of a vector field around it.
The circulation of vector field A around a closed path C is:
Circulation of A = v∫ A.dl
C
The curl of A , is a 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 to make circulation maximum. That is:
Curl
⎛ A.dl
⎜ v∫
∇XA = LimΔS →0 ⎜ C
A=
⎜ ΔS
⎝
⎞
⎟
⎟ aˆn
⎟
⎠ max
is the surface bounded by C and aˆn is the unit vector normal to
ΔS . aˆ n is determined by the right hand rule.
∇S
∂Az ∂Ay
−
∂y
∂z
∂A ∂A
( curl A ) y = x − z
∂z
∂x
∂A ∂A
( curl A ) z = y − x
∂x
∂y
( curl A ) x =
In Cartesian Coordinates:
aˆ x
∂
∇XA =
∂x
Ax
aˆ y
∂
∂y
Ay
aˆ z
∂
∂z
Az
In Cylindrical Coordinates:
aˆ ρ
ρ aˆφ
aˆ z
1 ∂
ρ ∂ρ
Aρ
∂
∂φ
ρ Aφ
∂
∂z
Az
aˆr
raˆθ
r sin θ aˆφ
∂
1
r 2 sin θ ∂r
Ar
∂
∂θ
rAθ
∂
∂φ
r sin θ Aφ
∇XA =
In Spherical Coordinates:
∇XA =
STOKES’S THEOREM
Consider an open surface S which is bounded by a closed path C.
The vector field A is defined everywhere on S and on C.
v∫ A .d l
C
=
∫ ∇ X A .d s
S
Stokes’s Theorem states that the circulation of a vector field
A around a closed path is equal to the surface integral of the curl
of A over the open surface S bounded by C.
If ∇XA = 0 , vector field
conservative field.
A is said to be irrotational or
If ∇XA = 0 , for a scalar V , ∇X [∇V ] = 0 , since curl of a
gradient is zero.
If ∇XA = 0 , then
v∫ A.dl
=0
C
.
HELMHOLTZ’S THEOREM
A vector field is determined within an additive constant if both its
divergence and its curl are specidied.