eed 2008 : electromagnetic theory
DESCRIPTION
EED 2008 : Electromagnetic Theory. Vectors Divergence and Stokes Theorem. Özgür TAMER. Vector integration. Linear integrals Vector area and surface integrals Volume integrals. Line Integral. The line integral is the integral of the tangential component of A along Curve L - PowerPoint PPT PresentationTRANSCRIPT
EED 2008: Electromagnetic Theory
Özgür TAMER
Vectors Divergence and Stokes Theorem
Vector integration Linear integrals Vector area and surface integrals Volume integrals
Line Integral The line integral is the
integral of the tangential component of A along Curve L
Closed contour integral (abca)
Circulation of A around L
b
aL
ldAldA
cos
A is a vector field
L
ldA
Surface Integral (flux) Vector field A containing
the smooth surface S Also called; Flux of A
through S
Closed Surface IntegralNet outward flux of A from S
A is a vector field
SS
n
S
SdAdSaAdSA
cos
S
SdA
Volume Integral Integral of scalar over the volume V
V
V
V
V
V
V
V
V
ddrdrf
dzddf
dxdydzfdVf
sin
2
V
Vector Differential Operator The vector differential operator (gradient
operator), is not a vector in itself, but when it operates on a scalar function, for example, a vector ensues.
zyx dz
d
dy
d
dx
daaa
zyx dz
d
d
d
d
daaa
aaa
dr
d
rd
d
dr
dr sin
Physical meaning of T :
A variable position vector r to describe an isothermal surface :
CzyxT ),,(
0dT
0 dTTdr
Since dr lies on the isothermal plane…
and
Thus, T must be perpendicular to dr.
Since dr lies in any direction on the plane,T must be perpendicular to the tangent plane at r.
dr
T
T is a vector in the direction of the most rapid change of T, and its magnitude is equal to this rate of change.
if A·B = 0The vector A is zeroThe vector B is zero = 90°
Gradient
Gradient1- Definition. (x,y,z) is a differentiable scalar field
2 – Physical meaning: is a vector that represents both the magnitude and the direction of the maximum space rate of increase of Φ
grad
x, y, z ctt
dr88888888888888
zyx dz
d
dy
d
dx
daaa
The operator is of vector form, a scalar product can be obtained as :
z
A
y
A
x
A
AAAdz
d
dy
d
dx
dA
zyx
zzyyxxzyx
)( aaaaaa
Output - input : the net rate of mass flow from unit volume
A is the net flux of A per unit volume at the point considered, countingvectors into the volume as negative, and vectors out of the volume as positive.
zzyyxx BABABABA
Divergence
Ain Aout
0 A
The flux leaving the one end must exceed the flux entering at the other end.The tubular element is “divergent” in the direction of flow.
Therefore, the operator is frequently called the “divergence” :
AA divDivergence of a vector
Divergence
Divergence
y ux zu
vv vdiv = v
x y z
v v8888888888888888888888888888
1 – Definition
2 – Physical meaning
(x, y, z)v88888888888888
is a differentiable vector field
The divergence of A at a given point P is the outward flux per unit volume as the volume shrinks about P.
(x, y, z)V88888888888888
(x dx, y, z)V88888888888888
x x+dx
Divergence
(a) Positive divergence, (b) negative divergence, (c) zero divergence.
Divergence To evaluate the divergence of
a vector field A at point P(x0,y0,x0), we let the point surrounded by a differential volume
After some series expansions we get;
Divergence Cylindrical Coordinate System
Spherical Coordinate System
Divergence Properties of the divergence of a vector field
It produces a scalar field The divergence of a scalar V, div V, makes no
sense
Curl
1 – Definition. The curl of a is an axial (or rotational) vector whose magnitude is the maximum circulation of A per unit area as the area lends 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
2 – Physical meaning: is related to the local rotation of the vectorfield:
curl v88888888888888
v88888888888888
, =2 v ω r curl v ω8888888888888888888888888888888888888888888888888888888888888888888888
is the fluid velocity vectorfield 0curl v88888888888888
0curl v88888888888888
If
What is its physical meaning?
Assume a two-dimensional fluid element
uv
x
y xx
vv
yy
uu
O A
B
Regarded as the angular velocity of OA, direction : kThus, the angular velocity of OA is ; similarily, the angular velocity of OB is
x
vk
y
uk
y
u
x
vk
vuyx
kji
0
0u
Curl
The angular velocity of the fluid element is the average of the two angular velocities :
uv
x
y xx
vv
yy
uu
O A
B
k
y
u
x
v
2
1
y
u
x
vk
vuyx
kji
0
0u
ku 2
This value is called the “vorticity” of the fluid element,which is twice the angular velocity of the fluid element.This is the reason why it is called the “curl” operator.
Curl
Curl
Cartesian Coordinates
Curl Cylindrical Coordinates
Curl Spherical Coordinates
Considering a surface S having element dS and curve C denotes the curve :If there is a vector field A, then the line
integral of A taken round C is equal to the surface integral of × A taken over S :
SSC
dSAdSAdlA )(
Two-dimensional system
C S
xyyx dxdy
y
A
x
AdyAdxA )(
Stokes’ Theorem
Stokes’ Theorem Stokes's theorem states that ihe circulation of
a vector field A around a (closed) pth L is equal to the surface integral of the curl of A over the open surface S bounded by L provided that A and are continuous on S
A
Laplacian1 – Scalar Laplacian. The Laplacian of a scalar field V, written as . is the divergence of the gradient of V.
The Laplacian of a scalar field is scalar
V2
VVLaplacianV 2
V Gradient of a scalar is vectorDivergence of a vector is scalar
Laplacian In cartesian coordinates
In Cylindrical coordinates
In Spherical Coordinates
zyx
zyxzyx
az
Va
y
Va
x
VV
az
Va
y
Va
x
Va
za
ya
xV
2
2
2
2
2
22
2
Laplacian: physical meaning
E
E
v : maximum in E ( (E) > average value in the surrounding)
v : minimum in E ( (E) < average value in the surrounding)
As a second derivative, the one-dimensionalLaplacianoperator is related to minima andmaxima: when the second derivative ispositive (negative), the curvature isconcave (convexe).
In most of situations, the 2-dimensionalLaplacianoperator is also related to localminima and maxima. If vE is positive:
x
(x)
concave
convex
Laplacian
A scalar field V is said to be harmonic in a given region if its Laplacian vanishes in that region.
02 V
Laplacian Laplacian of a vector: is defined as the
gradient of the divergence of A minus the curl of the curl of A;
Only for the cartesian coordinate system;
A2
3. Differential operators
Summary
Operator grad div curl Laplacian
is a vector a scalar a vectora scalar
(resp. a vector)
concernsa scalar field
a vector field
a vector field
a scalar field
(resp. a vector field)
Definition v88888888888888
v88888888888888
2 2 ( v) 88888888888888
resp.
The divergence theorem states that the total outward flux of a vector field A through the closed surface S is the same as the volume integral of the divergence of A
The theorem applies to any volume v bounded by the closed surface S
Gauss’ Divergence Theorem
The tubular element is “divergent” in the direction of flow.
uu div The net rate of mass flow from unit volume
Ain Aout
AA div
We also have : The surface integral of the velocity vector u givesthe net volumetric flow across the surface dSnudSu
dS
udSuThe mass flow rate of a closed surface (volume)
Gauss’ Divergence Theorem
Gauss’ Divergence Theorem
Stokes’ Theorem
SSC
dSAdSAdrA
dS
AdSA
Classification of Vector Fields A vector field is characterized by its
divergence and curl
0 ,0
0 ,0
0 ,0
0 ,0
AA
AA
AA
AA
Solenoidal Vector Field: A vector field A is said to be solenoidal (or divergenceless) if
Such a field has neither source nor sink of flux, flux lines of A entering any closed surface must also leave it.
Classification of Vector Fields
0 A
A vector field A is said to be irrotational (or potential) if
In an irrotational field A, the circulation of A around a closed path is identically zero.
This implies that the line integral of A is independent of the chosen path
An irrotational field is also known as a conservative field
Classification of Vector Fields
0 A
Stokes formula: vector field global circulation
2
C S(C) V dr curl V dS
88888888888888888888888888888888888888888888888888888888
Theorem. If S(C)is any oriented surface delimited by C:
2
2
2
x
yC
S(C)
dS88888888888888
Sketch of proof. Vy
2
2
y y 3
3x x
y 2 3x
V V V(P) V(P) O( )
2 x 2 x
V V V(P) V(P) O( )
2 y 2 y
V V O( )
x y
V dr
V dr
8888888888888888888888888888
8888888888888888888888888888
… and then extend to any surface delimited by C.
Vx
P
Divergence Formula: global conservation laws
xx x
VV (x+dx,y,z).dydz - V (x,y,z).dydz = dxdydz
x
S V(S)div dV V dS V
888888888888888888888888888888888888888888
Theorem. If V(C)is the volume delimited by S
Sketch of proof. Flow through the oriented elementary planes x = ctt and x+dx = ctt:
extended to the vol. of the elementary cube:
V(x, y, z)88888888888888
V(x dx, y, z)88888888888888
x x+dx
-Vx(x,y,z).dydz + Vx(x+dx,y,z).dydz
Other expression:
and then extend this expression to the lateral surface of the cube.
yx zVV V+ dxdydz
x y z