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