Chapter 1 Vector Analysis

Gradient 梯度 , Divergence 散度 , Rotation, Helmholtz’s Theory

1. Directional Derivative 方向导数 & Gradient

2. Flux通量 & Divergence

3. Circulation 环量 & Curl旋度

4. Solenoidal 无散 & Irrotational 无旋 Fields

5. Green’s Theorems

6. Uniqueness 唯一性 Theorem for Vector Fields

7. Helmholtz’s Theorem

8. Orthogonal 正交 Curvilinear 曲线 Coordinate坐标

1. Directional Derivative & Gradient

The directional derivative of a scalar at a point indicates the

spatial rate of change of the scalar at the point in a certain direction.

l

PP

l lP Δ

)()(lim

0Δ

The directional derivative of scalar

at point P in the direction of l is defined as

Pl

P

l

lΔ P

The gradient is a vector. The magnitude 幅度 of the gradient

of a scalar field at a point is the maximum directional derivative at

the point, and its direction is that in which the directional derivative

will be maximum.

zyx zy

eeexgrad

zyx zyx

eee

grad

In rectangular coordinate system 直角坐标系 , the gradient of a

scalar field can be expressed as

Where “grad” is the observation of the word “gradient”.

In rectangular coordinate system, the operator 算符 is denoted as

Then the grad of scalar field can be denoted as

The surface integral 面积分 of the vector field A evaluated over a

directed surface S is called the flux through the directed surface S,

and it is denoted by scalar , i.e.

2. Flux & Divergence

S

d SA

The flux could be positive, negative, or zero.

The direction of a closed surface is defined as the outward normal

on the closed surface. Hence, if there is a source in a closed surface, the

flux of the vectors must be positive; conversely, if there is a sink, the flux

of the vectors will be negative.

The source a positive source; The sink a negative source.

A source in the closed surface produces a positive integral, while a

sink gives rise to a negative one.

From physics we know that

S

q

0

d

SE

If there is positive electric charge in the closed surface, the flux will

be positive. If the electric charge is negative, the flux will be

negative. In a source-free region where there is no charge, the flux

through any closed surface becomes zero.

The flux 通量 of the vectors through a closed surface can reveal

the properties of the sources and how the sources existed within the

closed surface.

The flux only gives the total source in a closed surface, and it

cannot describe the distribution 分布 of the source. For this reason,

the divergence is required.

VS

V Δ

d limdiv

0Δ

SAA

Where “div” is the observation of the word “divergence, and V is the

volume closed by the closed surface. It shows that the divergence of a

vector field is a scalar field, and it can be considered as the flux through

the surface per unit volume.

In rectangular coordinates, the divergence can be expressed as

z

A

y

A

x

A zyx

Adiv

We introduce the ratio 比率 of the flux of the vector field A at the

point through a closed surface to the volume enclosed by that surface,

and the limit 极限 of this ratio, as the surface area is made to become

vanishingly small at the point, is called the divergence of the vector field

at that point, denoted by divA, given by

Using the operator , the divergence can be written

as AA div

SV

V

d d div SAA

Divergence Theorem

SV

V

d d SAAor

From the point of view of mathematics, the divergence theorem

states that the surface integral 面积分 of a vector function over a

closed surface can be transformed into a volume integral 体积分

involving the divergence of the vector over the volume enclosed by the

same surface. From the point of the view of fields, it gives the

relationship between the fields in a region a 区域 nd the fields on the

boundary 边界 of the region.

The line integral of a vector field A evaluated along a closed curve

is called the circulation of the vector field A around the curve, and it

is denoted by , i.e.

3. Circulation 环量 & Curl 旋度

l

d lA

If the direction of the vector field A is the same as that of the line

element dl everywhere along the curve, then the circulation > 0. If

they are in opposite direction, then < 0 . Hence, the circulation can

provide a description of the rotational property of a vector field.

From physics, we know that the circulation of the magnetic flux

density B around a closed curve l is equal to the product of the

conduction current I enclosed by the closed curve and the permeability

磁导率 in free space, i.e.

where the flowing direction of the current I and the direction of the

directed curve l adhere to the right hand rule. The circulation is

therefore an indication of the intensity of a source.

Il 0

d lB

However, the circulation only stands for the total source, and it

is unable to describe the distribution of the source. Hence, the

rotation is required.

Sl

S Δ

d lim curl max

0Δn

lAeA

Where en the unit vector at the direction about which the circulation of

the vector A will be maximum, and S is the surface closed by the closed

line l.

The magnitude of the curl vector is considered as the maximum

circulation around the closed curve with unit area.

Curl is a vector. If the curl of the vector field A is denoted by

. The direction is that to which the circulation of the vector A will be

maximum, while the magnitude of the curl vector is equal to the

maximum circulation intensity about its direction, i.e.

Acurl

In rectangular coordinates, the curl can be expressed by the

matrix as

zyx

zyx

AAAzyx

eee

A curl

or by using the operator

as AA curl

Stokes’ Theorem

lS

d d) curl( lASA

lS

d d)( lASAor

A surface integral can be transformed into a line integral by using

Stokes’ theorem, and vise versa.

The gradient, the divergence, or the curl is differential operator.

They describe the change of the field about a point, and may be

different at different points.

lS

d d)( lASA

From the point of the view of the field, Stokes’ theorem establishes

the relationship between the field in the region and the field at the

boundary of the region.

They describe the differential properties of the vector field. The

continuity of a function is a necessary condition for its

differentiability. Hence, all of these operators will be untenable where

the function is discontinuous.

The field with null-divergence is called solenoidal field (or called

divergence-free field), and the field with null-curl is called irrotational

field (or called lamellar field).

4. Solenoidal 无散 & Irrotational 无旋Fields

The divergence of the curl of any vector field A must be zero, i.e.

0)( A

which shows that a solenoidal field can be expressed in terms of the

curl of another vector field, or that a curly field must be a solenoidal

field.

0)(

Which shows that an irrotational field can be expressed in terms

of the gradient of another scalar field, or a gradient field must be

an irrotational field.

The curl of the gradient of any scalar field must be zero,

i.e.

5. Green’s Theorems 格林定理 The first scalar Green’s

theorem: S

V

, ne

SVS

nV

2 dd)(

SV

V

2 d)(d)( S

or

where S is the closed surface bounding the volume V, the second order

partial derivatives of two scalar fields and exist in the volume V,

and is the partial derivative of the scalar in the direction of , the

outward normal to the surface S.

n

ne

SV

Snn

V

22 dd)(

SV

V

22 d d)( S

The first vector Green’s theorem:

SV

V

d d])()[( SQPQPQP

where S is the closed surface bounding the volume V, the direction of

the surface element dS is in the outward normal direction, and the

second order partial derivatives of two vector fields P and Q exist in

the volume V.

The second scalar Green’s theorem:

The second vector Green’s theorem:

SV

V

d][d]()([ SPQQPQPPQ

all Green’s theorems give the relationship between the fields in

the volume V and the fields at its boundary S. By using Green’s

theorem, the solution of the fields in a region can be expanded in

terms of the solution of the fields at the boundary of that region.

Green theorem also gives the relationship between two scalar

fields or two vector fields. Consequently, if one field is known, then

another field can be found out based on Green theorems.

6. Uniqueness Theorem for Vector Fields

For a vector field in a region, if its divergence, rotation, and

the tangential 切向 component or the normal 法向 component

at the boundary are given, then the vector field in the region will

be determined uniquely.

The divergence and the rotation of a vector field represent the

sources of the field. Therefore, the above uniqueness theorem shows

that the field in the region V will be determined uniquely by its

source and boundary condition.

The vector field in an unbounded space is uniquely determined

only by its divergence and rotation if

)0( ,11

R

7. Helmholtz’s Theorem

)()()( rArrF

V

Vd)(

π4

1)(

rr

rFr V

)()(

V

d

π4

1

rr

rFrAwhere

A vector field can be expressed in terms of the sum of an

irrotational 无旋 field and a solenoidal 无散 field.

If the vector F(r) is single valued everywhere in an open space,

its derivatives are continuous, and the source is distributed in a

limited region , then the vector field F(r) can be expressed asV

0)( 1

)1

εR

|(| rF

The properties of the divergence and the curl of a vector field are

among the most essential in the study of a vector field.

8. Orthogonal Curvilinear Coordinates 正交曲线坐标系

Rectangular coordinates (x, y, z)

z

x

y

z = z0

x = x0

y = y0P0

ze

xeye

O

Cylindrical coordinates (r, , z)柱坐标系

y

z

x

P0

0

= 0

r = r0

z = z0

re

zee

O

The relationships between the

variables r, , z and the variables

x, y, z are

cosrx

sinry

zz

22 yxr

x

yarctan

zz

and

Spherical coordinates (r, , )球坐标系

x

z

y

= 0

0

0

r = r0

= 0

ere

e

P0

O

r0

The relationships between the

variables r, , and the variables

x, y, z are cossinrx

sinsinry

cosrz

222 zyxr

z

yx 22

arctan

x

yarctan

and

The relationships among the coordinate components of the

vector A in the three coordinate systems are

z

y

x

z

r

A

A

A

A

A

A

100

0cossin

0sincos

z

y

xr

A

A

A

A

A

A

0cossin

sinsincoscoscos

cossinsincossin

z

rr

A

A

A

A

A

A

010

sin0cos

cos0sin

* In rectangular coordinate system, a vector

zyx cba eeeA

where a, b, and c are constants. Is A a constant vector？

where a, b, and c are constants. Is A is a constant vector?

If A is a constant vector, how about a, b, and c?

zr cba eeeA

* In cylindrical coordinate system, a vector

eeeA cba r

* In spherical coordinate system, a vector

Questions