If f is a scalar function, then 1 2 3, ,ξ ξ ξ

4-1 gradient of a scalar

But grad ; hence

An alternative way to conclude that is a vector is to note

that is a scalar for all recall that is a vector ,and

invoke the appropriate quotient law.

j( )

ji j i j if f x f x

x x

e

ξ ξ ξ

j jff fx

e

( )ii

ff

ε

ξ

if

j

jf

df d

jd

4 、 tensor calculus

is the convariant component of if

ξthi f

iif

f

ε

jd

4-2 Derivative of a vector ; christoffel` symbol; covariant derivative

Consider the partial derivative of a vector F. with F = ,

we have

write

the contravriant component of the derivative with respect to of the base vector. Note that

i iij j j

iFF

εFε

ξ ξ ξ

j

Fξ

kiij kj

εε

ξ

2 jij j ii

x

εε

ξξ ξ ξ

iFiε

jthk

christoffel system of the second kind

k kij ji

We can now write

( )k iki ij j

iFF

Fε

ξ ξIntroduce the notation

,ij ij

F

εF

ξThis means that ----- called the convariant derivative of --- is definded as the contravariant component of the vector comparing (4-2-1) and (4-2-2) then gives us the formula

,iFj

iFthi j

F

,

ii k ij kjj

FF F

ξ

,( )j jij ij

d d F d

FF ξ ξ ε

ξ

Although is not necessarily a tensor, is one , foriF

j

ξijF,

(4-2-1)

(4-2-2)

A direct calculation of is more instructive; with F= ,we have

Now , whence

The covariant derivative of writing as , is defined as the covariant component of ; hence

jiF ,

kjkiji FgF ,,

iFthi

iFiε

,i( )

kj i

i j kj j j

FF F

F εε ε

ξ ξ ξk ki i

ε ε

( )k

k k l kii ij l ijj j

εεε ε ε ε

kk iijj

εε

,ki

i j k ijj

FF F

And therefore

consequently

And while this be the same as (4-2-3) it shows the explicit addition to needed to provide the covariant derivative of

jF

,Fi j

(4-2-3)

iF

ij

F

Other notations are common for convariant derivatiives; they are, in approximate order of popularity

| ;F F D F Fi j i j j i j i

Although is not a third-order tensor, the superscript can nevertheless be lowered by means of the operation

kij

[ , ]k k ikp ij kp pj j

ig ij p g

ε εε ε

ξ ξand the resultant quantity , denoted by , is the Christoffel symbol of the first kind. The following relations are easily verified

],[ pij

2

[ , ]ji

k kj i k jix x

ij k

εεε ε

ξ ξξ ξ ξ

( ) [ , ] [ , ]ji

i j j ik k k

gij

ik j jk ik

εεε ε ε ε

ξ ξ ξξ

1[ , ] { }

2

g gg jk ijp ikg ij kkp ij j i k

(4-2-4)

1{ }

2

1{[ , ] [ , ] [ , ] [ , ] [ , ] [ , ]}

21

{[ , ] [ , ]}2[ , ]

jkikj i k

ggg ij

kj i ij k ki j ji k jk i ik j

ij k ji k

ij k

[Prove] ：

4-3 covariant derivatives of Nth –order Tensors

Let us work out the formula for the covariant derivatives of . write ..ijAk

..ij kk i j

AA ε ε ε

By definition

.. ,

.. ( )

ij kk p i j

ij kk i jp

Ap

A

Aε ε ε

ξ

ε ε εξ

This leads directly to the formula

.... , .. .. ..

ijij rj j iji ir rkk p k rp k rp r pkp

AA A A A

4-4 divergence of a vector A useful formula for

will be developed for general coordinate systems. We have

But, by determinant theory

Hence

And therefore

,( )

i pi iii ip

FF divF F F

p

[ , ]

1 [ - ]

2

1

2

i isip

ps ipis isi s

is isp

g ip s

g ggg

gg

ξ ξξ

ξ is isp p

gggg

ξ ξ

( )1 1

2

ggiip p pg g

ξ ξ

1( )ii

F F gg

ξ

4-5 Riemann-Christoffel Tensor

Since the order of differentiation in repeated partial differentiation of Cartesian tensor is irrelevant, it follows that the indices in repeated covariant differentiation of general tensors in Euclidean 3-D may also be interchanged at will. Thus, identities like

jiij ,, and

jikijk ff ,, Eq (4-5-1) is easily verified directly, since

2

, ,p

ij p ijji

ξ ξ

(4-5-1)

However, the assertion of (4-5-1) in Euclidean 3-D leads to some nontrivial information. By direct calculation it can be shown that

ppkijjikijk fRff .,,

.

p pkjp p pr rkiR

kij ri kj rj kii j

ξ ξWith help of (4-2-4) it can be shown that , the Riemann-christoffel tensor, is given by

Rpkij

2 2 221

[ ] [ ]2

r m r mpkij rm pj ki pi kj

g g ggpj pi kjkiR gk i p j j pk i

ξ ξ ξ ξ ξ ξ ξ ξ

But since the left-hand side of vanishes for all vectors , it follows that kf

0pkijR

Although (4-5-2) represents 81equations, most of them are either identities or redundant, since . Only six distinct nontrivial conditions are specified by (4-5-2), and they may be written as

(4-5-2)

031312331232312311223 RRRRRRpkij

pkij pkji kpij ijpkR R R R

(4-5-3)

[Note]

, ,( )

p pkj p pr rki

k ij k ji rj kj rj ki pi jf f f

ξ ξ[prove]

2

,,

,

,

( ) ( ) ( )

( ) ( ) ( )

k i p pk ij kj pi iji

pp p rk kki p kj kp r ij kpi i i r

pp p prk kk ji kj p kj pj r ji kpi j j p

pp pp pk ki

k ij p ki kjj j ji

ff f

ff ff f f

ff ff f f f

f fff f

ξ

ξ ξ ξ ξ

ξ ξ ξ ξ

ξξ ξ ξ ξ

2,

,

, ,

(

pjk

p p pr kkj pi r ij ij kpi p

pk j p pp p p p prk

k ji kj ki kj pj r ji ji krj i i ji

p pjk kjp pr r

k ij k ji kj pi r p jk pj r pj i

p pkj p rki

ri kj rji j

ff f

f fff f f

f f f f f f

ξ

ξ ξξ ξ ξ

ξξ

ξ ξ)p r

ki pf

Since is antisymmetrical in i and j as well as in p and k, no information is lost if (4-5-2) is multiplied by . Consequently , a set of six equations equivalent to (4-5-3) is given neatly by

pkijRspk tijε ε

0stSWhere is the symmetrical, second-order tensor stS

1

4spk tijst

pkijS R ε ε

The tensor is related simply to the Ricci tensor ijS

pij ij ij pR S g S

.p

ij ijpR R

So that (4-5-3) is also equivalent to the assertion 0ijR

The familiar divergence theorem relating integrals over a volume V and its boundary surface S can obvious be written in tensor notation as

Where Ni is the unit outward normal vector to S . Similar stokes ， theorem

for integrals over a surface S and its boundary line C is just

Where tk is the unit tangent vector to C , and the usual handedness rules apply for direction of Ni and ti

,i if dv f N dsi i

v s

,ijk kf N ds f t dsck j i k

s ε

4-6 Integral Relations

GTRgR 82

1

ij ijR R

ijij

R g R

i i

k lx xi i

jk j jk jl

kijklR

1,

2j jkkk j

g ggjk

x xx

2 jiij

ds g dx dx

,iig jk

jk

廣義相對論

, ,i

j kl j lk jkl iA A R A jikl ijkl

R R

0i i ijkl klj ljkR R R

,i

iji j j

AA A

x

,

ii i

jj jA

A Ax

PgUUPT

ijkl klijR R

klijR

12

1

!4

321111

2

1 22

nnN

nnnnnnnn

0ijklR

ijlk ijklR R