Conformal Anomalies for Interacting Scalar Fields on Curved Manifolds with Boundary

a r X i v :h e p -t h /0311100v 5 22 O c t 2004

CONFORMAL ANOMALIES FOR INTERACTING SCALAR FIELDS

ON CURVED MANIFOLDS WITH BOUNDARY

GEORGE TSOUPROS THE SCHOOL OF PHYSICS,PEKING UNIVERSITY,

BEIJING 100871,

PEOPLE’S REPUBLIC OF CHINA

Abstract.The trace anomaly for a conformally invariant scalar ?eld theory on a curved manifold of positive constant curvature with boundary is considered.In the context of a perturbative evaluation of the theory’s e?ective action explicit calculations are given for those contributions to the conformal anomaly which emerge as a result of free scalar propagation as well as from scalar self-interactions up to second order in the scalar self-coupling.The renormalisation-group behaviour of the theory is,sub-sequently,exploited in order to advance the evaluation of the conformal anomaly to third order in the scalar self-coupling.As a direct consequence the e?ective action is evaluated to the same order.In e?ect,complete contributions to the theory’s conformal anomaly and e?ective action are evaluated up to fourth-loop order.

I.Introduction

The fundamental physical signi?cance of bounded manifolds has been amply demon-strated in the framework of Euclidean Quantum Gravity and,more recently,in the context of the holograpic principle and the AdS/CF T correspondence.An issue of im-mediate importance on such manifolds is the evaluation of the e?ective action and,for that matter,of the conformal anomaly relevant to the dynamical behaviour of quantised ?elds.This issue has hitherto received attention,almost exclusively,at one-loop level in the absence of matter-to-matter related interactions.

The present work extends the exploration of the dynamical behaviour of scalar ?elds beyond one-loop order [1],[2],[3],[4]to the study of higher loop-order contributions to the conformal anomaly for a self-interacting massless scalar ?eld conformally coupled to the background geometry of the bounded manifold.Such an issue has been studied on a general manifold without boundary [5],[6],[7]as well as in the speci?c case of de Sitter space [8],[9].In addition to the trace anomaly emerging at one-loop order as a result of the background curvature and presence of the boundary there is a further anomalous contribution stemming from the self-interaction of the scalar ?eld.Pivotal to such an analysis is the perturbative evaluation of the gravitational component of the renormalised e?ective action that is,of the renormalised vacuum e?ective action.The stated additional contributions which that component of the e?ective action perturbatively generates to the conformal anomaly are closely associated with the renormalisation-group behaviour

2

of the theory.The structure of the renormalisation-group functions on a general curved manifold with boundary has been discussed in[11],[12].The results hitherto obtained[1], [2],[3],[4]allow for an explicit calculation of the scalar-interaction-related contributions to the anomalous trace up to second order in the scalar self-coupling.On the basis of these results,the renormalisation group will,subsequently,be exploited in order to attain an extension of such an evaluation to order three in the scalar self-coupling. Speci?cally,it will be shown through use of dimensional analysis that to that order the anomalous contribution which emerges exclusively as a result of scalar self-interactions is proportional to the renormalisation-group function for the scalar self-coupling in,both, the interior and boundary of the manifold.In e?ect,the conformal anomaly will be successively evaluated to?rst,second,third and fourth loop-order.Such an evaluation, directly,reveals information about the renormalised vacuum e?ective action at four-loop order.Its exact form at that order can be elicited without use of a renormalisation procedure.As was also the case with the renormalisation of the theory such an evaluation will be simultaneously accomplished in the interior and the boundary of the manifold order by order in perturbation.

For reasons of technical convenience the analysis will be performed on C4,a segment of the Euclidean sphere bounded by a hypersurface of positive extrinsic curvature,with homogeneous Dirichlet-type boundary conditions.Such a choice allows for a direct use of the results hitherto attained on such a manifold[1],[2],[3],[4].The results obtained herein have a general signi?cance for bounded manifolds of the same topology both in terms of the general structure of the e?ective action and in terms of the interaction between boundary and surface terms.Notwithstanding that,such results as are obtained herein on C4deserve attention in their own merit due to their additional signi?cance for the Hartle-Hawking approach to the quantisation of closed cosmological models.

As expected on the basis of the results already obtained,the perturbative evaluation of the theory’s conformal anomaly involves substantially complicated expressions which render the associated calculations arduous.However,as is also the case with the theory’s perturbative renormalisation,the underlying premise which allows for the evaluation of the conformal anomaly is based on the conceptually simple techniques which were initiated and developed in[1],[2].

The contribution to the conformal anomaly which emerges from free propagation on a curved manifold is the exclusive result of the gravitational backreaction on the manifold’s geometry and has a distinct character from that which emerges from matter-to-matter interactions.For that matter,the analysis in the context of free scalar propagation in Sec.III is independent from the ensuing analysis which pursues the scalar self-coupling-related contributions to the conformal anomaly.A brief outline of the techniques relevant to the evaluation of the free-propagation-related component of the conformal anomaly on a general bounded manifold is presented in Sec.III as the incipient point of an analysis which advances from the general to the concrete case of the bounded manifold of positive constant curvature stated herein.Appendix II amounts,mostly,to a citation of results directly relevant to the evaluation of the trace anomaly.However,a study of the analysis relevant to a certain mathematical aspect of the theory in Appendix I and in the following segment of Appendix II is recommended prior to that of sections IV and V.

3 II.E?ective Action and Trace Anomaly on C4

The scalar component of the bare action de?ning a theory for a conformal,massless ?eldΦspeci?ed on C n-a manifold of positive constant curvature embedded in a(n+1)-dimensional Euclidean space with embedding radius a and bounded by a(n?1)-sphere of positive constant extrinsic curvature K(diverging normals)-at n=4is[4] S[Φ0]= C d4η 12n(n?2)4!Φ40?1

?ην?ην

?

a2

As stated,the bare action(1)is associated by choice with the homogeneous Dirichlet

conditionΦ|?C

4=0for the scalar?eld.

The gravitational component of the bare action on C n at n→4is

S g= C d4η 116πG0(R+K2)+α0R2+θ0RK2 (3)+ ?C d3η (?1)

4

background metric,at J=0.Speci?cally,with W[gμν;J]being the generating functional for connected Green functions,it is

<0,out|Tμν|0,in>=2?gδ

5 importance as that e?ected by vacuum scalar e?ects in?g.1(b)[6].The sum-total of their contributions results,through renormalisation,in the one-loop e?ective action associated with free propagation.Such contributions to the bare gravitational action on a general manifold with boundary as are represented here by the diagrams in?g.1(a)and?g.1(b) have been the object of extensive study in Euclidean Quantum Gravity through use of zeta-function techniques and heat kernel asymptotic expansions[13].In the context of the theory pursued herein the one-loop e?ective action and associated trace anomaly can be obtained from these general results on a bounded manifold M by specifying the geometry to be that of C4with homogeneous Dirichlet conditions on?C4and the coupling between matter and gravity to be a conformal coupling between a scalar?eld and the stated geometry.The outline of the associated calculation leading to the trace anomaly follows.

III.Trace Anomaly and Free Scalar Propagation on C4

The one-loop e?ective action W0associated with the free scalar propagation in?g.1(b) on a general manifold M is,generally,given by the expression

(4)W0=

1

2?

2

ζ(0)ln(μ2)

On the grounds of general theoretical considerations outlined in the previous paragraph the mean value of the stress energy-momentum tensor in some vacuum state is

(7)=2

?g

δW

6

In e?ect,the trace of the renormalised stress tensor in (7)receives a contribution from W 0which can be seen from (6)to relate to ζ(s )through

[13](9)

M

d 4x

√2

;t →0+

of the supertrace (11)

G (t )=

C

trK D (x,x,t )d 4

x =

n

e ?λn t =T re ?tD

of the heat kernel

K D (x,x ′

,t )=

n

e ?λn t

associated with the bounded elliptic operator D in (4)through the heat equation (12)

(?

Γ(s )

∞

t s ?1G (t )dt

The asymptotic expansion in (10)yields,in the context of (13),the result (14)

ζ(0)=A 4

which,as (9)reveals,reduces the issue of the conformal anomaly due to free propagation of matter on M to the issue of evaluating the constant coe?cient A 4in (10).

The general asymptotic expansion in (10)is characterised by the exclusive presence of even-order coe?cients A 2k on any manifold M for which ?M =0.The presence of a non-trivial ?M has the e?ect of generating an additional boundary-related component for each even-order coe?cient as well as non-vanishing boundary-related values for all A 2k +1in (10).In general,the coe?cients for the supertrace of the heat kernel associated

7 with the relevant elliptic operator on a bounded four-dimensional manifold M admit,in the context of(10),the form

(15)A2k= M a(0)2k√hd3x

(16)A2k+1= ?M a(1)2k+1√

n(n?2)

2

1

90

1

90

8

?C d3η(?0KΦ20)

in(1)does not vanish.Such a non-vanishing e?ect arises as a result of the boundary condition

(21)K D(η,η′,t=0)=δ(4)(η?η′)

-imposed on the solution to the heat equation on C4

?

(22)(

9 the?ve coe?cientsβi which respectively multiply the geometry-related expressions in (27)depend only on the same operator and the conditions speci?ed on the boundary. If,in the case ofΛ>0,a boundary condition imposed on(25)is that of a compact four-geometry then the solution to(25)can be either the spherical cap C4or the Eu-clidean four-sphere S4.The former case emerges if the remaining boundary condition corresponds to the speci?cation of the induced three-geometry as a three-sphere.The latter case emerges if the remaining boundary condition corresponds to the absence of a boundary.In addition,the former case reduces to a disk D at the limit of boundary three-spheres small enough to allow for their embedding in?at Euclidean four-space. These three solutions are aspects of the Hartle-Hawking no-boundary proposal for the quantisation of closed universes[17].For the stated boundary conditions these solutions to(25)also coincide with the corresponding solutions to the Euclidean Einstein?eld equations in the presence of a massless scalar?eld conformally coupled to gravity on the additional Dirichlet condition of a constant?eld on?C4,in the present case of C4as well as in that of D.Such a coincidence is a consequence of a vanishing stress tensor for the conformal scalar?eld[17].

In e?ect,the constant coe?cient A4(D,C4)in the heat kernel asymptotic expansion for a conformal scalar?eld on C4,the corresponding A4(D,S)on S4,as well as the corresponding coe?cient A4(D,D)on D are expected to be inherently related.Withθ0 being the maximum colatitude on C4,which for that matter speci?es?C4,the stated relation is[14]

(28)A4(D,C4)=A4(D,S)(1

4

cosθ0+

1

8

β1cosθ0sin2θ0

where,in conformity with(9),(14)and(19),it is

(29)A4(D,S)=?

1

180

and the value of the coe?cientβ1for the present case of C4withΦ|?C

4

=0is[14] (31)β1=

29

10

In the context of (9)and (14)the result which (28)-(31)signify relates to the conformal anomaly through (32)

C

d 4η(C )

r +

?C

d 3η(?C )

r =A 4(D,C 4)

In order to arrive at a local expression for the trace anomaly on C 4use will be made of

the stated fact that on any bounded manifold the local interior coe?cients a (0)

2k ,associated through (15)with the asymptotic expansion of the supertrace of the heat kernel in (10),are speci?ed by the same local invariants as in the unbounded manifold of the same local geometry.This,in turn,reveals in the context of (10)that the local asymptotic expansion of the heat kernel associated with the operator D in (17)exclusively in the interior of C 4is in coincidence with the local asymptotic expansion of the heat kernel for the same operator on S 4so that (23)yields (33)

c (0)2k =s (0)

2k

with s (0)

2k being the local coe?cients a (0)

2k if M in (15)is speci?ed as S 4.Setting k =2

and integrating in the interior of C 4yields (34)

C

d 4ηc (0)

4=

C

d 4ηs (0)

4

and,through (32)and (23)(35)

(C )

r =s (0)

4

In the context of (15),however,(19)amounts to (36)

S

d 4ηr =?

1

90

190

1

11 Finally,substituting(38)in(32)yields

(39) ?C d3η(?C)r=A4(D,C4)+13sin2θ04cosθ04?23)

with[1]

(40) C d4η=a42π2(?13cosθ04+2

a34πA4(D,C4)+

1

a34π

(?

1

3

cosθ04+

2?ν =μ? λ+∞ k=1∞ i=k a kiλi

π41

212 ∞ N=0 1331(N+1)(N+4) C3

12

N ′

N ′=0

1

N ′0)+12

Γ(1+1

N ′0

)

N ′

0)(N +1)(N +2)]×

3

π

2

F 2

? 3π5+π3

2

N (cosθ04)]

2

?

C

d 4η+

λ2

13

1

[N ′2

?N 2

+3(N ′

?N )]2

×

Γ(

1

3

1

N ′0

)Γ(N ′+1+1

Γ(N ′+3+

1

? C

d 4η+

λ2

131

[N ′2?N 2+3(N ′?N )]2

×

Γ(

1

31N ′0

)Γ(N ′+1+

1

Γ(N ′+3+

1

?

?C

d 3η+F.T.

where the ?nite terms remain to be assessed.In this expression N is the quantum number associated with the angular momentum ?owing through the singular part of the propagator and N ′is the corresponding quantum number associated with the boundary

part of the propagator.The exact nature of the transform-space cut-o?N ′

0has been cited and analysed in [1]and summarised in Appendix I.The n-dependent quantities F,B and H are complicated functions of Gegenbauer polynomials.They have been derived in [2],[3]and reproduced in [4].For the sake of completeness they are given below at n →4.

F (4)=

N ′cosθ04

m 1=0

(sinθ04)2m 1C

m 1+

3

2

N ′?m 1

(cosθ0

4)m 1

with θ0

n being the angle specifying ?C n in the (n +1)-dimensional embedding Euclidean space and with

m 01=Ncosθ0n ;l 01=N ′cosθ0

n

being the degrees of spherical harmonics de?ned on ?C n .

B (4)=2 m 1(sinθ04)m 1+l 1+2(cosθ0

4)C

m 1+32

N ?m 1?1

(cosθ0

4)

C

l 1+3

13

H(4)=

2

k=1 π0C m k+1+32k

m k?m k+1

(cosθ4?k)C l k+1+32k

l k?l k+1

(cosθ4?k)[sinθ4?k]m k+1+l k+1+1dθ4?k×

2π0e i(±m3?l3)θ1

?a

which,by virtue of(8),reveals the relation between the component W I

r of the renor-

malised gravitational e?ective action generated by vacuum diagrams with vertices and the higher loop-order contributions to the trace of the renormalised stress tensorI

r to be

(46) C d4ηI r=a?W I r

14

This expression relatesI

r and W I

r

through the background geometry and can be

seen to have both formal and perturbative signi?cance.

The result of the perturbative evaluation of W,expressed to O( 3)as W(3)

I in(43),

contains three divergences at the dimensional limit?→0.They are all of orderλ2.As shown in[4]the RK2-related divergence cancels against the corresponding countertem contained in the bare gravitational couplingθ0in(3)to generate theθ12λ2term in the latter’s expansion in terms ofλ.Likewise,the R2-related divergence de?nes the corresponding O(λ2)term in the relevant expansion forα0and the surface,RK-related divergence,de?nes the corresponding O(λ2)term in the relevant expansion forζ0.As

W(3)

I r amounts to the?nite terms in(43)it is imperative that the latter be assessed.

The assessment of the stated?nite parts involves,in fact,the entire renormalisation program up to O(λ2).As the evaluation of the relevant counterterms necessitates only the divergent parts of the associated diagrams in the context of the minimal subtraction and mass-independent renormalisation the?nite parts of those diagrams were not cited in[4].The assessment of those parts simply necessitates the expansion of all?-dependent quantities in the diagramatic contributions about?=0.In[4]such an expansion was performed only in the relevantΓ-functions for it is only such expansions which provide the divergent parts as poles in?.The additional contributions to the stated?nite parts will emerge from expansions of the form

(47)(μa)?=1+?ln(μa)+O(?2)

at the?→0limit in all radius-dependent terms.It is obvious that the pole structure of the theory remains intact in the context of such expansions on account of the presence of the?0-term.It should,in addition,be recalled that the renormalisation scheme of minimal subtraction,invoked herein,is characterised by the perturbative absence of ?nite parts in all counterterms.

The results relevant to the?nite terms of the vacuum e?ective action in(43)have been derived in Appendix II and can be placed in the context of the ensuing analysis simply by inspection.In addition,the mathematically important commutativity between the ?nite summations in these results and integration over C4is the subject of Appendix I, a study of which is recommended.In e?ect,the entire three-loop contribution to the

scalar-interaction-related sector of the renormalised gravitational e?ective action W(3)

I r ,

which is also the entire contribution to O(λ2),amounts to

(48)W(3)

I r

=(1c)+(1d)+(1e)

In order to arrive at a local expression for the trace anomaly in both the interior

and boundary of C4the nature of W(3)

I r as the sum-total of contributions from both the

interior and boundary,explicitly featured in(48),will be exploited in order to recast (46)in the form

(49) C d4η(C)I r+ ?C d3η(?C)I r=a?

15

with L(3)

C (η)and L(3)

?C

(η)being the interaction-related e?ective Lagrangian evaluated to

O( 3)in the interior and boundary of C4respectively.As partial di?erentiation with respect to the embedding radius a,explicitly featured only in the logarithms of the Lagrangian,commutes with volume integration this yields

(50)(C)

I r =a

?

?a

L(3)

?C

Again,attention is invited to the fact that all explicit dependence of W(3)

I r on a enters

exclusively through the logarithms.The additional implicit dependence is evident in the relations[1]

(52a)R=

12

3a

cotθ04

as well as

(52c) C d nη=a42π2(?13cosθ04+2

16

The expression in(53)manifests an exclusive particularity for the component of this theory’s trace anomaly which is generated by the presence of interactions.Speci?cally, to O(λ2)the trace anomaly in the interior of C4receives a constant value.In Appendix I the case was also made to the e?ect that in the interior,as is also the case in the surface, of C4the e?ective lagrangian remains respectively constant to any order in perturbation.

Consequently,(C)

I r is expected to receive a constant value in the interior of C4to

any order.The origin of the constancy of both the trace anomaly and e?ective lagrangian can be traced to the high degree of symmetry of C4which allows for the reduction of the eigenvalue problem on it to that on S4with its characteristic summations over spherical harmonics of degree N[1].

Finally,a comparison between(48)and(51)reveals that

(54)(?C)

I r =λ2RK

1

sinθ04

∞

N=0C3(N)

This is the desired local expression for the O(λ2)contribution to the trace anomaly on ?C4.It emerges as a contribution from the RK-sector of the vacuum e?ective action which amounts to a convergent series over N and,as stated,it is also constant.

Both(53)and(54)have emerged as a result of an explicit calculation to O(λ2).In addition,this calculation reveals the announced result for the contribution to the trace anomaly to O(λ).The?nite contribution,associated with the diagram of?g.(1c),is obviously vanishing as that diagram’s contribution to the gravitational e?ective action features no explicit dependence on the embedding radius a[4].

In what follows,the renormalisation-group behaviour of the theory will be invoked in order to evaluate the contribution to the trace anomaly to O(λ3)without resorting to explicit diagramatic calculations.

V.Trace Anomaly and Renormalisation Group

In generating perturbatively the vacuum e?ective action W the summation over vac-uum diagramatic contributions without external wavefunctions also amounts perturba-tively to the theory’s zero-point proper function.For that matter,the latter coincides identically with W.In e?ect,(43)is also the O( 3)contribution to the theory’s zero-point function.

In addition to being generated by the background curvature the divergences contained in W I are also the result of the scalar self-interactions.To O(λ2)they are represented as simple poles at the dimensional limit in(43).The divergences inherent in?g.1(d)and ?g.1(e)are the exclusive result of the diagramatic loop-structures respectively and are, consequently,primitive.On the contrary,the divergence inherent in?g.1(c)is the result of the replacement of the bare self-couplingλ0byλthrough(42)and is,for that matter, an overlapping divergence[4].Consequently,the cancellation of all divergences up to, at least,O(λ2)necessitates,as stated,the counterterms contained in(42)in addition to the counterterms contained in the gravitational bare couplings.Moreover,(43)reveals that all divergences inherent in W I to O(λ2)are contained as single poles in the R2,RK2 and RK sectors of the bare gravitational action in(3).Therefore,the above-mentioned gravitational bare couplings are theα0,θ0,andζ0in(3).No additional mass-type

17 counterterms are required.In particular,to O(λ2)radiative contributions to the two-point function will generate a contribution toξ0in(1)[4].However,such a contribution

does not a?ect the renormalised zero-point function W I

r at the stated order.This is

the case because any contribution stemming from the RΦ20sector of the bare action

necessarily enters W I

r through the contributions which the semi-classical propagator

receives from radiative e?ects to the two-point function.Consequently,contributions

generated by that sector to O(λ2)will necessarily enter W I

r at higher orders.As an

immediate consequence the renormalised zero-point function W I

r satis?es,to O(λ2)the

renormalisation-group equation (55) μ??λr+γα(λr)??θr+γζ(λr)?

?μ

(56b)γα(λr)=μ

?αr

?μ

(56d)γζ(λr)=μ

?ζr

18

As a direct consequence of the scalar vacuum contributions to the R2,RK2and RK

sectors of the bare gravitational action in(3)the component W I

r of the renormalised

vacuum e?ective action stemming from the scalar self-interactions can be seen on dimen-sional grounds-at least to O(λ2)-to,necessarily,depend onλr,aμand the renormalised gravitational self-couplingsαr,θr andζr only.The above-stated dependence is a conse-

quence of dimensional analysis and is,for that matter,a mathematical property of W I

r .

Consequently,the mathematical statement

(57)dW(3)

I r =

?W(3)

I r

?(aμ)

d(aμ)+

?W(3)

I r

?θr

dθr+

?W(3)

I r

?a W(3)

I r

(λ,aμ)=μ′ ?W(3)I r?αr?αr?θr?θr?ζr?ζr

?a

=?β(λr)

?W I

r

?λr

Again,as was the case in(49),both the integral over C4and the vacuum e?ective action

W I

r in(60)can be resolved respectively into their interior and surface components so

that

C d4η(C)I r+ ?C d3η(?C)I r

19 (61)=?β(λr)

?

?λr

L C C d4η?β(λr)?

?λr

L C

and

(64)(?C)

I r =?β(λr)

?

?μ

=(1?λr

?

4π2(V c+1)

∞

N=1 C3N2(N+3)2K2

20

Consequently,(65a)yields

(66)β(λr)=?λ2

92N(cosθ04) 2R

It is worth emphasising that the dependence on geometry in(65a)and(65b)is ex-clusively speci?c to C4.Such a dependence on a general manifold would contradict the principles of locality and covariance as a result of which all renormalisation constants must be space-time independent[7].However,both R and K are constants on C4. There is,for that matter,no contradiction.Moreover,it is readily seen through(52a) and(52b)that even the dependence on the embedding radius a cancels out leaving a trivial dependence on the boundary-de?ning angleθ04.

The renormalised vacuum e?ective action W I

r

has been evaluated to O(λ2)in Appen-

dix II.The result W(3)

I r has been cited in(48).The derivative of L(3)

C

associated with(63)

can be elicited through direct di?erentiation with respect toλof W(3)

I r in the interior of

C4and the replacement of all?nite summations over N′by the same constants C1(N)and C2(N)as those in(53).Since such di?erentiation features all possible O(λ)terms for the partial derivative of the vacuum e?ective action with respect to the scalar self-coupling in the interior of C4and(66)features the O(λ2)contribution to the beta function it follows that the complete O(λ3)contribution to the trace anomaly in the interior of C4 is the result of the substitution in(63)of,both,(66)and the stated partial derivative. It can readily be con?rmed by inspection that such a contribution also signi?es the com-plete O( 4)to the trace anomaly in the interior of C4.Moreover,since to O(λ2)the

vacuum e?ective action W(3)

I r receives contributions in the RK2and R2sectors it can be

seen,through(66)and(63),that to O(λ3)the contribution to the trace anomaly in the interior of C4receives contributions exclusively in the,already established,RK2-sector as well as in the new K4-sector which emerges,for that matter,at loop-orders no lower than four.In e?ect,to O(λ3)to the trace anomaly in the interior of C4has the structure

(67)(C)

I r =λ3RK2

∞

N=0C3(N)?λ3K4∞ N=0C4(N)

As was the case to O(λ2)the contributions to the two sectors in(67)respectively amount to the constant values of the associated convergent series over the quantum number N and can be read o?the results in Appendix II.

As a direct consequence of the above-stated sectors in the trace anomaly this calcula-tion also reveals in the context of(46)that to O(λ3)the renormalised vacuum e?ective action in the interior of C4develops,itself,a new sector proportionate to K4.Equiva-lently,to the stated order the theory contains a divergence proportionate to K4.This is also a qualitatively new result obtained through the renormalisation-group invariance of the theory without recourse to explicit diagramatic calculations.

Finally,the derivative of L(3)

?C

associated with(64)can be,likewise,elicited through di-

rect di?erentiation of the surface components of W(3)

I r

featured in Appendix II.Again,the

[批处理]计算时间差的函数etime

[批处理]计算时间差的函数etime 计算时间差的函数etime 收藏 https://www.sodocs.net/doc/294670863.html,/thread-4701-1-1.html 这个是脚本代码[保存为etime.bat放在当前路径下即可：免费内容: :etime <begin_time> <end_time> <return> rem 所测试任务的执行时间不超过1天// 骨瘦如柴版setlocal&set be=%~1:%~2&set cc=(%%d-%%a)*360000+(1%%e-1%%b)*6000+1%%f-1% %c&set dy=-8640000 for /f "delims=: tokens=1-6" %%a in ("%be:.=%")do endlocal&set/a %3=%cc%,%3+=%dy%*("%3>> 31")&exit/b ---------------------------------------------------------------------------------------------------------------------------------------- 计算两个时间点差的函数批处理etime 今天兴趣大法思考了好多bat的问题，以至于通宵 在论坛逛看到有个求时间差的"函数"被打搅调用地方不少(大都是测试代码执行效率的) 免费内容: :time0

::计算时间差（封装） @echo off&setlocal&set /a n=0&rem code 随风@https://www.sodocs.net/doc/294670863.html, for /f "tokens=1-8 delims=.: " %%a in ("%~1:%~2") do ( set /a n+=10%%a%%100*360000+10%%b%%100*6000+10%% c%%100*100+10%%d%%100 set /a n-=10%%e%%100*360000+10%%f%%100*6000+10%%g %%100*100+10%%h%%100) set /a s=n/360000,n=n%%360000,f=n/6000,n=n%%6000,m=n/1 00,n=n%%100 set "ok=%s% 小时%f% 分钟%m% 秒%n% 毫秒" endlocal&set %~3=%ok:-=%&goto :EOF 这个代码的算法是统一找时间点凌晨0:00:00.00然后计算任何一个时间点到凌晨的时间差(单位跑秒) 然后任意两个时间点求时间差就是他们相对凌晨时间点的时间数的差 对09这样的非法8进制数的处理用到了一些技巧，还有两个时间参数不分先后顺序，可全可点， 但是这个代码一行是可以省去的(既然是常被人掉用自然体

to与for的用法和区别

to与for的用法和区别 一般情况下, to后面常接对象; for后面表示原因与目的为多。 Thank you for helping me. Thanks to all of you. to sb.表示对某人有直接影响比如，食物对某人好或者不好就用to; for表示从意义、价值等间接角度来说，例如对某人而言是重要的，就用for. for和to这两个介词，意义丰富，用法复杂。这里仅就它们主要用法进行比较。 1. 表示各种“目的” 1. What do you study English for? 你为什么要学英语？ 2. She went to france for holiday. 她到法国度假去了。 3. These books are written for pupils. 这些书是为学生些的。 4. hope for the best, prepare for the worst. 作最好的打算，作最坏的准备。 2．对于 1．She has a liking for painting. 她爱好绘画。 2．She had a natural gift for teaching. 她对教学有天赋/ 3．表示赞成同情，用for不用to. 1. Are you for the idea or against it? 你是支持还是反对这个想法？ 2. He expresses sympathy for the common people.. 他表现了对普通老百姓的同情。 3. I felt deeply sorry for my friend who was very ill. 4 for表示因为，由于（常有较活译法） 1 Thank you for coming. 谢谢你来。 2. France is famous for its wines. 法国因酒而出名。 5．当事人对某事的主观看法，对于（某人），对…来说（多和形容词连用）用介词to，不用for.. He said that money was not important to him. 他说钱对他并不重要。 To her it was rather unusual. 对她来说这是相当不寻常的。 They are cruel to animals. 他们对动物很残忍。 6．for和fit, good, bad, useful, suitable 等形容词连用，表示适宜，适合。 Some training will make them fit for the job. 经过一段训练，他们会胜任这项工作的。 Exercises are good for health. 锻炼有益于健康。 Smoking and drinking are bad for health. 抽烟喝酒对健康有害。 You are not suited for the kind of work you are doing. 7. for表示不定式逻辑上的主语，可以用在主语、表语、状语、定语中。 1．It would be best for you to write to him. 2．The simple thing is for him to resign at once. 3．There was nowhere else for me to go. 4．He opened a door and stood aside for her to pass.

of与for的用法以及区别

of与for的用法以及区别 for 表原因、目的 of 表从属关系 介词of的用法 （1）所有关系 this is a picture of a classroom （2）部分关系 a piece of paper a cup of tea a glass of water a bottle of milk what kind of football，American of soccer？ （3）描写关系 a man of thirty 三十岁的人 a man of shanghai 上海人 （4）承受动作 the exploitation of man by man．人对人的剥削。 （5）同位关系 It was a cold spring morning in the city of London in England． （6）关于，对于 What do you think of Chinese food？ 你觉得中国食品怎么样？ 介词 for 的用法小结 1. 表示“当作、作为”。如: I like some bread and milk for breakfast. 我喜欢把面包和牛奶作为早餐。What will we have for supper? 我们晚餐吃什么?

2. 表示理由或原因,意为“因为、由于”。如: Thank you for helping me with my English. 谢谢你帮我学习英语。 Thank you for your last letter. 谢谢你上次的来信。 Thank you for teaching us so well. 感谢你如此尽心地教我们。 3. 表示动作的对象或接受者,意为“给……”、“对…… (而言)”。如: Let me pick it up for you. 让我为你捡起来。 Watching TV too much is bad for your health. 看电视太多有害于你的健康。 4. 表示时间、距离,意为“计、达”。如: I usually do the running for an hour in the morning. 我早晨通常跑步一小时。We will stay there for two days. 我们将在那里逗留两天。 5. 表示去向、目的,意为“向、往、取、买”等。如: let’s go for a walk. 我们出去散步吧。 I came here for my schoolbag.我来这儿取书包。 I paid twenty yuan for the dictionary. 我花了20元买这本词典。 6. 表示所属关系或用途,意为“为、适于……的”。如: It’s time for school. 到上学的时间了。 Here is a letter for you. 这儿有你的一封信。 7. 表示“支持、赞成”。如: Are you for this plan or against it? 你是支持还是反对这个计划? 8. 用于一些固定搭配中。如: Who are you waiting for? 你在等谁? For example, Mr Green is a kind teacher. 比如,格林先生是一位心地善良的老师。

延时子程序计算方法

学习MCS-51单片机，如果用软件延时实现时钟，会接触到如下形式的延时子程序：delay:mov R5,#data1 d1:mov R6,#data2 d2:mov R7,#data3 d3:djnz R7,d3 djnz R6,d2 djnz R5,d1 Ret 其精确延时时间公式：t=（2*R5*R6*R7+3*R5*R6+3*R5+3）*T （“*”表示乘法，T表示一个机器周期的时间）近似延时时间公式：t=2*R5*R6*R7 *T 假如data1,data2,data3分别为50，40，248，并假定单片机晶振为12M，一个机器周期为10-6S，则10分钟后，时钟超前量超过1.11秒，24小时后时钟超前159.876秒（约2分40秒）。这都是data1,data2,data3三个数字造成的，精度比较差，建议C描述。

上表中e=-1的行（共11行）满足（2*R5*R6*R7+3*R5*R6+3*R5+3）=999，999 e=1的行（共2行）满足（2*R5*R6*R7+3*R5*R6+3*R5+3）=1，000，001 假如单片机晶振为12M，一个机器周期为10-6S，若要得到精确的延时一秒的子程序，则可以在之程序的Ret返回指令之前加一个机器周期为1的指令（比如nop指令）， data1,data2,data3选择e=-1的行。比如选择第一个e=-1行，则精确的延时一秒的子程序可以写成： delay:mov R5,#167 d1:mov R6,#171 d2:mov R7,#16 d3:djnz R7,d3 djnz R6,d2

djnz R5,d1 nop ；注意不要遗漏这一句 Ret 附： #include"iostReam.h" #include"math.h" int x=1,y=1,z=1,a,b,c,d,e(999989),f(0),g(0),i,j,k; void main() { foR(i=1;i<255;i++) { foR(j=1;j<255;j++) { foR(k=1;k<255;k++) { d=x*y*z*2+3*x*y+3*x+3-1000000; if(d==-1) { e=d;a=x;b=y;c=z; f++; cout<<"e="<

用c++编写计算日期的函数

14.1 分解与抽象 人类解决复杂问题采用的主要策略是“分而治之”，也就是对问题进行分解，然后分别解决各个子问题。著名的计算机科学家Parnas认为，巧妙的分解系统可以有效地系统的状态空间，降低软件系统的复杂性所带来的影响。对于复杂的软件系统，可以逐个将它分解为越来越小的组成部分，直至不能分解为止。这样在小的分解层次上，人就很容易理解并实现了。当所有小的问题解决完毕，整个大的系统也就解决完毕了。 在分解过程中会分解出很多类似的小问题，他们的解决方式是一样的，因而可以把这些小问题，抽象出来，只需要给出一个实现即可，凡是需要用到该问题时直接使用即可。 案例日期运算 给定日期由年、月、日（三个整数，年的取值在1970－2050之间）组成，完成以下功能： （1）判断给定日期的合法性； （2）计算两个日期相差的天数； （3）计算一个日期加上一个整数后对应的日期； （4）计算一个日期减去一个整数后对应的日期； （5）计算一个日期是星期几。 针对这个问题，很自然想到本例分解为5个模块，如图14.1所示。 图14.1日期计算功能分解图 仔细分析每一个模块的功能的具体流程： 1. 判断给定日期的合法性： 首先判断给定年份是否位于1970到2050之间。然后判断给定月份是否在1到12之间。最后判定日的合法性。判定日的合法性与月份有关，还涉及到闰年问题。当月份为1、3、5、7、8、10、12时，日的有效范围为1到31；当月份为4、6、9、11时，日的有效范围为1到30；当月份为2时，若年为闰年，日的有效范围为1到29；当月份为2时，若年不为闰年，日的有效范围为1到28。

图14.2日期合法性判定盒图 判断日期合法性要要用到判断年份是否为闰年，在图14.2中并未给出实现方法，在图14.3中给出。 图14.3闰年判定盒图 2. 计算两个日期相差的天数 计算日期A （yearA 、monthA 、dayA ）和日期B （yearB 、monthB 、dayB ）相差天数，假定A 小于B 并且A 和B 不在同一年份，很自然想到把天数分成3段： 2.1 A 日期到A 所在年份12月31日的天数； 2.2 A 之后到B 之前的整年的天数（A 、B 相邻年份这部分没有）； 2.3 B 日期所在年份1月1日到B 日期的天数。 A 日期 A 日期12月31日 B 日期 B 日期1月1日 整年部分 整年部分 图14.4日期差分段计算图 若A 小于B 并且A 和B 在同一年份，直接在年内计算。 2.1和2.3都是计算年内的一段时间，并且涉及到闰年问题。2.2计算整年比较容易，但

常用介词用法(for to with of)

For的用法 1. 表示“当作、作为”。如: I like some bread and milk for breakfast. 我喜欢把面包和牛奶作为早餐。 What will we have for supper? 我们晚餐吃什么? 2. 表示理由或原因,意为“因为、由于”。如: Thank you for helping me with my English. 谢谢你帮我学习英语。 3. 表示动作的对象或接受者,意为“给……”、“对…… (而言)”。如: Let me pick it up for you. 让我为你捡起来。 Watching TV too much is bad for your health. 看电视太多有害于你的健康。 4. 表示时间、距离,意为“计、达”。如: I usually do the running for an hour in the morning. 我早晨通常跑步一小时。 We will stay there for two days. 我们将在那里逗留两天。 5. 表示去向、目的,意为“向、往、取、买”等。如: Let’s go for a walk. 我们出去散步吧。 I came here for my schoolbag.我来这儿取书包。 I paid twenty yuan for the dictionary. 我花了20元买这本词典。 6. 表示所属关系或用途,意为“为、适于……的”。如: It’s time for school. 到上学的时间了。 Here is a letter for you. 这儿有你的一封信。 7. 表示“支持、赞成”。如: Are you for this plan or against it? 你是支持还是反对这个计划? 8. 用于一些固定搭配中。如: Who are you waiting for? 你在等谁? For example, Mr Green is a kind teacher. 比如,格林先生是一位心地善良的老师。 尽管for 的用法较多，但记住常用的几个就可以了。 to的用法: 一：表示相对，针对 be strange (common, new, familiar, peculiar) to This injection will make you immune to infection. 二：表示对比，比较 1：以-ior结尾的形容词，后接介词to表示比较，如：superior ,inferior,prior,senior,junior 2: 一些本身就含有比较或比拟意思的形容词，如equal，similar，equivalent，analogous A is similar to B in many ways.

Excel中如何计算日期差

Excel中如何计算日期差： ----Excel中最便利的工作表函数之一——Datedif名不见经传，但却十分好用。Datedif能返回任意两个日期之间相差的时间，并能以年、月或天数的形式表示。您可以用它来计算发货单到期的时间，还可以用它来进行2000年的倒计时。 ----Excel中的Datedif函数带有3个参数，其格式如下： ----=Datedif(start_date,end_date,units) ----start_date和end_date参数可以是日期或者是代表日期的变量，而units则是1到2个字符长度的字符串，用以说明返回日期差的形式（见表1）。图1是使用Datedif函数的一个例子，第2行的值就表明这两个日期之间相差1年又14天。units的参数类型对应的Datedif返回值 “y”日期之差的年数（非四舍五入） “m”日期之差的月数（非四舍五入） “d”日期之差的天数（非四舍五入） “md”两个日期相减后，其差不足一个月的部分的天数 “ym”两个日期相减后，其差不足一年的部分的月数 “yd”两个日期相减后，其差不足一年的部分的天数

表1units参数的类型及其含义 图1可以通过键入3个带有不同参数的Datedif公式来计算日期的差。units的参数类型 ----图中：单元格Ex为公式“=Datedif(Cx,Dx,“y”)”得到的结果（x=2,3,4......下同） ----Fx为公式“=Datedif(Cx,Dx,“ym”)”得到的结果 ----Gx为公式“=Datedif(Cx,Dx,“md”)”得到的结果 现在要求两个日期之间相差多少分钟，units参数是什么呢？ 晕，分钟你不能用天数乘小时再乘分钟吗？ units的参数类型对应的Datedif返回值 “y”日期之差的年数（非四舍五入） “m”日期之差的月数（非四舍五入） “d”日期之差的天数（非四舍五入） “md”两个日期相减后，其差不足一个月的部分的天数 “ym”两个日期相减后，其差不足一年的部分的月数 “yd”两个日期相减后，其差不足一年的部分的天数 假设你的数据从A2和B2开始，在C2里输入下面公式，然后拖拉复制。 =IF(TEXT(A2,"h:mm:ss")

of和for的用法

of 1....的,属于 One of the legs of the table is broken. 桌子的一条腿坏了。 Mr.Brown is a friend of mine. 布朗先生是我的朋友。 2.用...做成的;由...制成 The house is of stone. 这房子是石建的。 3.含有...的;装有...的 4....之中的;...的成员 Of all the students in this class,Tom is the best. 在这个班级中,汤姆是最优秀的。 5.(表示同位) He came to New York at the age of ten. 他在十岁时来到纽约。 6.(表示宾格关系) He gave a lecture on the use of solar energy. 他就太阳能的利用作了一场讲演。 7.(表示主格关系) We waited for the arrival of the next bus. 我们等待下一班汽车的到来。

I have the complete works of Shakespeare. 我有莎士比亚全集。 8.来自...的;出自 He was a graduate of the University of Hawaii. 他是夏威夷大学的毕业生。 9.因为 Her son died of hepatitis. 她儿子因患肝炎而死。 10.在...方面 My aunt is hard of hearing. 我姑妈耳朵有点聋。 11.【美】(时间)在...之前 12.(表示具有某种性质) It is a matter of importance. 这是一件重要的事。 For 1.为,为了 They fought for national independence. 他们为民族独立而战。 This letter is for you. 这是你的信。

单片机C延时时间怎样计算

C程序中可使用不同类型的变量来进行延时设计。经实验测试，使用unsigned char类型具有比unsigned int更优化的代码，在使用时 应该使用unsigned char作为延时变量。以某晶振为12MHz的单片 机为例，晶振为12M H z即一个机器周期为1u s。一. 500ms延时子程序 程序: void delay500ms(void) { unsigned char i,j,k; for(i=15;i>0;i--) for(j=202;j>0;j--) for(k=81;k>0;k--); } 计算分析: 程序共有三层循环 一层循环n:R5*2 = 81*2 = 162us DJNZ 2us 二层循环m:R6*(n+3) = 202*165 = 33330us DJNZ 2us + R5赋值 1us = 3us 三层循环: R7*(m+3) = 15*33333 = 499995us DJNZ 2us + R6赋值 1us = 3us

循环外: 5us 子程序调用 2us + 子程序返回2us + R7赋值 1us = 5us 延时总时间 = 三层循环 + 循环外 = 499995+5 = 500000us =500ms 计算公式:延时时间=[(2*R5+3)*R6+3]*R7+5 二. 200ms延时子程序 程序: void delay200ms(void) { unsigned char i,j,k; for(i=5;i>0;i--) for(j=132;j>0;j--) for(k=150;k>0;k--); } 三. 10ms延时子程序 程序: void delay10ms(void) { unsigned char i,j,k; for(i=5;i>0;i--) for(j=4;j>0;j--) for(k=248;k>0;k--);

for和to区别

1.表示各种“目的”，用for （1）What do you study English for 你为什么要学英语？ （2）went to france for holiday. 她到法国度假去了。 （3）These books are written for pupils. 这些书是为学生些的。 （4）hope for the best, prepare for the worst. 作最好的打算，作最坏的准备。 2．“对于”用for （1）She has a liking for painting. 她爱好绘画。 （2）She had a natural gift for teaching. 她对教学有天赋/ 3．表示“赞成、同情”，用for （1）Are you for the idea or against it 你是支持还是反对这个想法？ （2）He expresses sympathy for the common people.. 他表现了对普通老百姓的同情。 （3）I felt deeply sorry for my friend who was very ill. 4. 表示“因为，由于”（常有较活译法），用for （1）Thank you for coming. 谢谢你来。

（2）France is famous for its wines. 法国因酒而出名。 5．当事人对某事的主观看法，“对于（某人），对…来说”，（多和形容词连用），用介词to，不用for. （1）He said that money was not important to him. 他说钱对他并不重要。 （2）To her it was rather unusual. 对她来说这是相当不寻常的。 （3）They are cruel to animals. 他们对动物很残忍。 6．和fit, good, bad, useful, suitable 等形容词连用，表示“适宜，适合”，用for。（1）Some training will make them fit for the job. 经过一段训练，他们会胜任这项工作的。 （2）Exercises are good for health. 锻炼有益于健康。 （3）Smoking and drinking are bad for health. 抽烟喝酒对健康有害。 （4）You are not suited for the kind of work you are doing. 7. 表示不定式逻辑上的主语，可以用在主语、表语、状语、定语中。 （1）It would be best for you to write to him. （2） The simple thing is for him to resign at once.

51单片机延时时间计算和延时程序设计

一、关于单片机周期的几个概念 ●时钟周期 时钟周期也称为振荡周期，定义为时钟脉冲的倒数（可以这样来理解，时钟周期就是单片机外接晶振的倒数，例如12MHz的晶振，它的时间周期就是1/12 us），是计算机中最基本的、最小的时间单位。 在一个时钟周期内，CPU仅完成一个最基本的动作。 ●机器周期 完成一个基本操作所需要的时间称为机器周期。 以51为例,晶振12M，时钟周期(晶振周期)就是(1/12)μs，一个机器周期包 执行一条指令所需要的时间，一般由若干个机器周期组成。指令不同，所需的机器周期也不同。 对于一些简单的的单字节指令，在取指令周期中，指令取出到指令寄存器后，立即译码执行，不再需要其它的机器周期。对于一些比较复杂的指令，例如转移指令、乘法指令，则需要两个或者两个以上的机器周期。 1.指令含义 DJNZ：减1条件转移指令 这是一组把减1与条件转移两种功能结合在一起的指令，共2条。 DJNZ Rn，rel ；Rn←（Rn）-1 ；若（Rn）=0，则PC←（PC）+2 ；顺序执行 ；若（Rn）≠0，则PC←（PC）+2+rel，转移到rel所在位置DJNZ direct，rel ；direct←（direct）-1 ；若（direct）= 0，则PC←（PC）+3；顺序执行 ；若（direct）≠0，则PC←（PC）+3+rel，转移到rel 所在位置 2.DJNZ Rn,rel指令详解 例：

MOV R7，#5 DEL:DJNZ R7,DEL; rel在本例中指标号DEL 1.单层循环 由上例可知，当Rn赋值为几，循环就执行几次，上例执行5次，因此本例执行的机器周期个数=1（MOV R7，#5）+2（DJNZ R7,DEL）×5=11，以12MHz的晶振为例，执行时间（延时时间）=机器周期个数×1μs=11μs，当设定立即数为0时，循环程序最多执行256次，即延时时间最多256μs。 2.双层循环 1）格式： DELL:MOV R7，#bb DELL1:MOV R6，#aa DELL2:DJNZ R6,DELL2; rel在本句中指标号DELL2 DJNZ R7,DELL1; rel在本句中指标号DELL1 注意：循环的格式，写错很容易变成死循环，格式中的Rn和标号可随意指定。 2）执行过程

excel中计算日期差工龄生日等方法

excel中计算日期差工龄生日等方法 方法1：在A1单元格输入前面的日期，比如“2004-10-10”，在A2单元格输入后面的日期，如“2005-6-7”。接着单击A3单元格，输入公式“=DATEDIF(A1,A2,"d")”。然后按下回车键，那么立刻就会得到两者的天数差“240”。 提示：公式中的A1和A2分别代表前后两个日期，顺序是不可以颠倒的。此外，DATEDIF 函数是Excel中一个隐藏函数，在函数向导中看不到它，但这并不影响我们的使用。 方法2：任意选择一个单元格，输入公式“="2004-10-10"-"2005-6-7"”，然后按下回车键，我们可以立即计算出结果。 计算工作时间——工龄—— 假如日期数据在D2单元格。 =DA TEDIF(D2,TODAY(),"y")+1 注意：工龄两头算，所以加“1”。 如果精确到“天”—— =DA TEDIF(D2,TODAY(),"y")&"年"&DATEDIF(D2,TODAY(),"ym")&"月"&DATEDIF(D2,TODAY(),"md")&"日" 二、计算2003-7-617:05到2006-7-713:50分之间相差了多少天、多少个小时多少分钟 假定原数据分别在A1和B1单元格,将计算结果分别放在C1、D1和E1单元格。 C1单元格公式如下： =ROUND(B1-A1,0) D1单元格公式如下： =(B1-A1)*24 E1单元格公式如下： =(B1-A1)*24*60 注意:A1和B1单元格格式要设为日期,C1、D1和E1单元格格式要设为常规. 三、计算生日，假设b2为生日

=datedif(B2,today(),"y") DA TEDIF函数，除Excel2000中在帮助文档有描述外，其他版本的Excel在帮助文档中都没有说明，并且在所有版本的函数向导中也都找不到此函数。但该函数在电子表格中确实存在，并且用来计算两个日期之间的天数、月数或年数很方便。微软称，提供此函数是为了与Lotus1-2-3兼容。 该函数的用法为“DA TEDIF(Start_date,End_date,Unit)”，其中Start_date为一个日期，它代表时间段内的第一个日期或起始日期。End_date为一个日期，它代表时间段内的最后一个日期或结束日期。Unit为所需信息的返回类型。 “Y”为时间段中的整年数，“M”为时间段中的整月数，“D”时间段中的天数。“MD”为Start_date与End_date日期中天数的差，可忽略日期中的月和年。“YM”为Start_date与End_date日期中月数的差，可忽略日期中的日和年。“YD”为Start_date与End_date日期中天数的差，可忽略日期中的年。比如，B2单元格中存放的是出生日期（输入年月日时，用斜线或短横线隔开），在C2单元格中输入“=datedif(B2,today(),"y")”（C2单元格的格式为常规），按回车键后，C2单元格中的数值就是计算后的年龄。此函数在计算时，只有在两日期相差满12个月，才算为一年，假如生日是2004年2月27日，今天是2005年2月28日，用此函数计算的年龄则为0岁，这样算出的年龄其实是最公平的。 本篇文章来源于：实例教程网(https://www.sodocs.net/doc/294670863.html,) 原文链接：https://www.sodocs.net/doc/294670863.html,/bgruanjian/excel/631.html

双宾语tofor的用法

1. 两者都可以引出间接宾语，但要根据不同的动词分别选用介词to 或for： (1) 在give, pass, hand, lend, send, tell, bring, show, pay, read, return, write, offer, teach, throw 等之后接介词to。 如： 请把那本字典递给我。 正：Please hand me that dictionary. 正：Please hand that dictionary to me. 她去年教我们的音乐。 正：She taught us music last year. 正：She taught music to us last year. (2) 在buy, make, get, order, cook, sing, fetch, play, find, paint, choose,prepare, spare 等之后用介词for 。如： 他为我们唱了首英语歌。 正：He sang us an English song. 正：He sang an English song for us. 请帮我把钥匙找到。 正：Please find me the keys. 正：Please find the keys for me. 能耽搁你几分钟吗(即你能为我抽出几分钟吗)? 正：Can you spare me a few minutes? 正：Can you spare a few minutes for me? 注：有的动词由于搭配和含义的不同，用介词to 或for 都是可能的。如： do sb a favou r do a favour for sb 帮某人的忙 do sb harnn= do harm to sb 对某人有害

for和of的用法

for的用法： 1. 表示“当作、作为”。如: I like some bread and milk for breakfast. 我喜欢把面包和牛奶作为早餐。 What will we have for supper? 我们晚餐吃什么? 2. 表示理由或原因,意为“因为、由于”。如: Thank you for helping me with my English. 谢谢你帮我学习英语。 Thank you for your last letter. 谢谢你上次的来信。 Thank you for teaching us so well. 感谢你如此尽心地教我们。 3. 表示动作的对象或接受者,意为“给……”、“对…… (而言)”。如: Let me pick it up for you. 让我为你捡起来。 Watching TV too much is bad for your health. 看电视太多有害于你的健康。 4. 表示时间、距离,意为“计、达”。如:

I usually do the running for an hour in the morning. 我早晨通常跑步一小时。 We will stay there for two days. 我们将在那里逗留两天。 5. 表示去向、目的,意为“向、往、取、买”等。如: Let’s go for a walk. 我们出去散步吧。 I came here for my schoolbag.我来这儿取书包。 I paid twenty yuan for the dictionary. 我花了20元买这本词典。 6. 表示所属关系或用途,意为“为、适于……的”。如: It’s time for school. 到上学的时间了。 Here is a letter for you. 这儿有你的一封信。 7. 表示“支持、赞成”。如: Are you for this plan or against it? 你是支持还是反对这个计划? 8. 用于一些固定搭配中。如:

英语形容词和of for 的用法

加入收藏夹 主题： 介词试题It’s + 形容词 + of sb. to do sth.和It’s + 形容词 + for sb. to do sth.的用法区别。 内容： It's very nice___pictures for me. A.of you to draw B.for you to draw C.for you drawing C.of you drawing 提交人：杨天若时间：1/23/2008 20:5:54 主题：for 与of 的辨别 内容：It's very nice___pictures for me. A.of you to draw B.for you to draw C.for you drawing C.of you drawing 答：选A 解析：该题考查的句型It’s + 形容词+ of sb. to do sth.和It’s +形容词+ for sb. to do sth.的用法区别。 “It’s + 形容词+ to do sth.”中常用of或for引出不定式的行为者，究竟用of sb.还是用for sb.，取决于前面的形容词。 1) 若形容词是描述不定式行为者的性格、品质的，如kind，good，nice，right，wrong，clever，careless，polite，foolish等，用of sb. 例： It’s very kind of you to help me. 你能帮我，真好。 It’s clever of you to work out the maths problem. 你真聪明，解出了这道数学题。 2) 若形容词仅仅是描述事物，不是对不定式行为者的品格进行评价，用for sb.，这类形容词有difficult，easy，hard，important，dangerous，（im）possible等。例： It’s very dangerous for children to cross the busy street. 对孩子们来说，穿过繁忙的街道很危险。 It’s difficult for u s to finish the work. 对我们来说，完成这项工作很困难。 for 与of 的辨别方法： 用介词后面的代词作主语，用介词前边的形容词作表语，造个句子。如果道理上通顺用of，不通则用for. 如： You are nice.(通顺，所以应用of)。 He is hard.(人是困难的，不通，因此应用for.) 由此可知，该题的正确答案应该为A项。 提交人：f7_liyf 时间：1/24/2008 11:18:42

to和for的用法有什么不同(一)

to和for的用法有什么不同（一） 一、引出间接宾语时的区别 两者都可以引出间接宾语，但要根据不同的动词分别选用介词to 或for，具体应注意以下三种情况： 1. 在give, pass, hand, lend, send, tell, bring, show, pay, read, return, write, offer, teach, throw 等之后接介词to。如： 请把那本字典递给我。 正：Please hand me that dictionary. 正：Please hand that dictionary to me. 她去年教我们的音乐。 正：She taught us music last year. 正：She taught music to us last year. 2. 在buy, make, get, order, cook, sing, fetch, play, find, paint, choose, prepare, spare 等之后用介词for 。如： 他为我们唱了首英语歌。 正：He sang us an English song. 正：He sang an English song for us. 请帮我把钥匙找到。 正：Please find me the keys. 正：Please find the keys for me. 能耽搁你几分钟吗(即你能为我抽出几分钟吗)? 正：Can you spare me a few minutes?

正：Can you spare a few minutes for me? 3. 有的动词由于用法和含义不同，用介词to 或for 都是可能的。如： do sb a favor＝do a favor for sb 帮某人的忙 do sb harm＝do harm to sb 对某人有害 在有的情况下，可能既不用for 也不用to，而用其他的介词。如： play sb a trick＝play a trick on sb 作弄某人 请比较： play sb some folk songs＝play some folk songs for sb 给某人演奏民歌 有时同一个动词，由于用法不同，所搭配的介词也可能不同，如leave sbsth 这一结构，若表示一般意义的为某人留下某物，则用介词for 引出间接宾语，即说leave sth for sb；若表示某人死后遗留下某物，则用介词to 引出间接宾语，即说leave sth to sb。如： Would you like to leave him a message? / Would you like to leave a message for him? 你要不要给他留个话？ Her father left her a large fortune. / Her father left a large fortune to her. 她父亲死后给她留下了一大笔财产。 二、表示目标或方向的区别 两者均可表示目标、目的地、方向等，此时也要根据不同动词分别对待。如： 1. 在come, go, walk, move, fly, ride, drive, march, return 等动词之后通常用介词to 表示目标或目的地。如： He has gone to Shanghai. 他到上海去了。 They walked to a river. 他们走到一条河边。