Inflation, holography, and the choice of vacuum in de Sitter space

a r X i v :h e p -t h /0205227v 2 29 M a y 2002UUITP-07/02

hep-th/0205227

In?ation,holography,

and the choice of vacuum in de Sitter space

Ulf H.Danielsson Institutionen f¨o r teoretisk fysik Box 803,SE-75108Uppsala,Sweden ulf@teorfys.uu.se Abstract A family of de Sitter vacua introduced in hep-th/0203198as plausible initial condi-tions for in?ation,are discussed from the point of view of de Sitter holography.The vacua are argued to be physically acceptable and the in?ationary picture provides a physical interpretation of a subfamily of de Sitter invariant vacua.Some speculations on the issue of vacuum choice and the connection between the CMBR and holography are also provided.

May 2002

1Introduction

In recent years there has been a signi?cant interest in the use of cosmology as a probe of high energy physics.A well established example is provided by in?ation where microscopic quantum?uctuations are magni?ed to macroscopic scales,and act as seeds for subsequent structure formation in the early universe.For a nice review see [1].

An intriguing possibility is that in?ation might provide a window towards physics beyond the Planck scale.In the standard in?ationary scenario the?uctuations start out with a linear size much smaller than the Planck scale.Nevertheless it is assumed that no new physics appear,and a natural vacuum for the quantum?uctuations is chosen with this in mind.In several recent works the rationality of this assumption has been questioned,[2-26].After all,in the real world we know that fundamentally new physics is to be expected at the Planck scale and beyond.The idea is that the?uctuation spectrum might leave an imprint on the CMBR that depends on the details of this transplanckian physics.Several examples exist in the literature of speci?c modi?cations of the high energy physics which also give a modi?ed spectrum. In[27](and more recently in[28])another point of view was taken with the focus on the choice of vacuum.Initial conditions(for a particular mode)were imposed when the wavelength was comparable to some fundamental length scale in the theory.It was shown that this leads to corrections of order H

four the properties of the vacua relevant for in?ation are discussed.Finally,section ?ve contains some conclusions and speculations on possible relations with holography. 2A de Sitter family of vacua

De Sitter space does not have a globally time like Killing vector and the choice of vacuum is therefore a highly non trivial issue[61].For the study of physics in de Sitter space there are several di?erent coordinate systems available.Below we will consider global coordinates(with spherical spatial sections)that cover the full de Sitter space, and planar coordinates(with?at spatial sections)that only cover half of de Sitter space.In the last section we will make use of yet another type of coordinates,the static coordinates,which are useful when the focus is on observations made by a particular observer.

The planar coordinates are the ones that are relevant for an in?ating cosmology, where only half of de Sitter space is physical.The Penrose diagram of de Sitter space has the shape of a square,and can be viewed as consisting of two parts.An in?ating cosmology in the upper right triangle,and a de?ating cosmology in the lower left. They are separated by a light like surface corresponding to the Big Bang or the Big Crunch respectively.We will,however,?rst investigate the choice of vacuum from the point of view global coordinates,closely following the analysis of[53][54].

De Sitter space in D dimensions can be de?ned through a hypersurface

?X20+D i=1X2i=1

the spherical harmonics on S3.For clarity we will put an hat on all operator valued ?elds.The vacuum of the theory is de?ned to obey

a n|? =0.(6) The vacuum in de Sitter space is not unique,instead,as was?rst discussed in[62],and later in[63][64][65],there is a whole family of possible vacua.More recent discussions, in the context of de Sitter holography,can be found in[53][54].Di?erent choices correspond to di?erent mode expansions,like the one in(4),where the di?erent choices are related through Bogolubov transformations.For our study of the various vacua it will be convenient to use the Wightman function de?ned by

G+(x,x′)= ?| φ(x) φ(x′)|? = nφn(x)φ?n(x′).(7)

The Wightman function is a useful building block from which the other Green’s functions can be obtained.Through the study of the Wightman function one can learn about the properties of the vacua that one is interested in.A vacuum that will play a special role is the Bunch-Davies vacuum(also known as the Euclidean vacuum)which is obtained by analytical continuation from the Euclidean sphere.It has a Wightman function given by

G+E(x,x′)= E|φ(x)φ(x′)|E =Γ(h+)Γ(h?)

2 ,(8)

where

h±=

39

x=(τ,?)is mapped to x A=(?τ,?A).?A is the point opposite to?on S3.Note that P(x,x′)=?P(x A,x′).For easy comparison with the case of planar coordinates below,it will be useful to write(10)as

φ(x)=AφE(x)+BφE(x A).(13) Using(7)one?nds,?nally,a Wightman function given by

G+(x,x′)=|A|2G+E(x,x′)+|B|2G+E(x′,x)+AB?G+E(x,x′A)+BA?G+E(x A,x′).(14) After this warm up,we will now turn to the case relevant for in?ation,i.e.the planar coordinates.

The metric in terms of planar coordinates is given by

ds2=dt2?a(t)2d x2,(15) where a(t)=e Ht,is the scale factor.In terms of the conformal timeη=?1

H2η2 dη2?d x2 .(16) The relation to the hypersurface of(1)is given through

1

η=?

i=1,2,3,(18)

H(X0+X4)

with X0+X4>0.As discussed in[54]the Bunch-Davies vacuum has a simple description in planar coordinates.It corresponds to a situation where there are no incoming particles onη→?∞.As discussed above,the in?ationary universe can be viewed as the upper right triangle of the full de Sitter space whereη→?∞corresponds to the Big Bang,whileη→0is I+.The lower half of de Sitter space can be covered if we also consider positiveη.In particular,both signs ofηare allowed in the de Sitter invariant distance,which in planar coordinates becomes

η2+η′2?(x?x′)2

P(x,x′)=

x=(?η,x).That is,a map between the two halves of de Sitter space.Let us now proceed with the quantization,which is formally very similar to the analysis in global coordinates.

In planar coordinates a massless scalar?eld can be decomposed as

φ(x)= d3k φk(x) a k+φ??k(x) a?k .(20)

We now assume that the modes can be written in terms of the modes de?ning the Bunch-Davies vacuum through

φk(x)=Aφk,E(x)+Bφk,E(

x′)+BA?G+E(

√kη ,(25)

and new modes given by

φk(η)=?ηH2k Ae?ikη 1?i kη .(26) We also need the conjugate momentum which is given by

πk(η)=φ′k(η)=ikηH

2k Ae?ikη?Be ikη

.(27)

As reviewed in the previous section,there is(up to an overall phase)a two parameter family of de Sitter invariant vacua.We will now focus on a particular one parameter subfamily such that:

A=e i(θ?β)

2β?i

2β

,(28)

whereβandθ(the overall phase)are both real.Note that de?nitions are such that |A|2?|B|2=1.The crucial properties of these vacua,parametrized byβ,is that they imply the existence of a particular moment in time when there is a very simple relation betweenφk andπk.One easily veri?es that

φk(ηk)=?ηk H2k e iθ

πk(ηk)=?ikφk(ηk),(29)

where

ηk=?

β

aH

we?nd that

p=

k

H ,we?nd that the

physical momentum,corresponding to the mode k,is equal toΛat conformal time ηk.The vacua de?ned by(28)therefore have the property that a mode k is created in a state of the form(29)at the timeηk when its physical size is given by the universal scaleΛ.

There are several ways to characterize a state of the form(29).As pointed out in [27],following[69],one can note that it corresponds to a state of minimum uncertainty. By construction it is also what might be called a local Minkowski vacuum de?ned through

a k|? =0,(32)

where a k is the annihilation operator appearing in(20).

The analysis in[27]proceeded further by calculating the perturbation spectrum that would result from these initial conditions.In the case ofΛ

2π 2 1?H H ,(33)

which shows that the natural size of a transplanckian correction is given by H

2.The various de Sitter invariant vacua are physically di?erent.An example of such a di?erence is that only the Bunch-Davies vacuum is thermal.This was discussed in[53]where it was shown that a freely falling detector will only come into thermal equilibrium in the Bunch-Davies vacuum.It is also quite obvious from the explicit expressions for the Wightman function that it does not have the analyticity properties characteristic for a thermal system.

3.At high energies(short distances)the de Sitter invariant vacua(except the Bunch-Davies)are in general di?erent from ordinary Minkowski space(even if there are no new singularities).This is expected and actually the whole point with the reasoning at the end of the previous section.It simply means that we encounter transplanckian e?ects when we probe short distances.The standard argument would have been that the Greens function should approach the usual Minkowski expression when the distance is small enough compared to the Hubble scale.However,for small distances we encounter new physics and there is no regime where the Greens function will take the Minkowski form.

To conclude,there is no reason of principle to limit one self to the Bunch-Davies vacuum,at least if one is interested in an in?ating cosmology where only half of de Sitter space is relevant.

5Conclusions and speculations on holography

In this paper we have observed that the vacua discussed in[27]are among the family of de Sitter invariant vacua explored in[53][54].We have also pointed out that there is no obvious reason to exclude these vacua from a physical point of view,and a better knowledge of high energy physics is needed in order to make the correct choice.It would be interesting if the question of how the vacuum is determined could be made in a holographic setting.In this respect one should note that the holographic counterpart of the CMBR-?uctuation spectrum is given by two point correlator in the holographic theory.

In the by now standard AdS/CFT holographic correspondence,the hologram is situated at the boundary of AdS.The boundary has a Lorentzian signature and the theory living there is in many cases a fairly standard CFT.A similar relation is supposed to hold also in the de Sitter case,where the boundary is either in the in?nite future,I+,or in the in?nite past,I?.A crucial di?erence from the AdS is that the boundary has an Euclidean signature.Since we are interested in applying the construction to in?ation we will focus on I+.As we approach I+,asη→0,it therefore becomes inappropriate to divide the?eld according to positive and negative frequency,see[57].Rather we should distinguish between the two possible asymptotic behaviors withφ～ηh±,implying that there are actually two operators in the boundary conjugate to a given bulk?eld.We denote these operators by O+k

and O?k in momentum space.For the two point functions one therefore?nds that φk(η) φ?k(η′) =ηh?η′h? O?k O??k +ηh+η′h+ O+k O+?k ～ O?k O??k ,(34) with h+=3and h?=0for a massless bulk?eld,as we approach the boundary. It follows,therefore,that the?uctuation spectrum relevant for the CMBR is,from a holographic point of view,directly given by the correlator O?k O??k .The main observation is that the holographic screen on I+corresponds to the frozen modes that have expanded outside of the in?ationary horizon.If in?ation never stopped, they would not be associated with anything measurable.In[33]such quantities on I+ were referred to as meta-observables.However,in the real universe,the modes re-enter through the horizon after the end of in?ation and act like seeds for the?uctuation spectrum of the CMBR.The study of the CMBR therefore corresponds to a study of the theory on I+,and the meta-observables actually turns into real observables accessible to measurements.This picture,however,breaks down for small scales on I+,corresponding to late times,when in?ation turns o?.

The sensitivity to the initial conditions of in?ation explored in[27]has a precise counterpart in the dependence on the vacuum of the holographic correlation functions as discussed in[53][54],and the question arises what holography might have to say about the choice of vacuum.The family of de Sitter invariant vacua all correspond to scale invariant vacua in the theory on I+.Thinking in terms of planar coordi-nates,identifying the comoving momentum k with the Euclidean momentum in the hologram,the time evolution(in conformal time)corresponds to an inverted renor-malization group?ow from large to small scales.As the?ow proceeds,new modes are integrated in requiring initial conditions.Clearly this happens,for some particular k,when the corresponding wavelength becomes of the same order as the cuto?.

To proceed further we need to know a bit more about the de Sitter holography.

A useful coordinate system that brings the focus on what a particular observer will experience,is the static coordinate system.(A discussion of the relation between planar and static coordinates from the point of view of dS/CFT can be found in[48]). The metric in static coordinates is given by

ds2=? 1?H2r2 dt2+dr2

H2η2 ?dη2+dρ2+ρ2d?2d?2 ,(36) the coordinate transformation that takes us between the planar and static coordinates is given by

ρ=ρ(t,r)=?Hrη(37)

η(t,r)=?e?Ht

H2r2?1.(38)

In planar coordinates it follows from the metric(16)that a source atηwill give rise to a blob of size?x=|η|in the boundary at I+.It is therefore natural to associate a cuto?of order|η|to a spacelike section at conformal timeη(the same all over I+).If we now make a coordinate transformation on the boundary,mapping the plane into a cylinder,several things will happen.All circles,independent of their radiusρin planar coordinates,will be mapped onto the circumference of a cylinder with a radius independent of the direction t along the cylinder.If we want a cuto?that is the same all over the cylinder(rather than all over the plane)we need to change the original cuto?in order to keepρ/ηconstant.This will imply a constant r through(37),with a cuto?on the cylinder that is proportional to1/r.From(38)we can read o?what a particular cuto?corresponds to in terms ofη.We?nd that late times(large t)along the cylinder corresponds to smallη.

The cylinder corresponds to a surface of radius r>1/H outside the cosmological horizon,but if we want the theory to precisely describe the degrees of freedom on the horizon we need to take r=1/H.It follows from,e.g.,(37)that the cuto?now is comparable to the radius of the cylinder,and there will be only one lattice point encoding all the degrees of freedom of the whole horizon.Luckily,due to the large central charge the theory has many degrees of freedom living at each of these lattice points.This analysis is in line with the analysis in[71].

The above discussion makes it clear that the horizon has,atη,a size?x=|η| implying that the resolution of the hologram is such that there is only one lattice point per horizon volume.The vacua that we have been discussing implies that the initial conditions are imposed,at timeη,on scales given by

H

?x=

supported by a grant from the Knut and Alice Wallenberg Foundation.The work was also supported by the Swedish Research Council(VR).

References

[1]A.R.Liddle and D.H.Lyth,“Cosmological in?ation and large-scale structure”,

Cambridge University Press2000.

[2]R.H.Brandenberger,“In?ationary cosmology:Progress and problems,”

arXiv:hep-ph/9910410.

[3]J.Martin and R.H.Brandenberger,“The trans-Planckian problem of in?ation-

ary cosmology,”Phys.Rev.D63,123501(2001)[arXiv:hep-th/0005209]. [4]J.C.Niemeyer,“In?ation with a high frequency cuto?,”Phys.Rev.D63,123502

(2001)[arXiv:astro-ph/0005533].

[5]R.H.Brandenberger and J.Martin,“The robustness of in?ation to changes in

super-Planck-scale physics,”Mod.Phys.Lett.A16,999(2001)[arXiv:astro-ph/0005432].

[6]A.Kempf,“Mode generating mechanism in in?ation with cuto?,”Phys.Rev.D

63,083514(2001)[arXiv:astro-ph/0009209].

[7]C.S.Chu,B.R.Greene and G.Shiu,“Remarks on in?ation and noncommutative

geometry,”Mod.Phys.Lett.A16,2231(2001)[arXiv:hep-th/0011241].

[8]J.Martin and R.H.Brandenberger,“A cosmological window on trans-Planckian

physics,”arXiv:astro-ph/0012031.

[9]L.Mersini,M.Bastero-Gil and P.Kanti,“Relic dark energy from trans-Planckian

regime,”Phys.Rev.D64,043508(2001)[arXiv:hep-ph/0101210].

[10]J. C.Niemeyer and R.Parentani,“Trans-Planckian dispersion and scale-

invariance of in?ationary perturbations,”Phys.Rev.D64,101301(2001) [arXiv:astro-ph/0101451].

[11]A.Kempf and J.C.Niemeyer,“Perturbation spectrum in in?ation with cuto?,”

Phys.Rev.D64,103501(2001)[arXiv:astro-ph/0103225].

[12]R.Easther,B.R.Greene,W.H.Kinney and G.Shiu,“In?ation as a probe of

short distance physics,”Phys.Rev.D64,103502(2001)[arXiv:hep-th/0104102].

[13]M.Bastero-Gil and L.Mersini,“SN1A data and CMB of Modi?ed Curvature

at Short and Large Distances,”Phys.Rev.D65(2002)023502[arXiv:astro-ph/0107256].

[14]L.Hui and W.H.Kinney,“Short distance physics and the consistency rela-

tion for scalar and tensor?uctuations in the in?ationary universe,”arXiv:astro-ph/0109107.

[15]R.Easther,B.R.Greene,W.H.Kinney and G.Shiu,“Imprints of short distance

physics on in?ationary cosmology,”arXiv:hep-th/0110226.

[16]M.Bastero-Gil,P.H.Frampton and L.Mersini,“Modi?ed dispersion relations

from closed strings in toroidal cosmology,”arXiv:hep-th/0110167.

[17]R.H.Brandenberger,S.E.Joras and J.Martin,“Trans-Planckian physics and

the spectrum of?uctuations in a bouncing universe,”arXiv:hep-th/0112122. [18]J.Martin and R.H.Brandenberger,“The Corley-Jacobson dispersion relation

and trans-Planckian in?ation,”arXiv:hep-th/0201189.

[19]J. C.Niemeyer,“Cosmological consequences of short distance physics,”

arXiv:astro-ph/0201511.

[20]F.Lizzi,G.Mangano,G.Miele and M.Peloso,arXiv:hep-th/0203099.

[21]G.Shiu and I.Wasserman,“On the signature of short distance scale in the cosmic

microwave background,”arXiv:hep-th/0203113.

[22]R.Brandenberger and P.M.Ho,“Noncommutative spacetime,stringy spacetime

uncertainty principle,and density?uctuations,”arXiv:hep-th/0203119. [23]S.Shankaranarayanan,“Is there an imprint of Planck scale physics on in?ation-

ary cosmology?,”arXiv:gr-qc/0203060.

[24]N.Kaloper,M.Kleban,https://www.doczj.com/doc/f310035506.html,wrence and S.Shenker,“Signatures of short

distance physics in the cosmic microwave background,”arXiv:hep-th/0201158.

[25]R.H.Brandenberger and J.Martin,“On signatures of short distance physics in

the cosmic microwave background,”arXiv:hep-th/0202142.

[26]S.F.Hassan and M.S.Sloth,“Trans-Planckian e?ects in in?ationary cosmology

and the modi?ed uncertainty principle,”arXiv:hep-th/0204110.

[27]U.H.Danielsson,“A note on in?ation and transplanckian physics,”arXiv:hep-

th/0203198.Accepted for publication by Physical Review D.

[28]R.Easther,B.R.Greene,W.H.Kinney and G.Shiu,“A generic estimate of

trans-Planckian modi?cations to the primordial power spectrum in in?ation,”

arXiv:hep-th/0204129.

[29]J.Maldacena,“The large N limit of superconformal?eld theories and super-

gravity,”Adv.Theor.Math.Phys.2,231(1998)[Int.J.Theor.Phys.38,1113 (1998)][arXiv:hep-th/9711200].

[30]S.S.Gubser,I.R.Klebanov and A.M.Polyakov,“Gauge theory correlators

from non-critical string theory,”Phys.Lett.B428,105(1998)[arXiv:hep-th/9802109].

[31]E.Witten,“Anti-de Sitter space and holography,”Adv.Theor.Math.Phys.2,

253(1998)[arXiv:hep-th/9802150].

[32]C.M.Hull,“Timelike T-duality,de Sitter space,large N gauge theories and

topological?eld theory,”JHEP9807(1998)021[arXiv:hep-th/9806146].

[33]E.Witten,“Quantum gravity in de Sitter space,”arXiv:hep-th/0106109.

[34]A.Strominger,“The ds/CFT correspondence,”arXiv:hep-th/0106113.

[35]V.Balasubramanian,P.Horava and D.Minic,“Deconstructing de Sitter,”JHEP

0105(2001)043[arXiv:hep-th/0103171].

[36]M.Li,“Matrix model for de Sitter,”JHEP0204,005(2002)[arXiv:hep-

th/0106184].

[37]S.Nojiri and S.D.Odintsov,“Conformal anomaly from dS/CFT correspon-

dence,”Phys.Lett.B519(2001)145[arXiv:hep-th/0106191].

[38]D.Klemm,“Some aspects of the de Sitter/CFT correspondence,”arXiv:hep-

th/0106247.

[39]S.Nojiri and S.D.Odintsov,“Quantum cosmology,in?ationary brane-world

creation and dS/CFT correspondence,”JHEP0112,033(2001)[arXiv:hep-th/0107134].

[40]T.Shiromizu,D.Ida and T.Torii,“Gravitational energy,dS/CFT correspon-

dence and cosmic no-hair,”JHEP0111,010(2001)[arXiv:hep-th/0109057]. [41]C.M.Hull,“de Sitter space in supergravity and M theory,”arXiv:hep-

th/0109213.

[42]M.Spradlin,A.Strominger and A.Volovich,“Les Houches lectures on de Sitter

space,”arXiv:hep-th/0110007.

[43]B.McInnes,“Exploring the similarities of the dS/CFT and AdS/CFT corre-

spondences,”Nucl.Phys.B627,311(2002)[arXiv:hep-th/0110062].

[44]V.Balasubramanian,J.de Boer and D.Minic,“Mass,entropy and holography

in asymptotically de Sitter spaces,”arXiv:hep-th/0110108.

[45]Y.S.Myung,“Entropy of the three dimensional Schwarzschild-de Sitter black

hole,”arXiv:hep-th/0110123.

[46]B.G.Carneiro da Cunha,“Three-dimensional de Sitter gravity and the corre-

spondence,”Phys.Rev.D65,104025(2002)[arXiv:hep-th/0110169].

[47]R.G.Cai,Y.S.Myung and Y.Z.Zhang,“Check of the Mass Bound Conjecture

in de Sitter Space,”arXiv:hep-th/0110234.

[48]U.H.Danielsson,“A black hole hologram in de Sitter space,”JHEP0203(2002)

020[arXiv:hep-th/0110265].

[49]S.Ogushi,“Holographic entropy on the brane in de Sitter Schwarzschild space,”

Mod.Phys.Lett.A17,51(2002)[arXiv:hep-th/0111008].

[50]A.C.Petkou and G.Siopsis,“dS/CFT correspondence on a brane,”JHEP0202,

045(2002)[arXiv:hep-th/0111085].

[51]R.G.Cai,“Cardy-Verlinde formula and asymptotically de Sitter spaces,”Phys.

Lett.B525(2002)331[arXiv:hep-th/0111093].

[52]A.M.Ghezelbash and R.B.Mann,“Action,mass and entropy of Schwarzschild-

de Sitter black holes and the de Sitter/CFT correspondence,”JHEP0201,005 (2002)[arXiv:hep-th/0111217].

[53]R.Bousso,A.Maloney and A.Strominger,“Conformal vacua and entropy in de

Sitter space,”arXiv:hep-th/0112218.

[54]M.Spradlin and A.Volovich,“Vacuum states and the S-matrix in dS/CFT,”

arXiv:hep-th/0112223.

[55]R.G.Cai,“Cardy-Verlinde formula and thermodynamics of black holes in de

Sitter spaces,”Nucl.Phys.B628(2002)375[arXiv:hep-th/0112253].

[56]A.M.Ghezelbash,D.Ida,R.B.Mann and T.Shiromizu,“Slicing and brane

dependence of the(A)dS/CFT correspondence,”arXiv:hep-th/0201004.

[57]S.R.Das,“Thermality in de Sitter and holography,”arXiv:hep-th/0202008.

[58]A.Strominger,“In?ation and the dS/CFT correspondence,”JHEP0111,049

(2001)[arXiv:hep-th/0110087].

[59]https://www.doczj.com/doc/f310035506.html,rsen,J.P.van der Schaar and R.G.Leigh,“de Sitter holography and the

cosmic microwave background,”JHEP0204(2002)047[arXiv:hep-th/0202127].

[60]E.Halyo,“Holographic in?ation,”arXiv:hep-th/0203235.

[61]G.W.Gibbons and S.W.Hawking,“Cosmological Event Horizons,Thermody-

namics,And Particle Creation,”Phys.Rev.D15,2738(1977).

[62]N.A.Chernikov and E.A.Tagirov,“Quantum theory of scalar?eld in de Sitter

space-time,”Ann.Inst.Henri Poincar′e,vol.IX,nr2,(1968)109.

[63]E.Mottola,“Particle Creation In De Sitter Space,”Phys.Rev.D31(1985)754.

[64]B.Allen,“Vacuum States In De Sitter Space,”Phys.Rev.D32(1985)3136.

[65]R.Floreanini,C.T.Hill and R.Jackiw,“Functional Representation For The

Isometries Of De Sitter Space,”Annals Phys.175(1987)345.

[66]K.Kirsten and J.Garriga,“Massless Minimally Coupled Fields In De Sitter

Space:O(4)Symmetric States Versus De Sitter Invariant Vacuum,”Phys.Rev.

D48(1993)567[arXiv:gr-qc/9305013].

[67]A.J.Tolley and N.Turok,“Quantization of the massless minimally coupled

scalar?eld and the dS/CFT correspondence,”arXiv:hep-th/0108119.

[68]A.J.Tolley and N.Turok,“Quantum?elds in a big crunch/big bang spacetime,”

arXiv:hep-th/0204091.

[69]D.Polarski and A.A.Starobinsky,“Semiclassicality and decoherence of cosmo-

logical perturbations,”Class.Quant.Grav.13,377(1996)[arXiv:gr-qc/9504030].

[70]R.Bousso,“Positive vacuum energy and the N-bound,”JHEP0011,038(2000)

[arXiv:hep-th/0010252];R.Bousso,“Bekenstein bounds in de Sitter and?at space,”JHEP0104,035(2001)[arXiv:hep-th/0012052].

[71]D.Kabat and G.Lifschytz,“De Sitter entropy from conformal?eld theory,”

JHEP0204(2002)019[arXiv:hep-th/0203083].

最新英语语言学树型图详细讲解

树形图详细讲解 1. Indicate the category of each word in the following sentences. a) The old lady suddenly left. Det A N Qual V b) The car stopped at the end of the road. Det N V P Det N P Det N c) The snow might have blocked the road. Det N Aux Aux V Det N d) He never appears quite mature. N Qual V Deg A 2. The following phrases include a head, a complement, and a specifier. Draw the appropriate tree structure for each. a) full of people AP A P N full of people b) a story about a sentimental girl NP NP PP Det N P NP Det A N a story about a sentimental girl c) often read detective stories VP Qual V NP A N often read detective stories

d) the argument against the proposals NP NP PP Det N P NP Det N the argument against the proposals e) move towards the window VP V PP P Det N move towards the window 3. Draw phrase structure trees for each of the following sentences. a) The jet landed. InflP(=S) NP Infl VP Det N Pst V The jet landed b) Mary became very ill. InflP(=S) NP Infl VP N Pst V AP Deg A Mary became very ill

语言学树形图课后问题详解Word版

树形图详细讲解 网上的相对理想的树形图答案，注意正两 点： 1. 短语和中心词在一竖线上 2. 含有形容词修饰语的名词短语的画法 NP Det N A N a little boy 1. Indicate the category of each word in the following sentences. a) The old lady suddenly left. Det A N Qual V b) The car stopped at the end of the road. Det N V P Det N P Det N c) The snow might have blocked the road. Det N Aux Aux V Det N d) He never appears quite mature. N Qual V Deg A 2. The following phrases include a head, a complement, and a specifier. Draw the appropriate tree structure for each. a) full of people AP A P N

full of people b) a story about a sentimental girl NP NP PP Det N P NP Det A N a story about a sentimental girl c) often read detective stories VP Qual V NP A N often read detective stories

高等数学应用案例讲解

高等数学应用案例案例1、如何调整工人的人数而保证产量不变 一工厂有x名技术工人和y名非技术工人，每天可生产的产品产量为 , (=（件） f2 ) x x y y 现有16名技术工人和32名非技术工人，如何调整非技术工人的人数，可保持产品产量不变？ 解：现在产品产量为(16,32)8192 f=件，保持这种产量的函数曲线为y (= x f。对于任一给定值x，每增加一名技术工人时y的变化量即为, 8192 ) dy。而由隐函数存在定理，可得 这函数曲线切线的斜率 dx 所以，当增加一名技术工人时，非技术工人的变化量为 dy。 当16,32 ==时，可得4-= x y dx 因此，要增加一个技术工人并要使产量不变，就要相应地减少约4名非技术工人。 下面给出一个初等数学解法。令 c：每天可生产的产品产量； x；技术工人数； y；非技术工人数； x?；技术工人增加人数； y?；在保持每天产品产量不变情况下，当技术工人由16名增加到17名时，非技术人员要增加（或减少）的人数。 由已知列方程：

（1）当技术工人为16名，非技术工人为32名时，每天的产品产量为c ，则有方程： c y x =?020 （1） （2）当技术工人增加了1名时，非技术工人应为（y y ?+0）名，且每 天的产品产量为c ，则有方程： c y y x x =?+??+)()(020 （2） 联立方程组（1）、（2），消去c 得： 即 [] 002020)/(y y x x x y -??+=????????+--=20200)(1x x x y 代入x y x ?,,00，得：46.3-≈-≈?y 名，即减少4名非技术工人。 比较这两种解法我们可以发现，用初等数学方法计算此题的工作量很大，究其原因，我们注意到下面之展开式： 从此展开式我们可以看到，初等数学方法不能忽略掉高阶无穷小： 0)x ( )1(31 04120→????? ???-+???? ???--∞=-∑n n n x x n x x （3） 而高等数学方法却利用了隐函数求导，忽略掉高阶无穷小（3），所以计算较容易。

高等数学同济大学版课程讲解函数的极限

高等数学同济大学版课程讲解函数的极限 Coca-cola standardization office【ZZ5AB-ZZSYT-ZZ2C-ZZ682T-ZZT18】

课 时 授 课 计 划 课次序号：03 一、课 题：§函数的极限 二、课 型：新授课 三、目的要求：1.理解自变量各种变化趋势下函数极限的概念； 2.了解函数极限的性质． 四、教学重点：自变量各种变化趋势下函数极限的概念． 教学难点：函数极限的精确定义的理解与运用． 五、教学方法及手段：启发式教学，传统教学与多媒体教学相结合． 六、参考资料：1.《高等数学释疑解难》，工科数学课程教学指导委员会编， 高等教育出版社； 2.《高等数学教与学参考》，张宏志主编，西北工业大学出版社． 七、作业：习题1–31（2），2（3），3，6 八、授课记录： 九、 授课 效果分析： 第三节函数的极限 复习 1.数列极限的定义：lim 0,N,N n n n x a n x a εε→∞ =??>?>-<当时， ； 2.收敛数列的性质：唯一性、有界性、保号性、收敛数列与其子列的关系． 在此基础上，今天我们学习应用上更为广泛的函数的极限．与数列极限不同的是，对 于函数极限来说，其自变量的变化趋势要复杂的多． 一、x →∞时函数的极限 对一般函数yf （x ）而言，自变量无限增大时，函数值无限地接近一个常数的情形与数列极限类似，所不同的是，自变量的变化可以是连续的． 定义1若?ε＞0，?X ＞0，当x ＞X 时，相应的函数值f （x ）∈U （A ，ε）（即|f (x )A |<ε），则称x →∞时，f （x ）以A 为极限，记为lim x →+∞ f （x ）A ． 若?ε＞0，?X ＞0，当x ＜X 时，相应的函数值f （x ）∈U （A ，ε）（即|f (x )A |<ε），则称

教你如何画语言学树型图

树形图详细讲解 1、 Indicate the category of each word in the following sentences、 a) The old lady suddenly left、 Det A N Qual V b) The car stopped at the end of the road、 Det N V P Det N P Det N c) The snow might have blocked the road、 Det N Aux Aux V Det N d) He never appears quite mature、 N Qual V Deg A 2、 The following phrases include a head, a plement, and a specifier、 Draw the appropriate tree structure for each、 a) full of people AP A P N full of people b) a story about a sentimental girl NP NP PP Det N P NP Det A N a story about a sentimental girl c) often read detective stories VP Qual V NP A N often read detective stories d) the argument against the proposals NP NP PP Det N P NP Det N the argument against the proposals e) move towards the window VP

高等数学基础例题讲解

第1章 函数的极限与连续 例1．求 lim x x x →． 解：当0>x 时，0 00lim lim lim 11x x x x x x x + ++ →→→===， 当0

最新高等数学(上)重要知识点归纳讲解学习

高等数学（上）重要知识点归纳 第一章 函数、极限与连续 一、极限的定义与性质 1、定义（以数列为例） ,,0lim N a x n n ?>??=∞ →ε当N n >时，ε<-||a x n 2、性质 (1) )()()(lim 0 x A x f A x f x x α+=?=→，其中)(x α为某一个无穷小。 (2)(保号性）若0)(lim 0 >=→A x f x x ，则,0>?δ当),(0δx U x o ∈时，0)(>x f 。 (3)*无穷小乘以有界函数仍为无穷小。 二、求极限的主要方法与工具 1、*两个重要极限公式 (1)1sin lim =??→? (2)e =? +? ∞ →?)11(lim 2、两个准则 (1) *夹逼准则 (2)单调有界准则 3、*等价无穷小替换法 常用替换：当0→?时 （1)??~sin （2）??~tan （3）??~arcsin （4)??~arctan （5）??+~)1ln( （6）?-?~1e （7）221 ~cos 1??- （8）n n ?-?+~11

4、分子或分母有理化法 5、分解因式法 6用定积分定义 三、无穷小阶的比较* 高阶、同阶、等价 四、连续与间断点的分类 1、连续的定义* )(x f 在a 点连续 )()()()()(lim 0lim 0 a f a f a f a f x f y a x x ==?=?=??-+→→? 2、间断点的分类?? ?? ? ? ?????????? ?其他震荡型（来回波动） ） 无穷型（极限为无穷大第二类但不相等）跳跃型（左右极限存在可去型（极限存在） 第一类 3、曲线的渐近线* a x x f A y A x f a x x =∞===→∞ →则存在渐近线：铅直渐近线：若则存在渐近线：水平渐近线：若,)(lim )2(,)(lim )1( 五、闭区间连续函数性质 1、最大值与最小值定理 2、介值定理和零点定理

高等数学试题及答案讲解

《高等数学》 一.选择题 1. 当0→x 时，)1ln(x y +=与下列那个函数不是等价的 （ ） A)、x y = B)、x y sin = C)、x y cos 1-= D)、1-=x e y 2. 函数f(x)在点x 0极限存在是函数在该点连续的（ ） A )、必要条件 B )、充分条件 C )、充要条件 D )、无关条件 3. 下列各组函数中，)(x f 和)(x g 不是同一函数的原函数的有（ ）. A)、()()() 222 1 ,21)(x x x x e e x g e e x f ---=-= B) 、(( )) ()ln ,ln f x x g x x ==- C)、()()x x g x x f --=-=1arcsin 23,12arcsin )( D)、()2 tan ,sec csc )(x x g x x x f =+= 4. 下列各式正确的是（ ） A ）、2ln 2x x x dx C =+? B ）、sin cos tdt t C =-+? C ）、 2arctan 1dx dx x x =+? D ）、2 11 ()dx C x x -=-+? 5. 下列等式不正确的是（ ）. A ）、 ()()x f dx x f dx d b a =??????? B ）、()()()[]()x b x b f dt x f dx d x b a '=???? ??? C ）、()()x f dx x f dx d x a =??????? D ）、()()x F dt t F dx d x a '=???? ??'? 6. 0 ln(1)lim x x t dt x →+=?（ ） A ）、0 B ）、1 C ）、2 D ）、4 7. 设bx x f sin )(=，则=''?dx x f x )(（ ） A ）、 C bx bx b x +-sin cos B ） 、C bx bx b x +-cos cos C ）、C bx bx bx +-sin cos D ）、C bx b bx bx +-cos sin

高等数学知识点归纳知识讲解

第一讲: 极限与连续 一. 数列函数: 1. 类型: (1)数列: *()n a f n =; *1()n n a f a += (2)初等函数: (3)分段函数: *0102()(),()x x f x F x x x f x ≤?=? >?; *0 ()(), x x f x F x x x a ≠?=?=?;* (4)复合(含f )函数: (),()y f u u x ?== (5)隐式(方程): (,)0F x y = (6)参式(数一,二): () () x x t y y t =?? =? (7)变限积分函数: ()(,)x a F x f x t dt = ? (8)级数和函数(数一,三): 0 (),n n n S x a x x ∞ ==∈Ω∑ 2. 特征(几何): (1)单调性与有界性(判别); (()f x 单调000,()(()())x x x f x f x ??--定号) (2)奇偶性与周期性(应用). 3. 反函数与直接函数: 1 1()()()y f x x f y y f x --=?=?= 二. 极限性质: 1. 类型: *lim n n a →∞; *lim ()x f x →∞ (含x →±∞); *0 lim ()x x f x →(含0x x ± →) 2. 无穷小与无穷大(注: 无穷量): 3. 未定型: 000,,1,,0,0,0∞ ∞∞-∞?∞∞∞ 4. 性质: *有界性, *保号性, *归并性 三. 常用结论: 11n n →, 1(0)1n a a >→, 1()max(,,)n n n n a b c a b c ++→, ()00! n a a n >→ 1(0)x x →→∞, 0lim 1x x x +→=, lim 0n x x x e →+∞=, ln lim 0n x x x →+∞=, 0 lim ln 0n x x x + →=, 0, x x e x →-∞ ?→?+∞→+∞ ?

高等数学求极限的常用方法(附例题和详细讲解)

高等数学求极限的14种方法 一、极限的定义 1.极限的保号性很重要：设 A x f x x =→)(lim 0 ， （i ）若A 0>，则有0>δ，使得当δ<-<||00x x 时，0)(>x f ； （ii ）若有,0>δ使得当δ<-<||00x x 时，0A ,0)(≥≥则x f 。 2.极限分为函数极限、数列极限，其中函数极限又分为∞→x 时函数的极限和0x x →的极限。要特别注意判定极限是否存在在： （i ）数列{} 的充要条件收敛于a n x 是它的所有子数列均收敛于a 。常用的是其推论，即“一个数列收敛于a 的充要条件是其奇子列和偶子列都收敛于a ” （ii ）A x x f x A x f x =+∞ →= -∞ →? =∞ →lim lim lim )()( (iii) A x x x x A x f x x =→=→?=→+ - lim lim lim 0 )( (iv)单调有界准则 （v ）两边夹挤准则（夹逼定理/夹逼原理） （vi ）柯西收敛准则（不需要掌握）。极限 ) (lim 0 x f x x →存在的充分必要条件是： εδεδ<-∈>?>?|)()(|)(,0,021021x f x f x U x x o 时，恒有、使得当 二．解决极限的方法如下： 1.等价无穷小代换。只能在乘除．． 时候使用。例题略。 2.洛必达（L’ho spital ）法则（大题目有时候会有暗示要你使用这个方法） 洛必达法则（定理） 设函数f(x ）和F(x ）满足下列条件： ⑴x→a 时，lim f(x)=0,lim F(x)=0; ⑵在点a 的某去心邻域内f(x ）与F(x ）都可导，且F(x ）的导数不等于0; ⑶x→a 时，lim(f'(x)/F'(x)）存在或为无穷大 则 x→a 时，lim(f(x)/F(x))=lim(f'(x)/F'(x)) 注： 它的使用有严格的使用前提。首先必须是X 趋近，而不是N 趋近，所以面对数列极限时候先要转化成求x

高等数学应用案例讲解.

高等数学应用案例 案例1、如何调整工人的人数而保证产量不变 一工厂有x 名技术工人和y 名非技术工人，每天可生产的产品产量为 y x y x f 2),(= （件） 现有16名技术工人和32名非技术工人，如何调整非技术工人的人数，可保持产品产量不变？ 解：现在产品产量为(16,32)8192f =件，保持这种产量的函数曲线为8192),(=y x f 。对于任一给定值x ，每增加一名技术工人时y 的变化量即为这函数曲线切线的斜率dx dy 。而由隐函数存在定理，可得 y f x f dx dy ????= 所以，当增加一名技术工人时，非技术工人的变化量为 x y y f x f dx dy 2-=????= 当16,32x y ==时，可得4-=dx dy 。 因此，要增加一个技术工人并要使产量不变，就要相应地减少约4名非技术工人。 下面给出一个初等数学解法。令 c ：每天可生产的产品产量； 0x ；技术工人数； 0y ；非技术工人数；

x ?；技术工人增加人数； y ?；在保持每天产品产量不变情况下，当技术工人由16名增加到17名时，非技术人员要增加（或减少）的人数。 由已知列方程： （1）当技术工人为16名，非技术工人为32名时，每天的产品产量为c ，则有方程： c y x =?020 （1） （2）当技术工人增加了1名时，非技术工人应为（y y ?+0）名， 且每天的产品产量为c ，则有方程： c y y x x =?+??+)()(020 （2） 联立方程组（1）、（2），消去c 得： )(020020y y x x y x ?+??+=?）（ 即 [] 002020)/(y y x x x y -??+=????????+--=20200)(1x x x y ????????????? ????? ???+--=200111x x y 代入x y x ?,,00，得：46.3-≈-≈?y 名，即减少4名非技术工人。 比较这两种解法我们可以发现，用初等数学方法计算此题的工作量很大，究其原因，我们注意到下面之展开式： ∑∞=--???? ???-+???? ???-?=???? ???+-110120020)1(32111n n n x x n x x x x x x 从此展开式我们可以看到，初等数学方法不能忽略掉高阶无穷

高等数学例题讲解(提高篇)

第1章 函数的极限与连续 例1．求下列极限： 1）)1ln(1 2 )(cos lim x x x +→ 2）βαβ αβα--→e e lim 解：1）原式 2 201 ln cos ln cos lim ln(1) ln(1) lim x x x x x x e e →++→==，而 21)2(lim 22sin 2lim )1ln()2sin 21ln(lim )1ln(cos ln lim 22 02 2022020-=-=-=+-=+→→→→x x x x x x x x x x x x 所以， e e x x x 1) (cos lim 2 1 ) 1ln(1 2= =- +→ 2）原式11 lim lim e e e e αβαββ βαβαβαβαβ--→→--==-- 令t =-βα，当βα→时，0→t ，所以， 1lim 1lim 1lim 00==-=--→→-→t t t e e t t t βαβαβα． 从而，ββ αβαβαe e e =--→lim ． 例2．求 lim(1) p x x mx →-，其中 m 、p 是正整数． 解：因为 mp mx mp mx x p mx mx mx ]) 1[(1) 1()1(1)(1 --- -= -=-， 令mx u -=，当0→x 时，0→u 1 1 111lim(1)lim lim [(1) ] [(1)] p mp x mp x x u mp mp mx u mx e e mx u -→→→--=== =-+． 例3．若()0f x >，0 lim ()(0)x x f x A A →=> 且0 lim x x → lim x x → 解：设 lim x x a →= a β=+，β是0x x →时的无穷小量， 22()2f x a a ββ=++ 222 lim ()lim(2)x x x x f x a a a ββ→→=++= 由题应有：2 A a = ，a = a = x x →= 例4．证明：半径为R 的圆面积2 R S π= 证：做圆的内接正n （3≥n ）边形，如图1-13所示，记AOP n ∠=α其面积为 n R n R n R R n OP AB n S n n n n π ααα2sin 22sin 2cos sin 22222==?=?= 当边数n 取3，4， ,5，对应的面积3S ，4S ， ,5S 构成了一数列}{n S ，图1-13

大学高等数学知识点知识讲解

大学高等数学知识点整理 公式，用法合集 极限与连续 一. 数列函数: 1. 类型: (1)数列: *()n a f n =; *1()n n a f a += (2)初等函数: (3)分段函数: *0102()(),()x x f x F x x x f x ≤?=?>?; *0 ()(), x x f x F x x x a ≠?=?=?;* (4)复合(含f )函数: (),()y f u u x ?== (5)隐式(方程): (,)0F x y = (6)参式(数一,二): () ()x x t y y t =??=? (7)变限积分函数: ()(,)x a F x f x t dt = ? (8)级数和函数(数一,三): 0 (),n n n S x a x x ∞ ==∈Ω∑ 2. 特征(几何): (1)单调性与有界性(判别); (()f x 单调000,()(()())x x x f x f x ??--定号) (2)奇偶性与周期性(应用). 3. 反函数与直接函数: 1 1()()()y f x x f y y f x --=?=?= 二. 极限性质: 1. 类型: *lim n n a →∞; *lim ()x f x →∞ (含x →±∞); *0 lim ()x x f x →(含0x x ± →) 2. 无穷小与无穷大(注: 无穷量): 3. 未定型: 000,,1,,0,0,0∞ ∞∞-∞?∞∞∞ 4. 性质: *有界性, *保号性, *归并性 三. 常用结论: 11n n →, 1(0)1n a a >→, 1()max(,,)n n n n a b c a b c ++→, ()00! n a a n >→

2020年中考数学专题复习：树状图(含解析)

例谈画树状图 一、显性放回 例1 现有形状、大小和颜色完全一样的三张卡片，上面分别标有数字“1”、“2”、“3”．第一次从这三张卡片中随机抽取一张，记下数字后放回；第二次再从这三张卡片中随机 抽取一张并记下数字．请用画树状图的方法表示出上述试验所有可能的结果，并求第二 次抽取的数字大于第一次抽取的数字的概率． 分析从题中文字“记下数字后放回”知本题属于“显性放回”．本题中的事件是摸 两次卡片，看卡片的数字，由此可以确定事件包括两个环节．摸第一张卡片，放回去，再摸第二张卡片，所以树状图应该画两层．第一张卡片的数字可能是1，2，3等3个中的一个，所以第一层应画3个分叉；再看第二层，由于放回，第二个乒乓球的数字可能是3个中的一个，所以第二层应接在第一层的3个分叉上，每个小分支上，再有3个分叉．画出树状图，这样共得到3x3＝9种情况，从中找出第二次抽取的数字大于第一次抽取的数字的情况，再求出概率． 解根据题意画树状图如图1． 所有可能的结果为： (1，1)，(1，2)，(1，3)，(2，1)，(2，2)，(2，3)，(3，1)，(3，2)，(3，3)． ∵有9种等可能的结果，第二次抽取的数字大于第一次抽取的数字的只有3种， ∴P(第二次抽取的数字大于第一次抽取的数字)＝1 3 ． 二、显性不放回 例2 一个不透明的布袋里装有4个大小、质地都相同的乒乓球，球面上分别标有数字1，－2，3，－4．小明先从布袋中随机摸出一个球(不放回去)，再从剩下的3个球中随机摸出第二个乒乓球． (1)共有_______种可能的结果； (2)请用画树状图的方法求两次摸出的乒乓球的数字之积为偶数的概率． 分析从文字条件“不放回去”知，本题属于“显性不放回”．本题中的事件是摸两个乒乓球，看乒乓球的数字，由此可以确定事件包括两个环节，所以树状图应该画两层．第一个乒乓球的数字可能是1，－2，3，－4等4个中的一个，所以第一层应画4个分叉；由于不放回，第二个乒乓球的数字可能是剩下的3个中的一个，所以第二层应接在第一层的4个分叉上，每个小分支上，再有3个分叉，画出树状图． 解根据题意画树状图如图2． (1)由图2可知，共有12种可能结果，分别为： (1，－2)，(1，3)．(1，－4)，(－2，1)，(－2，3)，(－2．－4)，(3，1)，(3，－2)，(3，－4)，(－4，1)，(－4，－2)，(－4，3)． 故答案为12．

高等数学(专升本)学习指南讲解

高等数学（专升本）学习指南 一、判断题 1．2(ln log )'x x += 11 ln 2 x x + ，是否正确（ 对 ） 解：对原式直接求导即可得到。 ()2ln 11(ln log )'ln ln 2ln 2 x x x x x x '??'+=+=+ ? ?? 2. 函数sin y x x =-在区间[0,2]π上是单调减少，是否正确（ 错 ） 解：函数y 的导数 1cos 0y x '=-≥ 其中，[]0,2x π∈ 所以函数y 在该区间是个递增函数。 3．22 32sin lim 12cos x x x x x x x →∞-+=--，是否正确（ 对 ） 解：原式：2232sin lim 2cos x x x x x x x →∞-+-- 原式分子sin x 有界，分母cos x 有界，其余项均随着x 趋于无穷而趋于无穷。 这样，原式的极限取决于分子、分母高阶项的同阶系数之比。 得到：22 2 232sin lim lim 12cos x x x x x x x x x x →∞→∞-+?=-- 4．设2tan y x =，则22tan sec dy x xdx =，是否正确（ 对 ） 解：对原式关于x 求导，并用导数乘以dx 项即可，注意三角函数求导规则。

()() 22'tan tan 2tan 2tan sec y x d x x dx x x '=== 所以， 22tan sec dy x x dx =，即22tan sec dy x xdx = 5. 221 2 x x e dx e c --=-+?，是否正确（ 对 ） 解：()22211222 x x x e dx e d x e C ---=- -=-+?? 6．(sin cos )'cos sin x x x x +=-，是否正确（对 ） 解：(sin cos )'sin cos cos sin x x x x x x ''+=+=- 7．函数2(2)y x =-在区间[0,4]上极小值是1，是否正确（错 ） 解：对y 关于x 求一阶导，并令其为0，得到()220x -=； 解得x 有驻点：x=2，代入原方程验证此为其极小值点。 8. sin lim x x x →∞=1，是否正确（ 错 ） 解：因为 1sin 1x -≤≤有界， 所以 sin lim 0x x x →∞= 9．设1arctan y x =，则2 1 1dy dx x =+，是否正确（错） 解：222 11 11arctan 111dy y x x x dx x '? ??? '== ?-=-= ? ?+? ?? ???+ ??? 所以，2 1 1dy dx x =-+ 10．2 21x dx x =+? c x ++)1ln(2 ，是否正确（ 对 ）

最全最实用的高等数学公式大全讲解

高数工本阶段公司 空间解析几何和向量代数： 。 代表平行六面体的体积为锐角时， 向量的混合积：例：线速度：两向量之间的夹角：是一个数量轴的夹角。 与是向量在轴上的投影：点的距离：空间ααθθθ??,cos )(][..sin ,cos ,,cos Pr Pr )(Pr ,cos Pr )()()(22 2 2 2 2 2 212121*********c b a c c c b b b a a a c b a c b a r w v b a c b b b a a a k j i b a c b b b a a a b a b a b a b a b a b a b a b a a j a j a a j u AB AB AB j z z y y x x M M d z y x z y x z y x z y x z y x z y x z y x z z y y x x z z y y x x u u ??==??=?=?==?=++?++++=++=?=?+=+?=-+-+-==

（马鞍面）双叶双曲面：单叶双曲面：、双曲面： 同号） （、抛物面：、椭球面：二次曲面： 参数方程：其中空间直线的方程：面的距离：平面外任意一点到该平、截距世方程：、一般方程：，其中、点法式：平面的方程： 1 1 3,,2221 1};,,{,1 302),,(},,,{0)()()(122 222222 22222 222 22220000002 220000000000=+-=-+=+=++??? ??+=+=+===-=-=-+++++= =++=+++==-+-+-c z b y a x c z b y a x q p z q y p x c z b y a x pt z z nt y y mt x x p n m s t p z z n y y m x x C B A D Cz By Ax d c z b y a x D Cz By Ax z y x M C B A n z z C y y B x x A 多元函数微分法及应用 z y z x y x y x y x y x F F y z F F x z z y x F dx dy F F y F F x dx y d F F dx dy y x F dy y v dx x v dv dy y u dx x u du y x v v y x u u x v v z x u u z x z y x v y x u f z t v v z t u u z dt dz t v t u f z y y x f x y x f dz z dz z u dy y u dx x u du dy y z dx x z dz - =??-=??=? -?? -??=-==??+??=??+??===??? ??+?????=??=?????+?????==?+?=≈???+??+??=??+??= ， ，　隐函数＋， ， 隐函数隐函数的求导公式： 　 时， ，当 ： 多元复合函数的求导法全微分的近似计算： 全微分：0),,()()(0),(),(),()],(),,([)](),([),(),(22

高等数学(同济大学版) 课程讲解 1.3函数的极限

课时授课计划 课次序号：03 一、课题：§1.3 函数的极限 二、课型：新授课 三、目的要求：1.理解自变量各种变化趋势下函数极限的概念； 2.了解函数极限的性质． 四、教学重点：自变量各种变化趋势下函数极限的概念． 教学难点：函数极限的精确定义的理解与运用． 五、教学方法及手段：启发式教学，传统教学与多媒体教学相结合． 六、参考资料：1.《高等数学释疑解难》，工科数学课程教学指导委员会编， 高等教育出版社； 2.《高等数学教与学参考》，张宏志主编，西北工业大学出版社． 七、作业：习题1–3 1（2），2（3），3，6 八、授课记录： 九、授课效果分析：

第三节 函数的极限 复习 1.数列极限的定义：lim 0,N,N n n n x a n x a εε→∞ =??>?>-<当时， ； 2.收敛数列的性质：唯一性、有界性、保号性、收敛数列与其子列的关系． 在此基础上，今天我们学习应用上更为广泛的函数的极限． 与数列极限不同的是，对 于函数极限来说，其自变量的变化趋势要复杂的多． 一、x →∞时函数的极限 对一般函数y =f （x ）而言，自变量无限增大时，函数值无限地接近一个常数的情形与数列极限类似，所不同的是，自变量的变化可以是连续的． 定义1 若?ε＞0，?X ＞0，当x ＞X 时，相应的函数值f （x ）∈U （A ，ε）（即|f (x )-A |<ε），则称x →+∞时，f （x ）以A 为极限，记为lim x →+∞ f （x ）=A ． 若?ε＞0，?X ＞0，当x ＜-X 时，相应的函数值f （x ）∈U （A ，ε）（即|f (x )-A |<ε），则称x →-∞时，f （x ）以A 为极限，记为lim x →-∞ f （x ）=A ． 例1 证明lim x =0． 证 0 -?ε＞00-＜εε， 即x ＞ 2 1 ε．因此，?ε＞0，可取X = 2 1 ε，则当x ＞X 0-＜ε，故由定义1得 lim x =0． 例2 证明lim 100x x →-∞ =． 证 ?ε＞0，要使100x -=10x ＜ε，只要x ＜l gε．因此可取X =｜l gε｜+1，当x ＜-X 时， 即有｜10x -0｜＜ε，故由定义1得lim x →+∞ 10x =0． 定义2 若?ε＞0，?X ＞0，当｜x ｜＞X 时，相应的函数值f （x ）∈U （A ，ε）（即|f (x )-A |<ε），则称x →∞时，f （x ）以A 为极限，记为lim x →∞ f （x ）=A ． 为方便起见，有时也用下列记号来表示上述极限： f （x ）→A （x →+∞）；f （x ）→A （x →-∞）；f （x ）→A （x →∞）．

《高等数学》课程规划详解

《高等数学》课程规划 第一章函数 基本要求：熟悉函数的基本性质、反函数、基本初等函数；掌握函数的性质、复合函数、初等函数。 重点：函数的性质、反函数。 难点：初等函数

第二章极限与连续 基本要求：了解复合函数的连续性、初等函数的连续性；熟悉极限的基本理论及定义，极限存在准则，闭区间上连续函数的性质；掌握两个重要极限、无穷小的比较、等价无穷小代换、连续函数的定义及运算、函数间断点的类型。 重点：函数的性质；极限的性质、四则运算法则；无穷小阶的比较；两个重要极限。 难点：函数在某点处的左、右极限；利用等价无穷小代换求极限；用两个重要极限求极限。

第三章导数与微分 基本要求：了解导数的概念及意义，微分形式的不变性；熟悉微分的定义，导数概念与微分概念的联系与区别；掌握复合函数、隐函数及含参数方程所确定函数的求导运算。 重点：导数概念、函数的可导性与连续性的关系；复合函数求导的链式法则；隐函数求导；由参数方程所确定的函数的导数；函数可微性与可导性的关系。 难点：导数与微分在几何和物理上的应用。

第四章中值定理与导数的应用 基本要求：熟悉微分中值定理；掌握洛必达法则、函数的单调性与曲线的凹凸性、函数极值、最值的求法；了解函数图形的描绘。 重点：洛比达法则；函数单调性、凹凸性的判定；函数极值、最值的求法。 难点：微分中值定理及其应用；描绘函数的图形（包括渐近线）。

第五章不定积分 基本要求：了解积分表的使用；熟悉不定积分的概念；掌握不定积分的运算。重点：不定积分的基本性质、基本积分公式；两类换元积分法和分部积分法。难点：原函数和不定积分的概念；有理函数的不定积分。

《高等数学》不定积分课后习题详细讲解

不定积分容概要

课后习题全解 习题4-1 1.求下列不定积分： 知识点：直接积分法的练习——求不定积分的基本方法。 思路分析：利用不定积分的运算性质和基本积分公式，直接求出不定积分！★(1) 思路: 被积函数52x- =，由积分表中的公式（2）可解。 解：5 3 22 2 3 x dx x C -- ==-+ ? ★ (2)dx - ? 思路:根据不定积分的线性性质，将被积函数分为两项，分别积分。 解：114 111 333 222 3 ()2 4 dx x x dx x dx x dx x x C -- -=-=-=-+ ???? ★(3)2 2x x dx + ?（） 思路:根据不定积分的线性性质，将被积函数分为两项，分别积分。 解：223 2 1 22 ln23 x x x x dx dx x dx x C += +=++ ??? （） ★(4)3) x dx - 思路:根据不定积分的线性性质，将被积函数分为两项，分别积分。 解：3153 2222 2 3)32 5 x dx x dx x dx x x C -=-=-+ ??

★★(5)4223311 x x dx x +++? 思路:观察到422223311311 x x x x x ++=+++后，根据不定积分的线性性质，将被积函数分项，分别积分。 解：42232233113arctan 11x x dx x dx dx x x C x x ++=+=++++??? ★★(6)2 21x dx x +? 思路:注意到222221111111x x x x x +-==-+++，根据不定积分的线性性质，将被积函数分项， 分别积分。 解：2221arctan .11x dx dx dx x x C x x =-=-+++??? 注：容易看出(5)(6)两题的解题思路是一致的。一般地，如果被积函数为一个有理的假分式，通常先将其分解为一个整式加上或减去一个真分式的形式，再分项积分。 ★(7)x dx x x x ?34134（-+-）2 思路:分项积分。 解：3411342x dx xdx dx x dx x dx x x x x --=-+-?? ???34134（-+-）2 223134ln ||.423 x x x x C --=--++ ★(8) 23(1dx x -+? 思路:分项积分。 解： 2231(323arctan 2arcsin .11dx dx x x C x x -=-=-+++?? ★★(9) 思路=11172488x x ++==，直接积分。 解：715888.15 x dx x C ==+? ★★(10)221(1)dx x x +? 思路:裂项分项积分。