lecture8 high order derivative and differential
2.3 Partial derivative and high order derivatives
When we fix y at y0 and let x varies from x0
to x0 +Δx, function z obtains a partial
increment at P0 with respect to x:
f ( x0 x , y0 ) f ( x0 , y0 ) x z .
Now let’s turn to consider the relation between the continuity and the existence of partial derivatives.
xy 2 x y2 Example 4. Let f ( x, y ) 0
2z 2z 2 ye y , x 2e y (1 y ) x 2e y x 2e y (2 y ); x 2 y 2
2z 2 xe y (1 y ), xy
2z 2 xe y (1 y ). yx
Theorem 1. If the mixed partial derivatives fxy (x ,y)
0.
Note: The partial derivatives of f exist, but we
know f is discontinuous at (0,0).
So we see that
partial derivatives exist Of course, continuity partial derivatives exist. continuity.
f d ( x 2 y y 3 ) 2 xy. x dx
f d ( x 2 y y 3 ) Similarly x2 3 y2 . y dy
Lecture 8
• Yankee bonds must meet the requirements of the SEC, just like U.S. domestic bonds.
• Many borrowers find this level of regulation burdensome and prefer to raise U.S. dollars in the Eurobond market.
Straight Fixed Rate Debt
• These are “plain vanilla” bonds with a specified coupon rate and maturity and no options attached.
• Since most Eurobonds are bearer bonds,
small…as long as there is no change in the default risk.
Question
• Your firm has just issued five-year floatingrate notes indexed to six-month U.S. dollar Libor plus ¼ percent. What is the amount of the first coupon payment your firm will pay U.S. $1,000 of the face value, if sixmonth Libor is currently 7.2%?
Lecture 7 DB
CONTAINS
PART
• part number • part description • unit cost
Generalisation
NAME ADDRESS EMPLOYEE NO. EMPLOYEE DATE HIRED
ISA
ISA
ISA
HOURLY EMPLOYEE
SALARIED EMPLOYEE
CONSULTANT
• EMPLOYEE NO •HOURLY RATE
•EMPLOYEE NO •ANNUAL SALARY •STOCK OPTION
•EMPLOYEE NO •CONTACT NO. •DAILY RATE
Data Modelling Exercise
University Student Recreation Centre Database Students can only use the centre if they have paid their recreation fees in full. The centre will also allow a faculty to purchase a membership as well. Members are allowed to check out sports equipment such as basketballs, softball bats and balls, tennis rackets, badminton rackets and table tennis rackets that can be used at the facility. When the members check out equipment, an equipment-issue form is completed listing the membership number and equipment being used. This form must list at least one piece of equipment in order to be retained in the file. Otherwise it is discarded. A member of staff is employed to monitor the checkout and the use of the sporting equipment. Every employee is assigned to one of two departments: maintenance or general staff. The centre has 10 tennis courts. These courts may be reserved up to one week in advance. Reservations can be made via the equipment checkout window. The centre also operates a small accessory shop where some sporting equipment and clothing is sold. The sporting goods include tennis balls, table tennis balls, bandages, etc. Sportswear bearing the university emblem and mascot as well as a limited assortment of some name-brand sportswear are sold. Finally, the centre sponsors a limited number of classes in officiating various sports . A general rule is that instructors often teach in more than one sport but there is never more than one class offered in a particular sport.
Lecture_8_TH_2014
• Offsets similar to a very low freq. noise (appear constant)
Tang, He Lecture 8 10
Practical Input‐Referred Offsets
Vt
Tang, He
AVt WL A
– With mismatch, Vod=certain value when Vid=0 – Define this “certain value”=VOS,out – More commonly used, input‐referred offset voltage
VOS ,in VOS ,out A
VOS ,in I D VOV (W / L) gm 2 W /L
• Proportional to VOV and W/L mismatch – Trade‐off with linearity, area and power
Tang, He Lecture 8 7
Offsets Analysis Due to Mismatch
Tang, He
Lecture 8
3
Offsets Analysis Due to Mismatch
• Calculate VOS,in (consider R mismatch)
RD1 RD , RD 2 RD RD I SS I SS VD1 VDD RD , VD 2 VDD RD RD 2 2
• Chopper stabilization – 1/f noise, offsets, drift
– Modulate the input signal up and demodulate the signal down to baseband – 1/f noise removed but thermal noise still existed – Suited for sensing μV‐level input signal circuits
概率论讲义Lecture8
Lecture8.March10,20101Convergence of Markov chain transition kernelsTheorem1.1[Convergence of transition kernels]Let X be an irreducible aperiodic Markov chain with countable state space S.If the chain is transient or null recurrent,thenlimn→∞Πn(x,y)=0∀x,y∈S.(1.1) If the chain is positive recurrent with stationary distributionµ,thenlimn→∞Πn(x,y)=µ(y)∀x,y∈S.(1.2)Theorem1.1is in fact equivalent to the Renewal Theorem,which was proved last time.When the Markov chain is positive recurrent,Theorem1.1admits an elegant proof by coupling, which is worth explaining.Proof of Theorem1.1with positive recurrence.When X is positive recurrent,the Markov chain admits a unique stationary probability distributionµ.Let X1be a copy of the Markov chain with initial distributionµ.Let X2be an independent copy of the Markov chain with initial distributionδx for some x∈S.Then we claim thatτ:=inf{n≥0:X1n= X2n}<∞almost surely.If the claim holds,then we can couple X1and X2by defining two new Markov chains˜X1and˜X2on the same probability space such that˜X i n=X i n for n≤τ, and˜X i n=X1n for n>τ,i.e.,the second Markov chain starts following the trajectory of the first Markov chain as soon as they ing the strong Markov property of the pair of independent chains(X1,X2),it is clear that˜X i is equally distributed with X i.The claim τ<∞almost surely implies that P(˜X1n=˜X2n)↓0as n→∞.Sinceµ(y)=P(˜X1n=y)for all n∈N,whileΠn(x,y)=P(˜X2n=y),we have1 2y∈S|Πn(x,y)−µ(y)|≤P(˜X1n=˜X2n)=P(τ>n),which decreases to0as n→∞.This in fact proves convergence ofΠn(x,·)toµin total variational distance.To verify thatτ<∞a.s.,we only need to check that(X1,X2)defines an irreducible re-current Markov chain.The fact that(X1,X2)is Markov is clear.By aperiodicity assumption, P x(X1n=x)>0for all n sufficiently large,and hence P x(X1n=y)for all n sufficiently large for any given y.By the independence of X1and X2,it follows that for any two pairs(x1,x2) and(y1,y2),P x1,x2(X1n=y1,X2n=y2)=P x1(X1n=y1)P x2(X2n=y2)>0for all n sufficiently large,which implies the irreducibility of(X1,X2).Clearlyµ×µis a stationary probability distribution for(X1,X2),which implies that(X1,X2)is a positive recurrent Markov chain,which verifies the claim thatτ<∞a.s.Finally,we give an account of what happens when the Markov chain has period d>1.A simple example is the simple random walk on Z d,which has period2.Thefirst observationis that the state space S can be partitioned into d disjoint classes S0,S1,···,S d−1,and the Markov chain simply marches through these d classes sequentially.Let us make this statement more precise.Lemma1.2For x,y∈S,let D x,y={n≥0:Πn(x,y)>0}.Then d divides m−n for any m,n∈D x,y.Proof.By irreducibility,there exists k∈N withΠk(y,x)>0.ThereforeΠk+m(y,y)≥Πk(y,x)Πm(x,y)>0and k+m∈D y.Similarly k+n∈D y,which implies that d divides m−n.By Lemma1.2,given an x∈S,each y∈S is associated with an r y∈{0,1,···,d−1}, where r y is the residue modulo d of any n∈D x,y.Let S i:={y∈S:r y=i}for i= 0,1,···,d−1.Then S0,···,S d−1gives a disjoint union of S,and clearly the Markov chain marches through S0,S1,···,S d−1in this order cyclically.Theorem1.3[Convergence of transition kernels:periodic case]Let X be an irre-ducible Markov chain with countable state space S and period d>1.If the chain is transient or null recurrent,thenlimΠn(x,y)=0∀x,y∈S.(1.3)n→∞If the chain is positive recurrent with stationary distributionµ,thenΠn(x,y)=dµ(y)∀x,y∈S.(1.4)limn→∞r x+n≡r y(mod d)Proof.Let us consider the transition kernel˜Π=Πd.Clearly˜Π(x,y)>0if and only if x,y belongs to the same S r for some0≤r≤d−1.Restricted to each S r,the associated Markov chain is irreducible,and furthermore,aperiodic.In fact,it is simply X n restricted to a d-periodic subsequence of times.We can then apply Theorem1.1.The result follows once we observe that the stationary distribution˜µr(·)for the Markov chain on S r with transition kernel˜Πmust equal dµ(·)restricted to S r.2Perron-Frobenius TheoremWhen the state space S isfinite,everything boils down to the study of thefinite-dimensional transition matrixΠ.WhenΠhas positive entries,Perron’s theorem asserts that1is the dominant eigenvalue with a positive eigenvector.Theorem2.1[Perron’s Theorem]Let P be an n by n matrix with positive entries.Then P has a dominant eigenvalueλsuch that(i)λ>0and the associated eigenvector h has positive entries.(ii)λis a simple eigenvalue.(iii)Any other eigenvalueκof P satisfies|κ|<λ.(iv)P has no other eigenvector with non-negative entries.Proof.Let T :={t ≥0:P v ≥tv for some v ∈[0,∞)n ,v ≡0}.Then min 1≤i,j ≤n P ij ∈T since P e i ≥P ii e i ,and T ⊂[0, 1≤i,j ≤n P ij ]since |P v |∞≤ P ij |v |∞.Furthermore,T is a closed set since if there exists t n →t and P v n ≥t n v n ,where without loss of generality we may assume |v n |1=1,we can find a subsequence n i such that v n i converges to a limitingnon-negative vector v ∞with |v ∞|1=1.Then we see that P v ∞≥tv ∞,which proves that T is closed.Let λ>0be the maximum in T ,and let P v ≥λv for some non-negative vector v =0.We claim that in fact P v =λv and v ∈(0,∞)n .Indeed,if P v −λv ≡0,then by the positivity assumption on entries of P ,we have P 2v −λP v >0,which implies that there exists some λ >λwith P 2v −λ P v ≥0.Since P v ≥0,this implies that λ ∈T ,contradicting our assumption that λis the maximum in T .Therefore P v =λv .Since P has positive entries and v ≡0,λv =P v >0,which proves (i).If P w =λw for an eigenvector w distinct from any constant multiple of v ,then there exists c ∈R such that w +cv ≥0,w +cv ≡0and w +cv has zero components.Since P (w +cv )=λ(w +cv )>0by the positivity of P ,this creates a contradiction.To conclude that λis a simple eigenvalue of P ,it remains to rule out the existence of a generalized eigenvector w with eigenvalue λ,i.e.,P w =λw +cv for some c =0.Changing w if necessary,we can assume c >0.Replacing w by w +bv if necessary,we can guarantee that w >0.Then P w =λw +cv implies that max T >λ,a contradiction.Let P w =κw for some κ=λ,where κand w could both be complex.Then|P w |=|κ||w |≤P |w |,(2.5)where |w |=|(w (1),···,w (n ))|=(|w (1)|,···,|w (n )|).Therefore |κ|∈T and |κ|≤λ.The inequality in (2.5)is in fact strict,which implies |κ|<λ,unless w =e iθw for some θ∈R and w ≥0,in which case κ=λ.By (i)–(iii)applied to P T ,which has the same eigenvalues as P ,there exists a non-negative and non-trivial w such that P T w =λw .Suppose that h be a non-negative eigenvector of P with eigenvalue λ =λ.Thenλ w,h = w,P h = P T w,h =λ w,h .Since h =h ,we have w,h =0,which is not possible if h is non-negative.Remark 2.2For a transition probability matrix Π,clearly |Πv |∞≤|v |∞for any vector v ,and Π1=1.Therefore 1is an eigenvalue of Πand all other eigenvalues κof Πhas |κ|≤1.When Πis the transition matrix of an irreducible aperiodic Markov chain,there exists n 0∈N such that for all n ≥n 0,Πn has positive entries.Therefore by Perron’s Theorem,1is a simple dominant eigenvalue of Πn for n ≥n 0.It is then easy to see that 1is also a simple dominant eigenvalue of Π,since the eigenvectors and generalized eigenvectors are the same for Πand Πn .Remark 2.3Perron’s Theorem applied to the transpose of a positive transition matrix Πimplies the existence of a stationary positive probability distribution µ.The fact that 1is a simple dominant eigenvalue of ΠT implies that starting from any probability measure νon {1,···,n },(ΠT )n νconverges to µexponentially fast.The case when P is only assumed to be non-negative is covered by Frobenius’s Theorem.Theorem2.4[Frobenius’Theorem]Let P be an n by n matrix with non-negative entries. Then P has an eigenvalueλwith the following properties:(i)λ>0and there exists an associated eigenvector with non-negative entries.(ii)Any other eigenvalueκof P satisfies|κ|≤λ.(iii)If|κ|=λ,thenκ=e2πik/mλfor some k,m∈N with m≤n.Remark2.5IfΠis the transition matrix of an irreducible Markov chain with period d,then Πd is of block diagonal form withΠd(i,j)>0if and only if i and j belong to the same class, and there are exactly d such classes.Restricted to each class S i⊂{1,···,n},Πd is a positive matrix and therefore by Perron’s Theorem has1as a simple dominant eigenvalue.Therefore Πd has1as the dominant eigenvalue with multiplicity d.Consequently,counting multiplicity,Πhas exactly d eigenvalues of modulus1,and any other eigenvalueκofΠhas|κ|<1.We claim that these eigenvalues are precisely e2πik/d with k=0,1,···,d−1.Indeed,(Πd)T has d linearly independent eigenvectors,which are just the restriction of the invariant measureµof the Markov chain to the d classes of states S i,0≤i≤d−1.Denote the restriction ofµto S i byµi.ThenΠTµi=µi+1for0≤i≤d−1andΠTµd−1=µ0.The space V spanned by(µi)0≤i≤d−1is preserved byΠT,and on V,if we chooseµi to be the basis vectors,thenΠT is a permutation matrix and its characteristic polynomial isλd−1=0.Therefore the set of eigenvalues ofΠT restricted to V is precisely e2πik/d for0≤k≤d−1.This exhausts the possible eigenvalues of modulus1forΠT,and hence alsoΠ.For a proof of the Frobenius theorem,see Lax[1,Chapter16].There are also infinite-dimensional versions of the Perron-Frobenius Theorem for compact positive operators.3Reversible Markov chainsWe now consider a special class of Markov chains called reversible Markov chains.Definition3.1[Reversible Markov chains]A Markov chain with countable state space S and transition matrixΠis called reversible,if it admits a stationary measureµ,called a reversible measure,which satisfiesµ(x)Π(x,y)=µ(y)Π(y,x)∀x,y∈S.(3.6)Remark.In physics literature,(3.6)is called the detailed balance condition.Heuristically,µ(x)Π(x,y)represents the probabilityflow from x to y in equilibrium.(3.6)requires that the probabilityflow from x to y equals theflow from y to x,which is a sufficient condition forµto be stationary.Reversing theflow can be interpreted as reversing the time direction.Not all stationary measures are reversible measures,as can be seen for the uniform measureµ≡1 for an asymmetric simple random walk on Z.Note that the notion of a reversible Markov chain is accompanied by a reversible measure.Theorem3.2[Cycle condition for reversibility]Let X be an irreducible Markov chain with countable state space S and transition matrixΠ.A necessary and sufficient condition for the existence of a reversible measure for X is that(i)Π(x,y)>0if and only ifΠ(y,x)>0.(ii)For any loop x 0,x 1,···,x n =x 0with n i =1Π(x i −1,x i )>0,we haven i =1Π(x i −1,x i )Π(x i ,x i −1)=1.(3.7)Proof.Suppose µis a reversible measure for X and µ≡0.Then by irreducibility and the stationarity of µ,µ(x )>0for all x ∈S .The detailed balance condition (3.6)clearly implies (i).Similarly,reversibility implies that the probability flow along the cycle x 0,x 1,···,x n =x 0,i.e.,µ(x 0) n i =1Π(x i −1,x i ),equals the probability flow along the reversed cycle x n =x 0,x n −1,···,x 0,which is just µ(x 0) 1i =n Π(x i ,x i −1).This yields (3.7).Conversely,if Πsatisfies (i)and (ii),then for a given x ∈S ,we can define µ(x )=1,and for any y ∈S with a path of states z 0=x,z 1,···,z n =y connecting x and y such that Π(z i −1,z i )>0,we can define µ(y )=n i =1Π(z i −1,z i )Π(z i ,z i −1).Conditions (i)and (ii)guarantee that our definition of µis independent of the choice of path connecting x and y .It is easy to check that µsatisfies the detailed balance condition (3.6).Example 3.31An irreducible birth-death chain is reversible,since Theorem 3.2(i)fol-lows from irreducibility,and the cycle condition (3.7)is trivially satisfied due to the lack of cycles.Similarly,any irreducible Markov chain on a tree is reversible.2A random walk on a connected graph G =(V,E )with vertex set V and edge set E is a Markov chain with state space V and transition matrix Π(x,y )=1/d x for all y ∈V with {x,y }∈E ,where d x is the degree of x in G .It is easily seen that µ(x )=d x is a reversible measure for the walk with unit measure flow across each edge.More generally,if each edge {x,y }∈E is assigned a positive conductance C x,y =C y,x ,and Π(x,y )=C x,y z :{x,z }∈EC x,z ,then µ(x )= z :{x,z }∈E C x,z is a reversible measure for the walk with mass flow C x,y across each edge {x,y }∈E .We briefly explain the usefulness of reversibility.Let µbe a reversible measure for the Markov chain with transition matrix Π.Then Πis a self-adjoint operator on the Hilbert space L 2(S,µ)with inner product f,g µ:= x f (x )g (x )µ(x ).Indeed,formally for any f,g ∈L 2(S,µ),f,Πg µ= x ∈Sf (x ) y ∈SΠ(x,y )g (y )µ(x )= x,y ∈Sf (x )g (y )Π(y,x )µ(y )= Πf,g µ.All information about the Markov chain are encoded in its transition matrix.The self-adjointness of Πon L 2(µ,S )allows one to use spectral theory to study its spectrum,which otherwise is not possible when the Markov chain is irreversible.References[1]x.Linear Algebra .John Wiley &Sons,Inc.New York,1997.。
Lecture8
In addition to governments and other politically motivated institutions which may decide to censor or promote certain works, the groups and social institutions to which Venuti refers would include various players in the publishing industries as a whole.
Translations are judges to be successful when they read ‘fluently’, giving the appearance that they have not been translated.
Venuti and the invisibility of translators
Domestication and Foreignization
Domestication: involves ‘an ethnocentric reduction of the foreign text to target language-culture values. This entails translating in a transparent, fluent, ‘invisible’ style in order to minimize the foreignness of the TT.
Invisibility: is term used by Venuti in 1995 ‘to describe the translator’s situation and activity in contemporary Anglo-American culture’. Venuti sees this invisibility as typically being produced:
华中科技大学费剑平 高级微观经济学课程讲义 Lecture08
example, good 1 in figure 5). Some examples of inferior goods are things that allow us to stay
alive without spending much money (such as Spam). If our income increases we switch to a
1
Microeconomic Theory: Lecturer. J. Ping Fei
Lecture 08 - 2
remember that the assumptions that we make affect the conclusions we reach even if the data we work with does not change. The assumptions we make as well as the models we work with will affect the conclusions we reach.
good that we like better but that we could not afford before.
A note on assumptions. Two people, with different assumptions about the way the world
is set up, can look at the same data and come up with widely different conclusions. We need to
b. Effects of changes in prices Now we want to see how demand changes when we change prices. Let’s say the price of
Lecture_8
12
Multi-period arithmetic attribution analysis
Arithmetic attribution method produces “time residuals” while linking over time
Q4 Q1 (Q4t Q1t )
Factor Modelling – Regression approach Uses regression analysis to determine factor contributions Risks of using correlated factors Difficult to define factors for specific assets classes, eg fund of hedge funds
Simplifications to remove the interaction effect Vary allocations first then select funds Or Select funds then vary allocations
Allocation | Benchmark Returns = Q2 – Q1
All these methods use some form of scaling to correct the period attributions so that when linked they yield the multi-period excess return
14
Extended compounded notional portfolio (CNP) method
Performance attribution analysis must follow the investment process and mirror the investment style
Lecture8
2. 我们的开幕式，将是中国和世界杰出的 艺术家展现的舞台，讴歌人类的共同理想， 以及我们独特的文化传统和奥林匹克运动。 Our Ceremonies will give China's greatest-and the world's greatest artists a stage for celebrating the common aspirations of humanity and the unique heritage of our culture and the Olympic Movement.
• These were truly exceptional Games! And now, in accordance with tradition, I declare the Games of the XXIX Olympiad closed, and I call upon the youth of the world to assemble four years from now in London to celebrate the Games of the XXX Olympiad. Thank you!
1. 上海
旅游
上海是世界上最大的港口城市之一， 已发展成为中国重要的经济、金融、贸易、 科技、信息和文化中心。 Shanghai is one of the world’s largest seaports, which has become China’s important center of economy, finance, trade, science and technology, information and culture.
• 4. 古往今来，旅游一直是人们增长知识、丰富阅历、 强健体魄的美好追求。在古代，中国先哲们就提出了 “观国之光”的思想，倡导“读万卷书，行万里路”， 游历名山大川，承天地之灵气，接山水之精华。 • From ancient times till now, tourism has represented the happy wish of people for more knowledge, varied experience and good health. In ancient times, ancient Chinese sages believed in the idea of “appreciating the landscape through sightseeing” and advised people to “travel ten thousand li and read ten thousand books”, find pleasure in enriching themselves mentally and physically through traveling over famous rivers and mountains.
高阶线性微分方程的一般理论 ppt课件
at b (1) ( n 1) 上的连续函数, 则对于任一 t0 [a, b] 及任意的 x0 , x0 , x0 ,
方程（4.1）存在 唯一解 x (t )，定义于区间 a t b 上，且满足初始条件：
n 1 d (t0 ) d (t0 ) (1) ( n 1) (t0 ) x0 , x0 , , x 0 n 1 dt dt
8
也是（4.2）的解，这里 c1 , c2 ,, ck 是任意常数。
ppt课件 PPT 课件PPT课件
证明
d nx d n 1 x dx a1 (t ) n 1 an1 (t ) an (t ) x 0 n dt dt dt
(n)
(4.2)
[c1 x1 (t ) c2 x2 (t ) ck xk (t )] a1 (t )[c1 x1 (t ) c2 x2 (t ) ck xk (t )]( n1)
x1 (t ) (t ) x1
x2 (t ) (t ) x2
xk (t ) (t ) xk
12
( k 1) ( k 1) x1( k 1) (t ) x2 (t ) xk (t )
称为这些函数的伏朗斯基行列式。
ppt课件 PPT 课件PPT课件
定理3 若函数 x1 (t ), x2 (t ), , xn (t ) 在区间 a t b 上线性相关， 上它们的伏朗斯基行列式 W (t ) 0。 则在 [a, b] 证明 由假设，即知存在一组不全为零的常数 c1, c2 ,, cn , （4.6）
d nx d n 1 x dx a1 (t ) n 1 an1 (t ) an (t ) x 0 n dt dt dt
Generation of higher order gauss-laguerre modes in single-pass 2nd harmonic generation
Preben Buchhave and Peter Tidemand-Lichtenberg
DTU Physics, Department of Physics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark *Corresponding author: pbu@fysik.dtu.dk
©2008 Optical Society of America
OCIS codes: (190.2620) Harmonic generation and mixing; (190.4410) Nonlinear optics, parametric; (190.4420) Nonlinear optics, transverse effects.
difference frequency generation. However, nonlinear effects such as depletion and phase shift were not included, and the method only applies to very low power beams. In this paper an alternative formulation is used, where the beams are expanded in higher order Gauss-Hermite (G-H) or Gauss-Laguerre (G-L) modes after passage of each thin slab. The choice of mode expansion depends on the symmetry of the problem. Since both G-H and G-L modes form complete orthogonal sets of functions, an expansion of the beams in these modes fully describes the result of the interaction, and diffraction due to the distortion of the wave front is included by the mix of the higher order modes. Besides forming an alternative way of performing a wave propagation calculation, this method has the advantage that it allows to follow the generation of higher order modes both spatially and temporally through the crystal. Furthermore it is possible to calculate the power generated in each of these modes. It is also possible to input higher order pump modes or superposition of modes and calculate the power conversion efficiency into the generated modes, a method that has recently been used in generation of squeezed light in higher order G-H modes [6]. In the calculations the phase matching may be adjusted at will; it is for example possible to adjust the phase matching to compensate at least partially for the Gouy phase shift in a particular mode and thus selectively generate light in one mode by pumping in another mode [7]. In the following, single pass second harmonic generation (SHG) in an efficient nonlinear crystal pumped by a pump beam of known mode composition is considered. As an example we shall look at a single mode Q-switched Nd:YAG laser pulse frequency doubled through a PPKTP crystal and investigate the formation of higher order G-L modes as the fields propagate through the crystal, as well as illustrate the temporal development of the pulse shape as a function of time. 2. Wave propagation with G-L expansion The calculations are performed in the following way: 1. The crystal is divided into thin slabs normal to the optical axis. 2. The three interacting beams that make up the total field incident on a slab interact on their way through the slab. The slab is assumed so thin that the nonlinear differential equations governing the interaction (the coupled wave equations) can be approximated by difference equations, and diffraction effects on the way through the slab are unimportant (plane wave approximation within a slab). Losses can be included in each slab. The beams may be focused anywhere inside or outside the crystal. Collinear beams are assumed, although the simulations can easily be expanded to include non-collinear beams and walk-off. 3. After passage of a slab, the field incident on the slab has been distorted due to the nonlinear interaction. The fields leaving the slab are expanded in G-L modes. 4. The beam waist radius and phase of the individual modes of the expansion are recalculated for the position of the next slab. The different phase shifts of the different order G-L modes implicitly take care of diffraction. 5. The G-L modes with the new phases and beam waists are added for each beam, and these new beams are used as input for the next slab; the calculation is repeated from 2. 6. The output beams are found by summing the G-L modes leaving the last slab. 2.1. Nonlinear interaction The coupled wave equations (CWE) for the 2nd order nonlinear interaction between three beams, described by the normalized electric fields a j ( x, y, z ) , where j identifies the field, are in the difference equation form given by [8]
电路分析英文Lecture8
– As input frequencies become very high, the reactance of a capacitor approaches 0Ω
Power delivered at any instant in time
p Vm sin wt Im sin(wt ) Using the trigoneme tric identity
sinAsinB cos(A - B) - cos(A B) 2
p Vm Im cos Vm Im cos(2wt )
26
Average Power and Power Factor
Av P eo r P w a V g m I e m c e r o V s m Im c o V sc I os
2
22
where V and I are rms values of the sinusoidal voltage and current respectively
11
Sinusoidal Response: Waveforms
Inductor
12
Sinusoidal Response: Capacitors
i C dv dt
where
v Vm sin wt
dv dt
Vmw cos wt
Vm w s in(wt
90 o )
Vmsin(ωt)
i wCVm sin(wt 90 o ) Im sin(wt 90 o )
7
8
Sinusoidal Response: Resistor
微积分英文专业词汇
微积分词汇第一章函数与极限Chapter1 Function and Limit集合set元素element子集subset空集empty set并集union交集intersection差集difference of set基本集basic set补集complement set直积direct product笛卡儿积Cartesian product开区间open interval闭区间closed interval半开区间half open interval有限区间finite interval区间的长度length of an interval无限区间infinite interval领域neighborhood领域的中心centre of a neighborhood 领域的半径radius of a neighborhood 左领域left neighborhood右领域right neighborhood映射mappingX到Y的映射mapping of X ontoY 满射surjection单射injection一一映射one-to-one mapping双射bijection算子operator变化transformation函数function逆映射inverse mapping复合映射composite mapping自变量independent variable因变量dependent variable定义域domain函数值value of function函数关系function relation值域range自然定义域natural domain 单值函数single valued function多值函数multiple valued function单值分支one-valued branch函数图形graph of a function绝对值函数absolute value符号函数sigh function整数部分integral part阶梯曲线step curve当且仅当if and only if(iff)分段函数piecewise function上界upper bound下界lower bound有界boundedness无界unbounded函数的单调性monotonicity of a function 单调增加的increasing单调减少的decreasing单调函数monotone function函数的奇偶性parity(odevity) of a function 对称symmetry偶函数even function奇函数odd function函数的周期性periodicity of a function周期period反函数inverse function直接函数direct function复合函数composite function中间变量intermediate variable函数的运算operation of function基本初等函数basic elementary function 初等函数elementary function幂函数power function指数函数exponential function对数函数logarithmic function三角函数trigonometric function反三角函数inverse trigonometric function 常数函数constant function双曲函数hyperbolic function双曲正弦hyperbolic sine双曲余弦hyperbolic cosine双曲正切hyperbolic tangent反双曲正弦inverse hyperbolic sine反双曲余弦inverse hyperbolic cosine反双曲正切inverse hyperbolic tangent极限limit数列sequence of number收敛convergence收敛于a converge to a发散divergent极限的唯一性uniqueness of limits收敛数列的有界性boundedness of a convergent sequence子列subsequence函数的极限limits of functions函数当x趋于x0时的极限limit of functions as x approaches x0左极限left limit右极限right limit单侧极限one-sided limits水平渐近线horizontal asymptote无穷小infinitesimal无穷大infinity铅直渐近线vertical asymptote夹逼准则squeeze rule单调数列monotonic sequence高阶无穷小infinitesimal of higher order低阶无穷小infinitesimal of lower order同阶无穷小infinitesimal of the same order作者：新少年特工2007-10-8 18:37 回复此发言--------------------------------------------------------------------------------2 高等数学-翻译等阶无穷小equivalent infinitesimal函数的连续性continuity of a function增量increment函数在x0连续the function is continuous at x0左连续left continuous右连续right continuous区间上的连续函数continuous function函数在该区间上连续function is continuous on an interval 不连续点discontinuity point第一类间断点discontinuity point of the first kind第二类间断点discontinuity point of the second kind初等函数的连续性continuity of the elementary functions定义区间defined interval最大值global maximum value (absolute maximum)最小值global minimum value (absolute minimum)零点定理the zero point theorem介值定理intermediate value theorem第二章导数与微分Chapter2 Derivative and Differential速度velocity匀速运动uniform motion平均速度average velocity瞬时速度instantaneous velocity圆的切线tangent line of a circle切线tangent line切线的斜率slope of the tangent line位置函数position function导数derivative可导derivable函数的变化率问题problem of the change rate of a function导函数derived function左导数left-hand derivative右导数right-hand derivative单侧导数one-sided derivatives在闭区间【a,b】上可导is derivable on the closed interval [a,b]切线方程tangent equation角速度angular velocity成本函数cost function边际成本marginal cost链式法则chain rule隐函数implicit function显函数explicit function二阶函数second derivative三阶导数third derivative高阶导数nth derivative莱布尼茨公式Leibniz formula对数求导法log- derivative参数方程parametric equation相关变化率correlative change rata微分differential可微的differentiable函数的微分differential of function自变量的微分differential of independent variable微商differential quotient间接测量误差indirect measurement error绝对误差absolute error相对误差relative error第三章微分中值定理与导数的应用Chapter3 MeanValue Theorem of Differentials and the Application of Derivatives罗马定理Rolle’s theorem费马引理Fermat’s lemma拉格朗日中值定理Lagrange’s mean value theorem驻点stationary point稳定点stable point临界点critical point辅助函数auxiliary function拉格朗日中值公式Lagrange’s mean value formula柯西中值定理Cauchy’s mean value theorem洛必达法则L’Hospital’s Rule0/0型不定式indeterminate form of type 0/0 不定式indeterminate form泰勒中值定理Taylor’s mean value theorem 泰勒公式Taylor formula余项remainder term拉格朗日余项Lagrange remainder term麦克劳林公式Maclaurin’s formula佩亚诺公式Peano remainder term凹凸性concavity凹向上的concave upward, cancave up凹向下的，向上凸的concave downward’concave down 拐点inflection point函数的极值extremum of function极大值local(relative) maximum最大值global(absolute) mximum极小值local(relative) minimum最小值global(absolute) minimum目标函数objective function曲率curvature弧微分arc differential平均曲率average curvature曲率园circle of curvature曲率中心center of curvature曲率半径radius of curvature渐屈线evolute渐伸线involute根的隔离isolation of root隔离区间isolation interval切线法tangent line method第四章不定积分Chapter4 Indefinite Integrals原函数primitive function(antiderivative)积分号sign of integration被积函数integrand积分变量integral variable积分曲线integral curve积分表table of integrals换元积分法integration by substitution分部积分法integration by parts分部积分公式formula of integration by parts 有理函数rational function真分式proper fraction假分式improper fraction第五章定积分Chapter5 Definite Integrals曲边梯形trapezoid with曲边curve edge窄矩形narrow rectangle曲边梯形的面积area of trapezoid with curved edge积分下限lower limit of integral积分上限upper limit of integral积分区间integral interval分割partition积分和integral sum可积integrable矩形法rectangle method积分中值定理mean value theorem of integrals函数在区间上的平均值average value of a function on an integvals牛顿－莱布尼茨公式Newton-Leibniz formula微积分基本公式fundamental formula of calculus换元公式formula for integration by substitution递推公式recurrence formula反常积分improper integral反常积分发散the improper integral is divergent反常积分收敛the improper integral is convergent无穷限的反常积分improper integral on an infinite interval无界函数的反常积分improper integral of unbounded functions绝对收敛absolutely convergent第六章定积分的应用Chapter6 Applications of the Definite Integrals元素法the element method面积元素element of area平面图形的面积area of a luane figure直角坐标又称“笛卡儿坐标(Cartesian coordinates)”极坐标polar coordinates抛物线parabola椭圆ellipse旋转体的面积volume of a solid of rotation 旋转椭球体ellipsoid of revolution, ellipsoid of rotation曲线的弧长arc length of acurve可求长的rectifiable光滑smooth功work 水压力water pressure引力gravitation变力variable force第七章空间解析几何与向量代数Chapter7 Space Analytic Geometry and Vector Algebra向量vector自由向量free vector单位向量unit vector零向量zero vector相等equal平行parallel向量的线性运算linear poeration of vector三角法则triangle rule平行四边形法则parallelogram rule交换律commutative law结合律associative law负向量negative vector差difference分配律distributive law空间直角坐标系space rectangular coordinates坐标面coordinate plane卦限octant向量的模modulus of vector向量a与b的夹角angle between vector a and b方向余弦direction cosine方向角direction angle向量在轴上的投影projection of a vector onto an axis数量积，外积，叉积scalar product,dot product,inner product曲面方程equation for a surface球面sphere旋转曲面surface of revolution母线generating line轴axis圆锥面cone顶点vertex旋转单叶双曲面revolution hyperboloids of one sheet旋转双叶双曲面revolution hyperboloids oftwo sheets柱面cylindrical surface ,cylinder圆柱面cylindrical surface准线directrix抛物柱面parabolic cylinder二次曲面quadric surface椭圆锥面dlliptic cone椭球面ellipsoid单叶双曲面hyperboloid of one sheet双叶双曲面hyperboloid of two sheets旋转椭球面ellipsoid of revolution椭圆抛物面elliptic paraboloid旋转抛物面paraboloid of revolution双曲抛物面hyperbolic paraboloid马鞍面saddle surface椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线space curve空间曲线的一般方程general form equations of a space curve空间曲线的参数方程parametric equations of a space curve螺转线spiral螺矩pitch投影柱面projecting cylinder投影projection平面的点法式方程pointnorm form eqyation of a plane法向量normal vector平面的一般方程general form equation of a plane两平面的夹角angle between two planes点到平面的距离distance from a point to a plane空间直线的一般方程general equation of a line in space方向向量direction vector直线的点向式方程pointdirection form equations of a line方向数direction number直线的参数方程parametric equations of a line两直线的夹角angle between two lines 垂直perpendicular直线与平面的夹角angle between a line and a planes平面束pencil of planes平面束的方程equation of a pencil of planes行列式determinant系数行列式coefficient determinant第八章多元函数微分法及其应用Chapter8 Differentiation of Functions of Several Variables and Its Application一元函数function of one variable多元函数function of several variables内点interior point外点exterior point边界点frontier point,boundary point聚点point of accumulation开集openset闭集closed set连通集connected set开区域open region闭区域closed region有界集bounded set无界集unbounded setn维空间n-dimentional space二重极限double limit多元函数的连续性continuity of function of seveal连续函数continuous function不连续点discontinuity point一致连续uniformly continuous偏导数partial derivative对自变量x的偏导数partial derivative with respect to independent variable x高阶偏导数partial derivative of higher order 二阶偏导数second order partial derivative 混合偏导数hybrid partial derivative全微分total differential偏增量oartial increment偏微分partial differential全增量total increment可微分differentiable必要条件necessary condition充分条件sufficient condition叠加原理superpostition principle全导数total derivative中间变量intermediate variable隐函数存在定理theorem of the existence of implicit function曲线的切向量tangent vector of a curve法平面normal plane向量方程vector equation向量值函数vector-valued function切平面tangent plane法线normal line方向导数directional derivative梯度gradient数量场scalar field梯度场gradient field向量场vector field势场potential field引力场gravitational field引力势gravitational potential曲面在一点的切平面tangent plane to a surface at a point曲线在一点的法线normal line to a surface at a point无条件极值unconditional extreme values条件极值conditional extreme values拉格朗日乘数法Lagrange multiplier method 拉格朗日乘子Lagrange multiplier经验公式empirical formula最小二乘法method of least squares均方误差mean square error第九章重积分Chapter9 Multiple Integrals二重积分double integral可加性additivity累次积分iterated integral体积元素volume element三重积分triple integral直角坐标系中的体积元素volume element in rectangular coordinate system柱面坐标cylindrical coordinates柱面坐标系中的体积元素volume element in cylindrical coordinate system 球面坐标spherical coordinates球面坐标系中的体积元素volume element in spherical coordinate system反常二重积分improper double integral曲面的面积area of a surface质心centre of mass静矩static moment密度density形心centroid转动惯量moment of inertia参变量parametric variable第十章曲线积分与曲面积分Chapter10 Line(Curve)Integrals and Surface Integrals对弧长的曲线积分line integrals with respect to arc hength第一类曲线积分line integrals of the first type对坐标的曲线积分line integrals with respect to x,y,and z第二类曲线积分line integrals of the second type有向曲线弧directed arc单连通区域simple connected region复连通区域complex connected region格林公式Green formula第一类曲面积分surface integrals of the first type对面的曲面积分surface integrals with respect to area有向曲面directed surface对坐标的曲面积分surface integrals with respect to coordinate elements第二类曲面积分surface integrals of the second type有向曲面元element of directed surface高斯公式gauss formula拉普拉斯算子Laplace operator格林第一公式Green’s first formula通量flux散度divergence斯托克斯公式Stokes formula环流量circulation旋度rotation,curl第十一章无穷级数Chapter11 Infinite Series一般项general term部分和partial sum余项remainder term等比级数geometric series几何级数geometric series公比common ratio调和级数harmonic series柯西收敛准则Cauchy convergence criteria, Cauchy criteria for convergence正项级数series of positive terms达朗贝尔判别法D’Alembert test柯西判别法Cauchy test交错级数alternating series绝对收敛absolutely convergent条件收敛conditionally convergent柯西乘积Cauchy product函数项级数series of functions发散点point of divergence收敛点point of convergence收敛域convergence domain和函数sum function幂级数power series幂级数的系数coeffcients of power series阿贝尔定理Abel Theorem收敛半径radius of convergence收敛区间interval of convergence泰勒级数Taylor series麦克劳林级数Maclaurin series二项展开式binomial expansion近似计算approximate calculation舍入误差round-off error,rounding error欧拉公式Euler’s formula魏尔斯特拉丝判别法Weierstrass test三角级数trigonometric series振幅amplitude角频率angular frequency初相initial phase矩形波square wave谐波分析harmonic analysis直流分量direct component 基波fundamental wave二次谐波second harmonic三角函数系trigonometric function system傅立叶系数Fourier coefficient傅立叶级数Forrier series周期延拓periodic prolongation正弦级数sine series余弦级数cosine series奇延拓odd prolongation偶延拓even prolongation傅立叶级数的复数形式complex form of Fourier series第十二章微分方程Chapter12 Differential Equation解微分方程solve a dirrerential equation常微分方程ordinary differential equation偏微分方程partial differential equation,PDE 微分方程的阶order of a differential equation 微分方程的解solution of a differential equation微分方程的通解general solution of a differential equation初始条件initial condition微分方程的特解particular solution of a differential equation初值问题initial value problem微分方程的积分曲线integral curve of a differential equation可分离变量的微分方程variable separable differential equation隐式解implicit solution隐式通解inplicit general solution衰变系数decay coefficient衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of first order非齐次non-homogeneous齐次线性方程homogeneous linear equation 非齐次线性方程non-homogeneous linear equation常数变易法method of variation of constant 暂态电流transient stata current稳态电流steady state current伯努利方程Bernoulli equation全微分方程total differential equation积分因子integrating factor高阶微分方程differential equation of higher order悬链线catenary高阶线性微分方程linera differential equation of higher order自由振动的微分方程differential equation of free vibration强迫振动的微分方程differential equation of forced oscillation串联电路的振荡方程oscillation equation of series circuit二阶线性微分方程second order linera differential equation线性相关linearly dependence线性无关linearly independce二阶常系数齐次线性微分方程second order homogeneour linear differential equation with constant coefficient二阶变系数齐次线性微分方程second order homogeneous linear differential equation with variable coefficient特征方程characteristic equation无阻尼自由振动的微分方程differential equation of free vibration with zero damping 固有频率natural frequency简谐振动simple harmonic oscillation,simple harmonic vibration微分算子differential operator待定系数法method of undetermined coefficient共振现象resonance phenomenon欧拉方程Euler equation幂级数解法power series solution数值解法numerial solution勒让德方程Legendre equation微分方程组system of differential equations 常系数线性微分方程组system of linera differential equations with constant coefficient。
lecture8射流扩散火焰
2015.06.18
1
预混与扩散火焰比较 (premixed flame v.s. diffusion flame)
预混火焰
未完全燃烧 的CO, H2与 外界O2反应 形成焰后氧 化区
稳定预混火焰锥
扩散火焰
• 空气通过对流和扩散进入火焰面，燃 料和空气边燃烧边混合。燃烧远比扩
柴油发动机(非预混)
• Large combustion devices such as
furnace, operate under non-premixed
conditions.
汽油发动机 (预混)
柴油发动机将燃油液滴喷射入高温压缩气室中。液 滴迅速挥发与空气混合，在局部预混的条件下发生 自点火。然后在大部非预混的条件下完成燃烧； Diesel engines inject the fuel spray into the compressed hot air chamber. It rapidly evaporates and mixes with air and then auto-ignition occurs under partly premixed conditions. The final stage of combustion occurs at non-premixed condition.
散混合快得多，因此扩散是制约扩散 火焰燃烧速度的关键步！
• Fuel is mixed with the surrounding air
by convection and diffusion during
combustion. Since combustion is much
2.6Higher-Order Derivatives
Problem Introduction
Acceleration of variable speed linear motion? Let s s(t), so instantaneous velocity v(t) s(t)
a is the rate: the change rate of v(t) to t
y ( 1 ) 1 x2
2 x
(1 x2 )2
y
2x
(1
x2
)2
2(3x2 1) (1 x2 )3
y 2x x0 (1 x2 )2
0;
y x0
2(3 x2 1) (1 x 2 )3
2.
x0
x0
Example 3
If y ex , find y(n).
y ex , y ex ,
（2） (sin kx)(n)
kn
sin(kx n )
2
（3） (cos kx)(n)
kn
cos(kx n )
2
(ex )(n) ex
（4）( x )(n) ( 1) ( n 1) x n
（5） (ln x)(n)
( 1) n 1
(n 1)! xn
1 (n) x
(1)n
2
2
In the similar way (cos x)(n) cos(x n )
2
Summary of Higher-Order Derivatives
The n-th order derivative of some special functions
（1） (a x )(n) a x ln n a (a 0)
lecture8
G ( y ) Pr(Y y ) Pr(r ( X ) y )
{ x:r ( x ) y}
f ( x)dx
If Y also has a continuous distribution, its p.d.f. g can be obtained by
( y) g ( y) dG dy
X s(Y ) (1 Y )1/ 2 for 0<Y<1.
ds( y) 1 dy 2(1 y)1/ 2 1 3 1/ 2 3(1 y ) (1 y ) , 0 y 1 1/ 2 2(1 y) 2 g ( y) 0, otherwise
18
Suppose we want to find the joint distribution of Y1 min(X1,, X n ) and Yn max(X1,, X n )
For - y1 yn , G ( y1 , yn ) Pr(Y1 y1 and Yn yn ) Pr(Yn yn ) Pr(Yn yn and Y1 y1 ) Pr(Yn yn ) Pr( y1 X 1 yn , , y1 X n yn ) Gn ( yn ) Pr( y1 X i yn )
g ( y) Pr(Y y) Pr(r ( X ) y)
x:r ( x ) y
f ( x)
2
Variable with A Continuous Distribution
Suppose X has a continuous distribution with p.d.f. f, and Y=r(X), a function of X. The d.f. G of Y can be derived as:
生产率与效率分析lecture8随机前沿效率
ln yi = xi + vi - ui
v i.i.d. N(0,v2) u 0
14
Stochastic Frontier
ln yi = xi + vi - ui
v i.i.d. N(0,v2) u 0 v & u independent v accounts for
xi
xJ
x
20
y
frontier output i exp(xi+vi), if vi>0
deterministic production function y=exp(x)
frontier output j
yJ
exp(xJ+vJ), if vJ <0
observed output j
yi
exp(xJ+vJ-uJ)
1
History (cont.)
Recall
technical efficiency (TE) allocative efficiency (AE)
Focus on TE only
For simplicity in explanation of stochastic frontiers (SF)
2
(& other reasons)
8
History (cont.)
So, usual case is observed output < frontier output
yO < yF ln yi < xi
That is, firm is inefficient
9
Lecture8
Alliteration (头韵)
Alliteration is the use of words that begin with the same sound in order to make a special communicative effect. Usually they are pleasing to ears because of the clever choice of the word by the advertiser. In addition, the repetition of the beginning sound emphasizes the meaning the advertisement wants to express.
It is also a comparison, but the comparison is implied, not expressed with the word “as” or “like”
Examples：
Something within you is Dior. （Christian Dior Perfume） 迪奥存在于您的内心深处。
Figures of speech
In perspective of phonetics In perspective of semantics In perspective of sentence structure
In perspective of phonetics
Alliteration (头韵) End Rhyme(尾韵) Alliteration+Rhyme(头韵+尾韵)
---William Shakespeare
Examples:
lecture8
3.2 Riemann Integral of Several Variables
Last time we defined the Riemann integral for one variable, and today we generalize to many variables. Definition 3.3. A rectangle is a subset Q of Rn of the form Q = [a1 , b1 ] × · · · × [an , bn ], where ai , bi ∈ R. Note that x = (x1 , . . . , xn ) ∈ Q ⇐⇒ ai ≤ xi ≤ bi for all i. The volume of the rectangle is (3.11) v (Q) = (b1 − a1 ) · · · (bn − an ), and the width of the rectangle is width(Q) = sup(bi − ai ).
R
≤�
�
v (R)
(3.33)
≤ �v (Q). We can take � → 0, so sup L(f, P ) = inf U (f, P ),
P P
(3.34)
which shows that f is integrable. We have shown that continuity is sufficient for integrability. However, continuity is clearly not necessary. What is the general condition for integrability? To state the answer, we need the notion of measure zero. Definition 3.9. Suppose A ⊆ Rn . The set A is of measure zero if for every � > 0, there exists a countable covering of A by rectangles Q1 , Q2 , Q3 , . . . such that � i v (Qi ) < �.
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du
dy f [ ( x)] x dx f ( u) ( x )dx
f ( u)du
结论：
无论 u是自变量还是中间变量，函数 y f ( x ) 的 微分形式总可写成
dy f ( u ) du
微分形式不变性
x 例1 设 y ln(1 3 ) ，求 dy 。
结论： (1) 函数可微必可导，可导必可微，可导与可微 是等价的。 (2) 函数在点 x0的微分就是函数的导数与自变量的增 量的乘积。即
dy f ( x0 )x
函数微分的定义
函数 y f ( x )在任意点 x的微分，称为函数
的微分，记作dy 或 df ( x )，即
dy df ( x ) f ( x )x
x
1 x (3 ) ln 3 ( x ) dx x 1 3
3 ln 3 91年数三考研题填空题 dx x 1 3
x
例2 已知方程 y 1 xe y 确定函数 y f ( x ) ，求 dy . 解： 将方程两边同时对 x 求导，得
y 0 e y xe y y y xe y y e y
数 y 对应的增量
y f ( x0 x) f ( x0 )
当 y f ( x )很复杂时，计算很麻烦， 寻求 y的一个既简单而有一定准确程度
的一近似表达式。
y dy Ax
问题：如何确定 A？
定理(可微的条件) 函数 f ( x )在点 x0 可微的充要条
件是函数 f ( x )在点 x0 处可导，且 A f ( x0 )。
高阶导数
二阶导数的定义： 如果函数 f ( x )的导数 f ( x )在点 x 处可导， 则称 f ( x )在点 x处的导数为函数 f ( x )在点 x
处的二阶导数. 记作
d y f ( x ), y 或 2 dx
2
类似地，
二阶导数的导数称为三阶导数,
3 d y f ( x ), y 或 dx 3 三阶导数的导数称为四阶导数, 4 d y (4) (4) f ( x ), y 或 4 dx
微分公式
1 (12) d(cot x ) 2 dx sin x 2 csc xdx
(13) d(sec x ) sec x tan xdx
(14) (csc x ) csc x cot x (14) d(csc x ) csc x cot xdx 1 1 (15)(arcsin x ) (15) d(arcsin x ) dx 2 1- x 1- x 2 1 1 (16) d(arccos x ) dx (16)(arccos x ) 2 2 1- x 1- x
导数ห้องสมุดไป่ตู้式
1 (17)(arctan x ) 1 x2
微分公式
1 (17) d(arctan x ) dx 2 1 x
1 1 (18)(arc cot x ) dx 2 (18) d(arc cot x ) 2 1 x 1 x
2. 函数和、差、积、商的微分法则
(1 xe y )y e y
解出 y，得
y ey e y dx y dy y 1 xe 1 xe
05级微积分(上)期末考试计算题第3题
1.基本初等函数的微分公式
dy f ( x )dx
导数公式
(1) C 0 (C为常数)
1 (2) ( x ) x
x0 x在这区间内，如果函数的增量 y f ( x0 x) f ( x0 )
可表示为 y Ax o(x ) 其中A是不依赖 x的常数，那么称函数 y f ( x ) 在点 x0是可微的，而 A x 叫做函数 y f ( x )在点
x0相应于自变量增量 x的微分，记作 dy，即
如果将自变量 x当作自已的函数 y x ，则得
dx xx x
于是有
dy f ( x )x
dy f ( x )dx
即函数的微分就是函数的导函数与自变量的微分的 乘积。
x 例1 设 y ln(1 3 ) ，求 dy 。
解：利用 dy ydx 得
1 x (3 ) dx dy [ln(1 3 )]dx x 1 3
dy Ax
于是有 y dy o( x )
由定义知： (1) dy是自变量的改变量 x 的线性函数； (2) A是与 x无关的常数，但与 f ( x )和 x0 有关。 (3) 当 x 很小时，y dy (线性主部)
设 y f ( x )在 x0 处自变量 x 有增量 x ，函
x x
x y e
例3 求 y sin x 的2 阶导数。 解：
y (sin x ) cos x
y [cos x] sin x
例4 求 y ln(1 x )的 2阶导数。 解：
1 y [ln(1 x )] 1 x 1 1 y 2 (1 x ) 1 x
(三)、除法的微分
u vdu udv d( ) v v2
u vu uv ( ) 2 v v
3. 复合函数的微分法则
如果函数 y f (u)对 u是可导的，则 (1)当 u 是自变量时，此时函数的微分为
dy f (u)du
(2)当 u 不是自变量，而是 u ( x )，为 x 的可导 函数时，则 y为 x 的复合函数：y f [ ( x )] 此时函数的微分为
y x 02
x0 x
x0
2 x0 x (x)2
(1)
(2)
(1)是 x 的线性函数，且为 y 的主要部分； (2)是 x 的高阶无穷小；
当 x很小时，
y 2 x0 x
称为函数在 x0的微分
可微的定义
定义 设函数 y f ( x )在某区间内有定义，x0 及
(9) d(sin x ) cos xdx (10) d(cos x ) sin xdx 1 (11) d(tan x ) dx 2 cos x sec 2 xdx
导数公式
1 (12) (cot x ) 2 sin x 2 csc x
(13) (sec x ) sec x tan x
微分公式
(1)d (C ) 0
(2) d( x ) x 1 dx
1 1 (3) d( ) 2 dx x x 1 (4) d( x ) dx 2 x x (5) d(loga ) 1 dx x ln a
1 1 (3) ( ) 2 x x 1 (4) ( x ) 2 x 1 x (5) (log a ) x ln a
e
1 3 x
(3cos x sin x)dx
例3 设函数 y y( x )由方程 2xy x y所确定，求 dy x 0 . 解： d (2 xy ) d ( x y ) 2002年数二考 研题填空题
2 xy ln 2d ( xy) dx dy
2 xy ln 2( ydx xdy) dx dy
x dy d[ln(1 3 )] 解：
1 x d (1 3 ) x 1 3
微分形式 不变性
1 x 1 x d (3 ) x (3 ) ln 3 d ( x ) x 1 3 1 3
3 x ln 3 91年数三考研题填空题 dx 1 3 x
例2 设 y e
1 3 x
cos x ，求 dy。
解：dy d (e1 3 x cos x )
cos xd (e1 3 x ) e1 3 x d (cos x )
cos x e1 3 x d (1 3 x) e1 3 x ( sin x )dx cos x e1 3 x (3)dx e1 3 x sin xdx
xy 2 ln 2 y 1 dy dx 1 x
dy x0 (ln 2 1)dx
(4) y 注意：导数的表示法，要区别开函数的四阶导数
与幂函数 y 。
4
一般地，我们定义 y f ( x ) 的 n阶导数为 y f ( x ) 的 n 1阶导数的导数，即
[ y( n1) ] y( n) (n 2, 3,4)
记作
f ( n) ( x ), y( n) 或 d y , n dx
导数公式
1 (6) (ln x ) x
微分公式
1 (6) d(ln x ) dx x
(7) (a x ) a x ln a (8) (e ) e
x x
(7) d(a x ) a x ln adx (8) d(e ) e dx
x x
(9) (sin x ) cos x (10) (cos x ) sin x 1 (11) (tan x ) cos 2 x sec 2 x
函数的微分
引例 设有一个边长为 x0 的正方形，其面积用 y表
2 y x 示，显然 。如果边长 x0取一个改变量 x ， 0 2 ( x ) x x 则面积增加多少？ 0
y1 x0
2
y2 ( x0 x)2
x0 x
x
y y2 y1 ( x0 x)2 x02
n
d f ( x) n dx
n
二阶和二阶以上的导数统称为高阶导数.
f ( x )称为零阶导数， f ( x )称为一阶导数。 相应地，
高阶导数的求法法则
1.直接法: 由高阶导数的定义逐步求高阶导数。 例1 求 y x 4的2阶导数。 解： y 4 x 3
y 12 x 2
例2 求 y e x的2阶导数。 解：因为(e ) e ，即函数求导后不变，所以