根据老师提供的英文数学参考书和章节，将analytical finance的数学基础做了一个提纲，
2 G) F- m7 A* X: U
- }  P* ~, [" R  ?8 V7 S5 P6 b因为想在国内先恶补一下这些知识，所以想先翻译出提纲，再去买中文书。
$ D" }+ a8 g; ]* R- G
/ k2 c( G0 a) A  L但是很多专业词汇翻译不懂或者翻译的不好，请达人对红色字体进行斧正和增添。
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非常感谢，如果可能，希望能建议基本参考书。
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-----------
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Oxford! R0 g% \; s9 D8 h2 Z. u! D: s
1.4 Differentiation of functions of a real variable 实变函数求导法( r) z5 ?" w, i6 O5 R- a6 O3 _& h& c
1.4.1 The derivative 导数" g5 {( @# t( v& T+ g9 `
1.4.2 The chain rule 连锁率, g" O  `/ N$ I+ N; P
1.4.3 Increasing and decreasing functions 递减函数，递增函数
$ a, U, X3 _" [% V/ o! j$ l1.4.4 Inverse functions 逆函数4 B- K  t8 A& g' ~$ G1 B* B- w5 y
1.4.5 Taylor's theorem and the local behavior of functions
+ {/ z3 G8 h7 B9 L$ u! }. q1.4.6 Complex valued functions 复值函数4 a2 k0 V2 j3 S! m9 b

* J' e; l. F% E7 c' V& e ) e& I. ?! E1 K! r2 L
1.5 Derivatives of functions of several real variables 多实变函数求导0 l- s' E+ K" b3 h& S9 x" D
1.5.1 Partial derivatives 偏导数9 L& d! ^/ `1 O' K; g  q4 G2 {
1.5.2 The Frechet derivative
$ \) u, {. o+ k8 l; E1.5.3 The chain rule连锁率
( E1 @4 g" p: U2 z2 S1.5.4 Applications to the transformation of differential operators.
9 F$ ~$ @& J, P: A* J1.5.5 Application to the dependency of functions 函数相关性的应用5 `( m( ]- y1 x. ~) m
1.5.6 The theorem on implicit functions 隐函数定理
- @9 v6 T5 @6 R0 i! X8 H1.5.7 Inverse mappings逆映象, 反演映射
: x. t& `' f% J; n6 M. y1.5.8 The nth variation and Taylor's theorem7 i7 N1 W* H: M
1.5.9 Applications to estimation of errors 误差评定的应用
7 ^+ U$ b" U% e% x7 F1.5.10 The Frechet differential弗雷谢微分4 @$ a9 j7 n7 i* \# ]6 Z

) S* Q2 u2 o9 [3 c3 M2 I& D$ B
- m& D( J+ s0 w1.6 Integration of functions of a real variable 实变函数积分; {( W  k7 i8 Y2 _
1.6.1 Basic ideas
- Q$ Z5 Y) o$ _: m* w/ g1.6.2 Existence of the integral 积分存在定理
0 p- y' j  E$ f0 D) Y  a2 s1.6.3 The fundamental theorem of calculus
! e6 R5 ?! p8 B/ w6 q! A1.6.4 Integration by parts 部分积分法* _/ I5 p2 p5 x  y) ~$ ^
1.6.5 Substitution 代入
9 w9 {" v" S1 f& N6 u5 B' u/ \1.6.6 Integration on unbounded intervals
; m. {  H5 ]3 Z/ U- u  K% i1.6.7 Integration of unbounded functions 无界函数积分
$ ]) @9 q1 ?# ~! O/ t/ r& ~( K1.6.8 The Cauchy principal value柯西主值/ h5 T/ x" }4 L  A
1.6.9 Application to arc length 弧长的应用+ X$ n. T' g& ?
1.6.10 Astandard argument from physics
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0 `2 u" h- F) ?: B1.7 Integration of functions of several real variables 多实变函数积分
+ j+ Q( J3 w- F2 @7 \% R5 `1.7.1 Basic ideas
' r5 y0 Q. @$ ]# a5 D  z1.7.2 Existence of the integral 积分的存在性7 M. `4 C  `1 H/ ?9 d/ {
1.7.3 Calculations with integrals 积分的计算+ N, w- A4 |% {, y& n
1.7.4 The principle of Cavalieri (iteratedintegration 迭代积分) 卡瓦列里原理3 O6 s$ o+ U9 U* e
1.7.5 Substitution 代入法0 S" P( x# _" ~, k! `! }! |( l
1.7.6 The fundamental theorem of calculus (theorem of Gauss-Stokes) 微积分基本定理 （高斯定理）
/ y# g" `9 [8 \' l! X1.7.7 The Riemannian surface measure 黎曼2 {5 ?' f1 Y: O2 i8 b
1.7.8 Integration by parts 部分积分法# p3 F: |3 H+ K( `
1.7.9 Curvilinear coordinates 曲线坐标
8 X* Q6 X( W+ V0 K7 q; i1.7.10 Applications to the center of mass and center of inertia
# T" L9 @( G& o; H5 S$ X1.7.11 Integrals depending on parameters 依靠参数的积分1 k+ K4 d$ B+ a# A: ~- x2 `

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4 E& R2 ^" N' p$ L4 f% m1.12Ordinary differential equations 普通微分方程6 |5 g  [2 M4 n; T( V
1.12.1 Introductory examples
! F/ G7 t! h4 a- m9 I  Z* z1.12.2 Basic notions
( M: `# T  n4 k, e( v' V1.12.3 The classification of differential equations
8 q# T& x6 n' ?6 G( L* G- r+ I1.12.4 Elementary methods of solution& n  R1 z8 `2 g1 |
1.12.5 Applications- v& I% [* D" X, @
1.12.6 Systems of linear differential equations and the propagator. .# m& C1 u4 [" x1 U5 c3 B( P
1.12.7 Stability
' _8 E  L5 e# X) g1.12.8 Boundary value problems and Green's functions
  `/ W' ]5 Q* Z8 u9 b' d1.12.9 General theory
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2.1Elementary algebra 基本代数
! l8 e2 U# o) V" Q# T$ a2.1.1 Combinatorics 组合/ h5 J5 o+ ~$ v" }: c, ^7 r' L
2.1.2 Determinants 行列式, w8 U  @% Q6 g- X+ p
2.1.3 Matrices 矩阵/ V2 W/ ^" m( q  |: F/ T
2.1.4 Systems of linear equations 线性方程体系4 V& y5 ]' j' |# F. ^
2.1.5 Calculations with polynomials 多项式的计算$ X# ~2 u4 t0 j3 ^
2.1.6 The fundamental theorem of algebra according to Gauss# O1 I3 g" q, v7 ]; {
2.1.7 Partial fraction decomposition
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2.2Matrices 矩阵. ~3 f5 g6 F6 d- a) Z& Y
2.2.1 The spectrum of a matrix& s+ j( ~$ W# P9 F8 [) I7 u$ x% M$ X5 Y
2.2.2 Normal forms for matrices 矩阵的普通形式6 c' R7 b& S* u9 k4 j6 a- t
2.2.3 Matrix functions 矩阵函数
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1 b5 O  x( b- j5. Calculus of Variations and Optimization4 o' f3 R) L& \
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5.1 Calculus of variations - one variable
9 `& W. r; y9 R5 B; S* _$ j5.1.1 The Euler—Lagrange equations欧拉-拉格朗日方程
' {! M0 f( `& G1 s8 l' {5.1.2 Applications4 D. x% `+ r3 o' j& p! r
5.1.3 Hamilton's equations 汉密尔顿方程
: ~' f: v" Y; X% G/ C5.1.4 Applications
) ~' ]8 U% O* K, H: a5.1.5 Sufficient conditions for a local minimum 局部极小的充分条件- w6 i; p# }' k( u) @: q6 m5 Y
5.1.6 Problems with constraints and Lagrange multipliers . .
1 G% A7 @! I7 Y8 n4 ?. n& L2 l5.1.7 Applications
8 a4 c+ C9 K; n3 B0 ^! G5.1.8 Natural boundary conditions 自然边界条件/ b: k8 e" u! P; H3 G
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5.2 Calculus of variations - several variables
. S0 _. b( w+ j) s0 Z" v& w5.2.1 The Euler-Lagrange equations 欧拉-拉格朗日方程; n! |- d4 W6 c* ^
5.2.2 Applications* j: j& E& x2 C3 t. K, R9 `3 t
5.2.3 Problems with constraints and Lagrange multipliers , r0 t+ K  W" j& D

+ n; i# b. b: x8 u0 m5.3Control problems
1 y6 M: U, j- o5.3.1 Bellman dynamical optimization 贝尔曼动态优化" O, z# y3 ?# G3 I) U# i1 ]) X
5.3.2 Applications
: ^* V% J$ U9 I5.3.3 The Pontryagin maximum principle
* z; E" p4 L0 w. ]" ]; l7 }5.3.4 Applications
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  m' u! }% Q. L* e2 f, u5.4Classical non-linear optimization 古典非线性优化
6 d* d' a9 M8 h/ z5.4.1 Local minimization problems 局部极小化问题
6 |' C$ o) a$ {  v! Z/ s0 B5.4.2 Global minimization problems and convexity.
* y0 x  [. p& X. L2 K3 w1 f5.4.3 Applications to Gauss' method of least squares . . .2 f" _9 g' a2 g" W; ^
5.4.4 Applications to pseudo- inverses
! @4 f' K/ I" b. X5 H6 |; L0 I5.4.5 Problems with constraints and Lagrange multipliers% G+ u% ]7 @+ m
5.4.6 Applications to entropy
8 Y; ]( s1 T: K+ T6 D4 p5.4.7 The subdifferential 次微分
2 u; \( e2 a3 x6 @( Q5.4.8 Duality theory and saddle points; u* Y# B" M5 V' M1 I

$ i8 [3 k& |7 E4 K$ C5.5Linearoptimization 线性优化( {- k% _% \' f6 V, _# {' B
5.5.1 Basic ideas# h4 K+ ]4 p7 o% k
5.5.2 The general linear optimization problem5 H8 z; m9 i  v$ L7 {9 ~; E3 J
5.5.3 The normal form of an optimization problem and the minimal test# x& ?) D1 w" K3 o( Z
5.5.4 The simplex algorithm/ u& T7 b: h0 V  P* {4 ?0 n
5.5.5 The minimal test
/ _: d' L5 k/ Q+ A9 q, A5.5.6 Obtaining the normal form4 C/ \' ?* z5 C5 f8 E
5.5.7 Duality in linear optimization 线性优化的二元性* b/ d0 \7 c" q% R! X: d
5.5.8 Modifications of the simplex algorithm" p/ ?$ S4 j; S$ I* e
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5.6Applications of linear optimization 线性优化的应用3 M  h; J% R3 o- q9 B
5.6.1 Capacity utilization 能力利用
  b8 h; E5 g$ }9 `5.6.2 Mixing problems
4 A3 T. c) A, c' e( X2 x5.6.3 Distributing resources or products 资源或产品的分配1 Y, _9 ~+ U5 H# [6 a. g; T9 R4 x
5.6.4 Design and shift planing
4 J* i0 a( ~7 u9 X+ W3 q" ?5.6.5 Linear transportation problems 线性运输问题2 z9 Y$ s- _; x- O" {" }

0 N& k7 W1 G+ j& G: ~) j: R7 t6.Stochastic Calculus - Mathematics of Chance 随机微积分9 d. `7 E& U, p# Z; M
6.1 Elementary stochastics 基本推测学
9 k* n$ W' {4 k6.1.1 The classical probability model 古典概率模型
$ c; t# [, i% o6.1.2 The law of large numbers due to Jakob Bernoulli . .
, y6 K& f& \0 N* _6.1.3 The limit theorem of de Moivre
3 C0 C+ ]- u4 k. e6.1.4 The Gaussian normal distribution 高斯正态分布
- J9 p: @$ i7 M7 J+ y6.1.5 The correlation coefficient 相关系数
. E( V8 g" c  w& D. K6.1.6 Applications to classical statistical physics 古典统计物理学的应用
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6.2 Kolmogorov's axiomatic foundation of probability theory
  R8 b- P0 _( L( O4 q  H9 m* H1 `6.2.1 Calculations with events and probabilities8 V* J1 L3 s; ~+ I3 n' X; V
6.2.2 Random variables 随机变量
0 q+ K5 r9 h  W6 ]6.2.3 Random vectors 随机向量
7 t7 e& H+ m5 }) n6.2.4 Limit theorems 极限定理  ]7 s+ o: }" z1 [6 ]
6.2.5 The Bernoulli model for successive independent trials
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6.3Mathematical statistics 数理统计学! R4 K8 S9 C) S0 L; {. X, T
6.3.1 Basic ideas7 }# \, }' ~7 \7 ?# T
6.3.2 Important estimators 重要估计1 s/ f" a$ j% Z; f, r/ \7 w
6.3.3 Investigating normally distributed measurements . & Q! c& d6 ~, w6 V7 v  |
6.3.4 The empirical distribution function 经验分布函数+ U) J, {6 Y8 l4 }
6.3.5 The maximal likelihood method 最大可能性方法
# v0 m9 S/ ~: [7 b, B: B6 R6.3.6 Multivariate analysis 多变量分析9 u# s6 h; E/ H0 p3 Q
: G7 A, q% l0 n+ h
6.4Stochastic processes 随机过程3 L' s" m3 {. N* r9 l
6.4.1 Time series 时间序列
0 p' \2 k( O& X( H' _6.4.2 Markov chains and stochastic matrices 马尔可夫链和随机矩阵
4 ?2 z1 {, g9 L, n" u( r6.4.3 Poisson processes泊松过程
2 g8 ^5 W  b# n8 Y& l6.4.4 Brownian motion and diffusion 布朗运动和扩散5 [  C0 `9 W; e( `1 I) w
6.4.5 The main theorem of Kolmogorov for general stochastic
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MathematicsHandbook  G$ k- {& z, @1 B  ]$ c0 j; [; F6 V
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2.2Combinatories 组合5 z( |0 z. ~: D! P: t6 _/ t
, }3 \: ]( [" k& j& z, o2 G5 a# W
2.3Finite sequences, sums, products, means 有限序列和，积，均值0 o. {* Z/ u# N
2.3.1 Notation for sums and products 和积的计算' q. b2 H  @+ U, r
2.3.2 Finite sequences 有限序列
8 J4 z, ^# G/ d, }5 a" ]' K; \2.3.3 Some sums of finite sequences 有限序列部分求和, `* |( c% @# E! I( j- x8 t
2.3.4 Means) X6 \* L" ^) r6 t9 L7 g
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2.4Algebra% P. `9 a9 S& e) ?
2.4.4 Linear algebra 线性代数, L: o. X! C$ L6 e

: I0 G. |* p+ D# G3.1Differential and integral calculus of functions of one and several variables 单和多变量微积分函数
2 O1 I7 Y8 I' f- @1 U) Q$ G3.1.5 Differentiation of functions of a real variable 单实变函数求导
  k) g" I6 l9 Z3 J5 S5 O# k3.1.7 Integral calculus for functions of one variable 单变量函数积分0 E' b5 y1 C/ W' i3 T$ V" \9 ]
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3.2 Calculus of variations and optimal processes 变量积分和优化过程3 l4 s$ B, K! _7 `/ T
3.2.1 Calculus of variations
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3.3Differential equations 微分方程# B& e- E2 g6 n
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5.Probability theory and mathematical statistics 概率论和数理统计
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6.1 The problem of linear optimization and the simplexalgorithm
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[ 本帖最后由 atlantiswind 于 2008-7-3 22:56 编辑 ]

just read english version' R; z8 U6 U# ?8 x8 n% o
翻译 sux

粗略看了下, 虽然不是红的, 发现这3个有误:
5 ?) S3 W. |: l8 JThe chain rule不叫连锁率, 是链式法则  @) p2 Y* k6 c/ m3 s
Integration by parts不是部分积分法, 是分部积分
4 T& h/ ?! N8 J3 NThe maximal likelihood method 不叫最大可能性方法, 是最(或极)大似然法, 用的是最(或极)大似然函数  C2 M; I5 G$ l4 z# w% {
其实中文翻译很无所谓, 同LS的, LZ最好看英文; D3 Y. S7 h# z

$ E$ j& \7 {' a7 n; H红色的偶大概看了下, 太多了... 大工程~偶闪了...

直接看英文的最好# B  d5 E! A# O5 X- ?+ s! `
否则脑子里形成定式了再投入英文教学环境中听课会很难的

主要我觉得手头的英文教程3 [4 G' l/ X8 d, Q
handbook of mathematics, e8 q& F2 R. L
Oxford Users Guide to Mathematics
$ X' ?; k2 ~  w- w3 [, V8 T太过简略～所以想找对应的中文书看看

按照我个人的经验，在美国上数学课的时候碰到在国内已经接触过的词语，都直接肯英文课本，啃过以后，如果跟以前学过的概念对的上号，自然就会知道就是这个东西。. w1 Z' u3 k" R. }: Y+ c( O" Z6 V
所以如果你不只是要学这些词语，而是有心要不这些课程的话，还是直接读英文吧，碰到的时候查一下，重在理解。看过后，如果你有学过相关的数学知识，能理解的话，自然能对的上的。不需要在开始看之前就担心这些。