常微分方程中的英文单词和短语.docx

常微分方程中的英文单词和短语常微分方程Ordinary Differential Equtions第一章 基本概念Chapter 1 Basic Concepts常微分方程ordinary differential equation / ode阶order线性linear非线性nonlinear偏微分方程partial differential equation / pde通解general solution特解particular solution初值条件initial value condition初值问题initial value problem柯西问题Cauchy problem几何解释geometry explanation积分曲线integral curve线素line element线素场field of line element方向场direction field等斜线isocline对称形式symmetric form奇异点singularity通积分general integral第二章 初等积分法Chapter 2 Elementary Integration Methods恰当方程exact equation可微函数differentiable function全微分total derivative偏导数partial derivative隐函数implicit function原函数primitive function变量可分离的方程equation of separable variables一阶线性方程first-order linear equation齐次homogeneous非齐次non-homogeneous积分因子法method of integration factor常数变易法method of constants variation初等变换法elementary method of transformation齐次方程homogeneous equation伯努利方程Bernoulli equation利卡提方程Riccati equation第三章 存在和唯一性定理Chapter 3 Existence and Uniqueness Theorems皮卡存在和唯一性定理Picard theorem of existence and uniqueness李普希兹条件Lipschitz condition逐次迭代法successive iterative method皮卡序列Picard sequence一致收敛uniform convergence奥斯古德唯一性条件Osgood uniqueness condition皮亚诺存在定理Peano existence theorem欧拉折线Eulers polygonal arc函数序列function sequence一致有界uniform bound等度连续equicontinuous子序列sub-sequence欧拉序列Euler sequence矩形区域rectangular area解的延伸extension of solution有界闭区域bounded closed region最大存在区间maximum interval of existence边界boundary有限闭区间finite closed interval开集open set有限半开区间finite semi-open interval拉格朗日公式Lagrange formula局部local有限覆盖定理finite covering theorem条形域bar area单调递减monotone decreasing扇形区域sectorial area比较定理comparability solution最小解minimum solution最大解maximum solution斜率slope第四章 奇解Chapter 4 Singular Solutions一阶隐式微分方程first-order implicit differential equation微分法differentiation克莱罗方程Clairaut equation参数parameter奇解singular solutionP-判别式p-test formulaP-判别曲线p-test curve包络envelope曲线族curve family相切tangencyc-判别式c-test formula非退化nonsingular切向量tangent vector第五章 高阶微分方程Chapter 5 Higher-Order Differential Equations轨线trajectory相平面phase plane相图phase diagram首次积分first intefral自变量independent-variable未知函数unknown function拓扑变换topological transform逆变换inverse transform邻域neighborhood开区域open regionn维线性空间n-dimensional linear spacen维向量值函数n-dimensional vector-valued function函数组function group标准微分方程normal differential equation欧氏模Euler module解对初值的连续依赖性continuous dependence of solution on initial value变分方程variational equation第六章 线性微分方程组Chapter 6 Linear Differential Equations存在和唯一性定理existence and uniqueness theorem齐次线性微分方程组homogeneous linear differential equations基本解组basic solutions朗斯基行列式Wronskian determinant刘维尔公式Liouville formula解矩阵solution matrix基解矩阵basic solution matrix常数矩阵constant matrix非齐次线性微分方程组non-homogeneous linear differential equations常数列向量constant arrange vector常数变异公式formula of variation of constant计算公式computing formula结构公式framework formula常系数线性微分方程组constant coefficient linear differential equations矩阵指数函数exponential function of matrix标准基解矩阵normal basic solution matrix若尔当标准型Jordan normal form矩阵函数matrix function实值解real-valued function复值解complex-value function代数余子式algebraic cofactor特征方程characteristic equation算子式operator method算子多项式operator polynomial拉普拉斯逆变换Laplace inverse transform第七章 微分方程的幂级数解法Chapter 7 Power Series Solutions of Differential Equations柯西定理Cauchy theorem解析解analytic solution优级数excellent series优函数excellent function强函数strong function幂级数power series幂级数解法method of power series奇点singularity勒让德方程Legendre equation勒让德多项式Legendre polynomial罗德里格斯公式Rodrigues formula正交orthogonal广义傅里叶级数generalized Fourier series广义幂级数generalized power series指标index正则奇点regular singular point指标方程index equation共轭复根conjugate complex root贝塞尔方程Bessel equation第一类贝塞尔函数Bessel function of the first kind渐进式asymptotic expression第八章 定性理论与分支理论Chapter 8 Qualitative Theory and Bifurcation Theory动力系统dynamical system相空间phase space增广相空间augmented phase space速度场velocity field向量场vector field拓扑结构图topological structure graph相图phase diagram几何理论geometrical theory平衡点equilibrium point奇点singular point闭轨closed orbit轨线的唯一性uniqueness of orbit抽象动力系统abstract dynamical system拓扑动力系统topological dynamical system微分动力系统differential dynamical system解的稳定性stability of solution利亚普诺夫稳定性Liapunov stability渐近稳定asymptotic stability渐近稳定域asymptotic stability field吸引域attraction domain全局渐近稳定globe asymptotic stability线性近似linear approximate临界情形critical state利亚普诺夫第一方法first method of Liapunov直接方法direct method李雅普诺夫函数Liapunov function正定函数positive definite function负定函数negative definite function平面上的动力系统dynamical system on plane极限环limit cycle常点constant point初等奇点elementary singular point高阶奇点higher-order singular point星形结点star node临界结点critical node鞍点saddle point单向结点one-way point退化结点improper node稳定结点stable focus不稳定结点unstable focus中心点center point拓扑等价topological equivalence双曲奇点hyperbolic singularity稳定极限环stable limit cycle不稳定极限环unstable limit cycle半稳定极限环semi-stable limit cycle轨道稳定性stability of orbit庞加莱-班迪克环域定理Poincar-Bendixson annulus theorem of theorem李纳方程Linard equation李纳作图法Linard construction李纳映射Linard mapping后继函数法subsequence function method单极限环single limit cycleK重极限环k-fold limit cycle鞍-结点saddle-node霍普夫分支Hopf bifurcation庞加莱分支Poincar bifurcation开折unfolding普适开折universal unfolding第九章 边值问题Chapter 9 Bounda