华东师范数学大纲

华东师范数学大纲 《线性代数》教学大纲 教学目的和要求: 线性代数是数学学科中的一门重要基础课程，也是高等院校大部分专业的主要基础理论课，对于培养面向21世纪人才起着重耍的作用。目前也是华东师范大学各专业的重要基础课之一本课程主要学习线性代数中行列式，矩阵，n维向量和线性方程组，向量空间，矩阵的特征值和特征向量，二次型，线性变换的基本概念，基本计算及有关的计算方法。为适应培养面向21世纪人才的需要，要求学生比校系统理解线性代数的基本概念，基本理论，掌握线性代数的基本计算方法。要求较好地理解线性代数这门课的抽象理论，具有严谨逻辑推理能力，空间想象能力，运算能力和综合运用所学的知识分析问题和解决问题的能力。 教学基本和学时分配: 第一章:行列式 )展开定理，行列式的计算，教学内容:行列式的定义，行列式的基本性质，行列式按行(列 克莱姆法则。教学要求:理解行列式的概念，掌握行列式的性质，会用行列式的性质和行列式按行(列)展开定理计算行列式，会用克莱姆法则解线性方程组。 第二章:矩阵 教学内容:矩阵的概念，矩阵的线性运算，矩阵的乘法，方阵的幂，方阵乘积的行列式，矩阵的转置，逆矩阵的概念和性质，矩阵可逆的充要条件，伴随矩阵，矩阵的初等变换，初等矩阵，矩阵的等价，矩阵的秩，初等变换求矩阵的秩和逆矩阵的方法，分块矩阵及其运算。教学要求:理解矩阵的概念，了解单位矩阵，对角矩阵，数量矩阵，三角矩阵，对称矩阵，正交矩阵，掌握矩阵的加法，数乘，乘法，转置及它们的运算法则，了解方阵的方幂和方阵乘积的行列式。理解逆矩阵的概念，掌握逆矩阵的性质以及矩阵可逆的充要条件，会用伴随矩阵求逆矩阵，了解矩阵的初等变换和初等矩阵的概念，理解矩阵秩的概念。掌握矩阵的初等变换，会用初等变换求矩阵的秩和逆矩阵，了解分块矩阵掌握分块矩阵的运算法则。 第三章:n维向量与线性方程组 教学内容:向量的概念、向量的线性组合和线性表示、向量的线性相关与线性无关、向量组的极大线性无关组、等价向量组、向量组的秩、向量组的秩和矩阵的秩之间的关系、齐次线性方程组有非零解的充要条件、非齐次线性方程组有解的充要条件、线性方程组解的性质和解的结构、齐次线性方程组及基础解系和通解，非齐次线性方程组的通解，行初等变换求线性方程组的方法。 教学要求:理解向量的概念、掌握向量的加法和数乘运算法则，理解向量的线性组合线性表示，向量组的线性相关线性无关的定义，掌握向量组线性相关、线性无关的有关性质和判别法。理解向量组的极大线性无关组和向量组秩的概念会求向量组的极大线性无关组及向量组的秩，了解向量组等价的概念，向量组的秩与矩阵秩的关系。理解齐次线性方程组有非零解的充要条件，掌握非齐次线性方程有解和无解的判别方法。理解齐次线性方程组的基础解系、通解的概念，掌握齐次线性方程组基础解系的求法。理解非齐次线性方程组解的结构和通解的概念，掌握非齐次线性方程组通解的求法，掌握用行初等变换求解线性方程组的方法。 第四章 向量空间 center pin checks and anchoring the inner cylinder. According to While coupling alignment should keep oil out of cavity and locate the rotor axis neck raised are within the prescribed limits, there should be no gaps, Horn, Horn and Horn 0.05mm feeler gauges fit into. Center coupling, condenser water filling and water-filled according to factory requirement. For coupling three supported rotor Center, the mouth under the coupling between the value should be as stipulated in the manufacturer, usually 0.15~0.25mm. In the case of coupling connection, if possible, lifting only one bearing end of the support bearing of the rotor, remove the bearing, fitted with special tiles and special lifting tools, disc rotors, check the journal flutter tile, should generally be less than 0.10MM. When the couplings and rotor comes when there is a conflict, the coupling Centre shall prevail. 6.4.5.2.2 shaft looking for: ... 6.4.5.3. Provisional bolt confirmed the positioning of the rotor size, as well as the coupling between the adjustment shim thickness accuracy, drawings to meet the technical requirements. Plane, cylindrical and clean the coupling bolt holes should be clean and without burrs, scrap iron and grease-free, relative position of the coupling according to the drawing requirements are. Coupling top screws, round distance apart, indicators to monitor movement of the aircraft, its value is allowed to pass the value axis moving, mounted adjustment pads, note pads installation direction guide pin coupling with process end closed. The back wheels slowly tightened, when 2mm or so, with standard white bars where two back wheel bolt hole measurement above and below, confirm two back wheel bolt hole concentricity. Then, the symmetrical horizontal mount two temporary bolts tightened, attention to monitor dial indicator, then install the symmetrical temporary bolts, this time sites slowly moving rotor, in the measured circumference of 8 equal parts back wheel rim rock, each measuring relative changes in value should be less than 0.02mm. 6.4.5.4 coupling reamed and paired bolt reaming die reamer, adjustable reamers, boring mill, in several ways. 6.4.5.4.1 General requirements before reaming or boring, 教学内容:空间的基、维数与坐标，子空间，n维向量空间的基变换和坐标变换、过渡矩阵、向量的内积线性无关向量组的正交化方法，规范正交基、正交矩阵及其性质教学要求:了解集合、数域的概念，了解n维向量空间，子空间、基、维数，坐标的概念，掌握求n维向量空间的基和维数，判别子空间，了解基变换与坐标变换公式，会求过渡矩阵，理解内积的概念，掌握线性无关向量组正交规范化的施密特(Schmidt)方法，了解规范正交基、正交矩阵的概念，以及它们的性质。 第五章 矩阵的特征值和特征向量 教学内容: 矩阵的特征值和特征向量的概念，性质及求法，相似变换，相似矩阵的概念及性质、矩阵可对角化的充要条件及相似对角矩阵，实对称矩阵的特征值、特征向量及相似对角矩阵。教学要求理解矩阵的特征值和特征向量的概念及性质 掌握求矩阵的特征值和特征向量的方法。理解相似矩阵的概念，性质及矩阵可相似对角阵的充要条件，掌握用相似变换化矩阵为对角阵的方法。了解实对称矩阵的特征值和特征向量的概念及性质，掌握用相似变换化实对称矩阵为对角阵的方法。 第六章 二次型 教学内容:二次型及其矩阵表示，合同变换与合同矩阵、二次型的秩、惯性定理、二次型的形与规范形、用矩阵的初等变换和配方法化二次型为标准型，正定二次型与正定矩阵及其判别法，用正交变换化二次型为标准形。教学要求:了解二次型的概念，掌握用矩阵表示二次型，了解二次型的秩，合同矩阵的概念，理解二次型的标准形、规范形概念以及惯性定理，掌握用矩阵的初等变换和配方法化二次为标准形。理解正定二次型与正定矩阵的概念，掌握正定二次型与正定矩阵判别法。掌握用正定矩阵化二次型为标准形的方法。 第七章 线性变换 教学内容:线性变换的概念及其运算，线性变换的性质，线性变换的矩阵，线性变换在不同基矩阵的关系，欧氏空间的正交变换与对称变换。教学要求:了解线性变换的概念及其运算的法则，了解线性变换在基下的矩阵，掌握线性变换运算与矩阵运算的运算法则，了解线性变换在不同基下的矩阵关系，了解正交变换，对称变换的概念及其性质。 《高等数学(B)》教学大纲 教学目的和要求:高等数学是高等院校大部分专业的一门重要基础理论课，是深入学习专业课程的必备基础。 随着数学在各学科中的应用日夜广泛，作为地理、环科、心理等专业的学生无论将来从事科研工作还是教学工作，都应该具备良好的数学基础和灵活应用数学的能力。本课程主要学习一元函数和多元函数的微积分学，以及无穷级数和常微分方程的主要内容，是将来进一步学习专业知识的必备的数学基础。为适应地理、环科等各类专业的特点和要求，使用本大纲应当遵循以下原则:强调基本概念的实际意义，而不追求概念的抽象性;强调基本理论的实际应用，而不追求理论的完备性;强调基本计算方法的实际操作，而不追求计算的技巧。 教学基本内容和学时分配: 第一章 函数(4学时) 教学内容 1、函数的概念 measured circumference of 8 equal parts back wheel rim rock, each measuring relative changes in value should be less than 0.0tention to monitor dial indicator, then install the symmetrical temporary bolts, this time sites slowly moving rotor, in the d, atnd below, confirm two back wheel bolt hole concentricity. Then, the symmetrical horizontal mount two temporary bolts tighteneThe back wheels slowly tightened, when 2mm or so, with standard white bars where two back wheel bolt hole measurement above a losed.o pass the value axis moving, mounted adjustment pads, note pads installation direction guide pin coupling with process end cements are. Coupling top screws, round distance apart, indicators to monitor movement of the aircraft, its value is allowed tfree, relative position of the coupling according to the drawing requir-es should be clean and without burrs, scrap iron and greasement shim thickness accuracy, drawings to meet the technical requirements. Plane, cylindrical and clean the coupling bolt holdjustlooking for: ... 6.4.5.3. Provisional bolt confirmed the positioning of the rotor size, as well as the coupling between the athan 0.10MM. When the couplings and rotor comes when there is a conflict, the coupling Centre shall prevail. 6.4.5.2.2 shaft e lessbearing, fitted with special tiles and special lifting tools, disc rotors, check the journal flutter tile, should generally bn the case of coupling connection, if possible, lifting only one bearing end of the support bearing of the rotor, remove the value should be as stipulated in the manufacturer, usually 0.15~0.25mm. Ifilled according to factory requirement. For coupling three supported rotor Center, the mouth under the coupling between the -nto. Center coupling, condenser water filling and waterfit i he rotor axis neck raised are within the prescribed limits, there should be no gaps, Horn, Horn and Horn 0.05mm feeler gaugespin checks and anchoring the inner cylinder. According to While coupling alignment should keep oil out of cavity and locate tcenter 2eral requirements before reaming or boring,.4.5.4 coupling reamed and paired bolt reaming die reamer, adjustable reamers, boring mill, in several ways. 6.4.5.4.1 Gen2mm. 6 华东师范大学 邻域;函数及其表示法 2、具有某些特性的函数 有界函数;单调函数;奇函数与偶函数;周期函数。 3、初等函数 反函数;复合函数;初等函数 第二章 极限和连续(14学时) 教学内容 1、数列及其极限 2、自变量趋于无穷大时的函数极限 自变量趋于无穷大时的函数极限;数列极限 3、自变量趋于有限值时的函数极限 函数极限的定义;左、右极限;函数极限和数列极限的关系。 4、极限的性质 收敛数列的性质;函数极限的性质。 5、无穷小量，无穷大量和极限的运算法则 无穷小量;无穷大量;无穷小量的四则运算;极限的四则运算法则;极限的复合运算法 则。 6、极限存在条件和两个重要极限 数列极限存在条件;函数极限存在条件;两个重要极限 7、无穷大量和无穷小量的比较 、连续函数 8 函数的连续性;间断点及其分类;连续函数的运算和初等函数的连续性。 9、闭区间上连续函数的性质 最大、最小值定理与有界性定理;介值定理与根的存在性定理 第三章 导数与微分(14学时) 教学内容 1、导数的定义 导数的定义;导函数;导数的几何意义和物理意义;可导性与连续性的关系。 2、求导法则 导数的四则运算法则;反函数的导数;复合函数的导数;基本求导法则与导数基本公式 3、隐函数的导数;参变量函数的导数;平面曲线的切线和法线及其方程;导数的应用 举例 4、微分 微分的概念;微分的基本公式及运算法则;一阶微分形式的不变性;微分在近似计算中 的应用 5、高阶导数 高阶导数的概念;某些简单函数的n阶导数 第四章 微分中值定理与导数的应用(14学时) 教学内容 1、中值定理 罗尔中值定理;拉格朗日中值定理;柯西中值定理 2、不定式的极限 of the rotor size, as well as the coupling between the adjustment shim thickness accuracy, drawings to meet the technical req lict, the coupling Centre shall prevail. 6.4.5.2.2 shaft looking for: ... 6.4.5.3. Provisional bolt confirmed the positioninga conftors, check the journal flutter tile, should generally be less than 0.10MM. When the couplings and rotor comes when there is ing end of the support bearing of the rotor, remove the bearing, fitted with special tiles and special lifting tools, disc robear be as stipulated in the manufacturer, usually 0.15~0.25mm. In the case of coupling connection, if possible, lifting only one ding to factory requirement. For coupling three supported rotor Center, the mouth under the coupling between the value shouldfilled accor-uld be no gaps, Horn, Horn and Horn 0.05mm feeler gauges fit into. Center coupling, condenser water filling and watering alignment should keep oil out of cavity and locate the rotor axis neck raised are within the prescribed limits, there shocenter pin checks and anchoring the inner cylinder. According to While coupl3reamers, boring mill, in several ways. 6.4.5.4.1 General requirements before reaming or boring, stablesuring relative changes in value should be less than 0.02mm. 6.4.5.4 coupling reamed and paired bolt reaming die reamer, adjurary bolts, this time sites slowly moving rotor, in the measured circumference of 8 equal parts back wheel rim rock, each meatempo symmetrical horizontal mount two temporary bolts tightened, attention to monitor dial indicator, then install the symmetricale bars where two back wheel bolt hole measurement above and below, confirm two back wheel bolt hole concentricity. Then, the d whitallation direction guide pin coupling with process end closed. The back wheels slowly tightened, when 2mm or so, with standarmonitor movement of the aircraft, its value is allowed to pass the value axis moving, mounted adjustment pads, note pads instindicators to free, relative position of the coupling according to the drawing requirements are. Coupling top screws, round distance apart,-ents. Plane, cylindrical and clean the coupling bolt holes should be clean and without burrs, scrap iron and greaseuirem 型与型不定式的极限;其它类型不定式的极限 3、函数的单调性和极值 函数单调性的判别法;函数极值的判别法;函数的最大值和最小值及其简单应用 4、函数图象的讨论 曲线的凸凹性与拐点;曲线的渐近线;函数作图 5、曲率 曲率的概念;曲率半径 6、方程的近似解(牛顿切线法) 第五章 不定积分(12学时) 教学内容 1、不定积分的概念与基本积分公式 原函数与不定积分;基本积分表;不定积分的性质 2、换元积分法 第一类换元积分法;第二类换元积分法 3、分部积分法 4、几类特殊函数的不定积分 有理函数的不定积分;三角函数有理式的不定积分;某些简单无理函数的不定积分 第六章 定积分(14学时) 教学内容 1、定积分的概念 定积分的定义;定积分的几何意义 2、牛顿-莱布尼兹公式和定积分的性质 牛顿-莱布尼兹公式;定积分的性质;积分上限函数及其导数 3、定积分的换元积分法与分部积分法 4、定积分的近似计算 矩形法;梯形法;抛物线法 5、定积分的应用 平面图形的面积;已知平行截面面积求立体体积和旋转体的体积;平面曲线的弧长;旋 转曲面面积;定积分在物理学上的某些应用(变力作功，压力，引力，函数的平均值)。 6、广义积分 无限区间上的广义积分;无界函数的广义积分 第七章 无穷级数(22学时) 教学内容 1、数项级数的收敛性及其性质 无穷级数的概念;级数收敛的条件;收敛级数的性质 2、正项级数 正项级数的收敛准则;比较判别法;比值判别法和根式判别法 3、任意项级数 交错级数及莱布尼茨判别法;任意项级数的绝对收敛和条件收敛;绝对收敛级数的性质 4、幂级数 函数项级数的收敛域与和函数的概念;幂级数及其收敛半径、收敛区间和收敛域;幂级 es should be clean and without burrs, scrap iron and greasement shim thickness accuracy, drawings to meet the technical requirements. Plane, cylindrical and clean the coupling bolt holdjustlooking for: ... 6.4.5.3. Provisional bolt confirmed the positioning of the rotor size, as well as the coupling between the athan 0.10MM. When the couplings and rotor comes when there is a conflict, the coupling Centre shall prevail. 6.4.5.2.2 shaft e lessbearing, fitted with special tiles and special lifting tools, disc rotors, check the journal flutter tile, should generally bn the case of coupling connection, if possible, lifting only one bearing end of the support bearing of the rotor, remove the value should be as stipulated in the manufacturer, usually 0.15~0.25mm. Ifilled according to factory requirement. For coupling three supported rotor Center, the mouth under the coupling between the -nto. Center coupling, condenser water filling and waterfit i he rotor axis neck raised are within the prescribed limits, there should be no gaps, Horn, Horn and Horn 0.05mm feeler gaugespin checks and anchoring the inner cylinder. According to While coupling alignment should keep oil out of cavity and locate tcenter 4eral requirements before reaming or boring,.4.5.4 coupling reamed and paired bolt reaming die reamer, adjustable reamers, boring mill, in several ways. 6.4.5.4.1 Gen2mm. 6measured circumference of 8 equal parts back wheel rim rock, each measuring relative changes in value should be less than 0.0tention to monitor dial indicator, then install the symmetrical temporary bolts, this time sites slowly moving rotor, in the d, atnd below, confirm two back wheel bolt hole concentricity. Then, the symmetrical horizontal mount two temporary bolts tighteneThe back wheels slowly tightened, when 2mm or so, with standard white bars where two back wheel bolt hole measurement above a losed.o pass the value axis moving, mounted adjustment pads, note pads installation direction guide pin coupling with process end cements are. Coupling top screws, round distance apart, indicators to monitor movement of the aircraft, its value is allowed tfree, relative position of the coupling according to the drawing requir- 华东师范大学 数在其收敛区间内的基本性质;简单幂级数的和函数的求法 5、幂级数的应用 泰勒级数;泰勒中值定理;初等函数的幂级数展开;近似计算 第九章 多元函数微分法及其应用(18学时) 教学内容 1、多元函数 多元函数的概念;二元函数的几何意义;二元函数的极限和连续的概念;有界闭区域上 多元连续函数的性质 2、多元函数的偏导数与全微分 偏导数;二阶偏导数;全微分;全微分在近似计算中的应用 3、复合函数和隐函数的微分法 复合函数的偏导数;一阶全微分形式不变性;隐函数的微分法 4、方向导数与梯度 方向导数与梯度的概念及其计算 5、多元函数微分学的几何应用 空间曲线的切线与法平面;曲面的切平面与法线 6、多元函数的泰勒公式与极值 二元函数的二阶泰勒公式;多元函数的极值和条件极值的概念;多元函数极值的必要条 件;二元函数极值的充分条件;极值的求法;拉格朗日乘数法;多元函数的最大值、最 小值及其简单应用 第十章 重积分及其应用(16学时) 教学内容 1、重积分的概念与性质 二重积分的概念、二重积分的性质;三重积分的概念和性质 2、二重积分的计算 化二重积分为累次积分;用极坐标计算二重积分 3、三重积分的计算 化三重积分为累次积分;用柱坐标变换计算三重积分;用球坐标变换计算三重积分 4、重积分的应用 曲面的面积;物体的重心 第十二章 常微分方程(18学时) 教学内容 1、一阶微分方程 微分方程的一般概念;可分离变量型微分方程;齐次型微分方程;一阶线性微分方程; 贝努利方程;全微分方程 2、二阶微分方程概念 可降阶的高阶微分方程;二阶线性微分方程解的性质和解的结构定理 3、二阶线性常系数微分方程 二阶线性常系数齐次方程;二阶线性常系数非齐次方程。 lict, the coupling Centre shall prevail. 6.4.5.2.2 shaft looking for: ... 6.4.5.3. Provisional bolt confirmed the positioninga conftors, check the journal flutter tile, should generally be less than 0.10MM. When the couplings and rotor comes when there is ing end of the support bearing of the rotor, remove the bearing, fitted with special tiles and special lifting tools, disc robear be as stipulated in the manufacturer, usually 0.15~0.25mm. In the case of coupling connection, if possible, lifting only one ding to factory requirement. For coupling three supported rotor Center, the mouth under the coupling between the value shouldfilled accor-uld be no gaps, Horn, Horn and Horn 0.05mm feeler gauges fit into. Center coupling, condenser water filling and watering alignment should keep oil out of cavity and locate the rotor axis neck raised are within the prescribed limits, there shocenter pin checks and anchoring the inner cylinder. According to While coupl5reamers, boring mill, in several ways. 6.4.5.4.1 General requirements before reaming or boring, stablesuring relative changes in value should be less than 0.02mm. 6.4.5.4 coupling reamed and paired bolt reaming die reamer, adjurary bolts, this time sites slowly moving rotor, in the measured circumference of 8 equal parts back wheel rim rock, each meatempo symmetrical horizontal mount two temporary bolts tightened, attention to monitor dial indicator, then install the symmetricale bars where two back wheel bolt hole measurement above and below, confirm two back wheel bolt hole concentricity. Then, the d whitallation direction guide pin coupling with process end closed. The back wheels slowly tightened, when 2mm or so, with standarmonitor movement of the aircraft, its value is allowed to pass the value axis moving, mounted adjustment pads, note pads instindicators to free, relative position of the coupling according to the drawing requirements are. Coupling top screws, round distance apart,-ents. Plane, cylindrical and clean the coupling bolt holes should be clean and without burrs, scrap iron and greaseuiremof the rotor size, as well as the coupling between the adjustment shim thickness accuracy, drawings to meet the technical req