一、课程名称：微积分（一）Calculus（一）、微积分（二）Calc

一、课程名称：微积分（一）Calculus（一）、微积分（二）Calculus（二）
二、课程编码：0700011、0700012
三、课时与学分：总学时176，学分11
四、先修课程：无
五、课程教学目标：
通过本课程的学习，要使学生获得本大纲所规定内容的基本概念，基本理论和基本技能。为今后学习后继课程以及进一步获得数学知识，奠定必要的数学基础。在传授知识的同时，努力培养学生的抽象思维和逻辑推理的能力，分析问题和解决问题的能力，准确理解和勇于创新的能力。
六、适用学科专业：理工科和经济学，管理学各专业
七、基本教学内容与学时安排
1．一元函数微积分学（56学时）
(1) 函数：区间、邻域，函数的概念，函数的基本性质(单调性、奇偶性、周期性、有界性)，基本初等函数，复合函数与反函数，初等函数、双曲函数。
(2) 极限：极限的定义(数列极限与函数极限)，单边极限，极限的性质(唯一性、有界性、局部比较、局部有界性)，两个准则，两个重要极限，无穷小量与无穷大量，极限运算法则，无穷小量比较。
(3) 连续：函数的连续性定义，左右连续的概念，间断点，初等函数的连续性，在闭区间上的连续函数的性质。
(4)导数与微分：切线斜率与瞬时速度问题，函数的导数(左、右导数的定义)，可导与连续的关系，导数的几何意义，导数的四则运算，复合函数的导数，反函数的导数，对数求导法，参变量函数的求导法和隐函数求导法、函数微分的定义，可微与可导的关系，微分的几何意义，微分运算法则，微分形式的不变性，微分在近似计算上的应用，高阶导数、相关变化率问题。
(5)微分中值定理及其应用：费尔马定理，罗尔定理，拉格朗日定理，柯西定理，洛比达法则，函数的增减性，函数的极值，函数的最大、最小值问题，曲线的凹向、拐点，渐近线，函数图形的描绘，台劳公式及其应用。
(6)不定积分：原函数与不定积分的概念，不定积分的性质，基本积分公式，换元积分法，分部积分法，有理函数积分法，三角函数的有理式的积分，简单无理函数的积分法。
(7)定积分：定积分的概念，定积分的存在条件与可积函数类简介，定积分的性质，变限积分函数及其求导定理，牛顿一莱布尼兹公式，定积分的换元法与分部积分法，广义积分的定义与计算。
(8)定积分的应用：平面图形的面积，平面曲线的弧长，平行截面积为已知的立体体积，绕坐标轴的旋转体的体积，沿直线变力作功计算，静压力计算，函数在区间上的平均值。
2．矢量代数、空间解析几何（12学时）
(1) 空间直角坐标系：空间直角坐标系的基本概念，两点间距离。矢量的模和方向，矢量的加减法，矢量与数量的乘法，基矢量，矢量的坐标表示，矢量的数量积，矢量积，混合积。
(2) 平面的点法式方程和一般方程，点到平面的距离，两平面的夹角，两平面的平行，垂直关系。直线的点向式方程和一般方程，参数式方程。两直线的夹角，两直线的平行、垂直关系、直线与平面的平行、垂直关系，平面束方程。
(3) 曲面的方程：球面方程，母线平行于坐标轴的柱面方程，旋转面方程。空间曲线方程，投影柱面与投影曲线。二次曲面分类。
3．多元函数的微积分学（50）
(1) 多元函数微分学：多元函数的概念，二元函数的极限与连续，有界闭区域上连续函数的重要性质，偏导数的概念，二元函数偏导数的几何意义，复合函数微分法，隐函数的微分法，方程组所确定的函数微分法，高阶偏导数。二元函数的全增量与全微分的概念，函数可微的必要条件与充分条件，微分的几何意义，用微分计算近似值。方向导数与梯度，二元函数的台劳公式，二元函数的极值，二元函数取得极值的必要条件和充分条件，条件极值的必要条件，拉格朗日乘数法。最大值最小值应用问题求解。
(2) 重积分：曲顶柱体的体积和平板的质量计算，二重积分的概念，二重积分的简单性质，二重积分在直角坐标下和在极坐标下的计算法。三重积分的概念及其简单性质，三重积分在直角坐标系、柱面坐标系、球面坐标系中的计算法，曲面的面积、体积计算，重积分在静力学中的应用(重心，转动惯量)。
(3) 曲线积分与曲面积分、场论初步：对弧长的曲线积分的定义、性质及计算法，曲线的质量。对坐标的曲线积分的定义、性质及计算法，格林公式，线积分与路径无关条件，沿曲线变力做功计算。对面积的曲面积分的概念，定义，性质及计算法。对坐标的曲面积分的定义，性质及计算法。各种积分间的关系，高斯公式和斯托克斯公式。场的概念，数量场的等值面，矢量场的矢量线，数量场的梯度，矢量场的散度与旋度。
4．级数与常微分方程（30学时）
(1) 数项级数：无穷级数的概念和基本性质，级数收敛的必要条件，正项级数的收敛判别法：比较原理、比阶法、比值法、根值法和积分法。交错级数的莱布尼兹判别法，绝对收敛与条件收敛。
(2) 幂级数：函数项级数的一般概念及其基本性质，和函数的连续性、函数项级数的逐项积分、逐项微分定理。幂级数的收敛域确定与和函数计算，初等函数的幂级数展开式。
(3) 付里叶级数：三角级数概念，三角函数族的正交性，函数的付里叶级数，付里叶级数的收敛定理， 上的函数展开为付里叶级数， 上的函数展开为正弦级数或余弦级数。
(4)常微分方程：微分方程一般概念，一阶微分方程：可分离变量方程，齐次微分方程，线性微分方程，贝努利方程。可降阶微分方程，线性微分方程解的结构与叠加原理，齐次常系数线性微分方程，非齐次常系数微分方程的解法。通过建立模型求解简单的微分方程应用问题。
组的通解的求法。

結果 (中文) 3:

First, the Course Name: Calculus (a), calculus (two) Calculus (two) Calculus ()
two, encoding: 0700011, 0700012
three, hours and credits: total hours 176, credit 11
four, the first course: no
five, curriculum teaching objectives: to get the students to obtain the basic concepts, basic theory and basic skills. In order to further study the following courses and to obtain the knowledge of mathematics, the necessary mathematical foundation is established. At the same time, efforts to develop students' ability to abstract thinking and logical reasoning, the ability to analyze problems and solve problems,Accurate understanding and ability to innovate. Six, applied disciplines: Science and engineering and economics, management of the professional
seven, the basic teaching content and class hour arrangement
1 (56 hours)
(1) function: interval, neighborhood, function concept, function of basic properties (monotonic, parity, periodic, boundedness), basic elementary function, composite function and inverse function, elementary function, hyperbolic function.
(2) limit: definition of limit (sequence limit and function limit), unilateral limit, limit of nature (uniqueness, boundedness, local comparison, local boundedness), two criteria, two important limits,Infinitesimal and infinite number, limit operation principle, infinite small comparison.
(3) continuous definition of function continuity, the continuity of the discontinuity, the continuity of the elementary functions, and the properties of the continuous functions on the closed interval.
(4) derivative and differential: tangent slope and the instantaneous speed problem, derivative of a function (left and right derivative definition), and continuous relationship, the geometric meaning of derivative, derivative of the four operations, the derivative of the composite function, derivative of the inverse function, the logarithmic derivative method, variable as a function of the amount of the derivative method and implicit function derivative method, differential function definition, micro and derivative and differential geometric meaning,Differential operation, differential form of invariance, differential in the approximate calculation of the application, the higher order derivatives, the relevant change rate.
(5) differential mean value theorem and its application: Fermat's theorem, Rolle theorem, Lagrange's theorem, Cauchy's theorem, Luo ratio rule, increasing and decreasing of functions, the extremum of function, function of maximum and minimum value problem, concave curve, inflection point, asymptote, graphic function description, Taylor formula and its application.
(6) indefinite integral: the concept of the original function and the indefinite integral, the nature of the indefinite integral, the basic integral formula, the integral method, the partial integral method, the rational function integral method,The integral method of the integral of the trigonometric function and the integral method of the simple irrational function.
(7) definite integral, definite integral, definite integral and existence conditions of the integrable function introduction class, nature of definite integral, variable limit integral function and derivative theorem, Newton Leibniz formula, the integral element method and subsection integral method, the generalized integral definition and calculation.
(8) definite integral application: area of a plane figure, the plane curve arc length, parallel to the cross-sectional area of known three-dimensional volume, around the axis of the rotating body volume, along a straight line variable masterpiece power calculation, static pressure calculation function in the interval average value. 2 vector algebra,Space analytic geometry (12 hours)
(1) space rectangular coordinate system: basic concept of space rectangular coordinate system, distance between two points. Vector and vector, vector and scalar multiplication, vector and vector, vector product, vector product, mixed product. (2) the points of the plane equation and the general equation, the distance to the plane, the angle between the two planes, the parallel and the vertical relationship between the two planes. The linear equation of the point and the equation of the equation. Two the angle of the straight line, the parallel and vertical relations of the two straight lines, the parallel and vertical relations between the straight line and the plane, the plane beam equation.
(3) the equation of the surface: the spherical equation, the cylinder equation of the bus parallel to the axis of the axis, the equation of the rotating surface. Space curve equation, projection cylinder and projection curve. Two surface classification. The concept of multivariate function, the limit and continuity of two variables, the important properties of continuous functions, the geometric meaning of partial derivative, the differential method, the higher order partial derivatives of the differential equations of the two elements, the derivative of the 3 variables, the geometric meaning of the derivative, the derivative method, the differential method, the higher order derivatives of the 1 variables. The concept of total increment of two element function and full differential,Necessary and sufficient conditions for a function to be differentiable, the geometric meaning of the differential, and the approximate value of the differential calculation. Directional derivative and gradient, two element function, the extreme value of the function of two yuan, the necessary and sufficient condition of the extreme value of the function of the two elements, and the necessary condition of conditional extreme value, Lagrange multiplier method. Application problem solving for maximum minimum value.
(2) integral: Calculation of cylinder volume and plate quality, the concept of double integral, the simple nature of the double integral, double integral in Cartesian coordinates and polar coordinates calculation method. The concept of three integral and its simple properties, the three points are in the Cartesian coordinate system,Calculation method in cylindrical coordinate system, spherical coordinate system, surface area, volume calculation, application of heavy integral in statics (center of gravity, moment of inertia).
(3) curvilinear integral and surface integral and field preliminary: the arc length of the curve integral definition, properties and calculation method, the curve of quality. To coordinate the definition of the integral curve, properties and calculation method and Green's formula, the path of a line integral irrelevant condition, along the curve change force acting on a calculation. Concepts, definitions, properties, and computational methods of surface integrals in the area. Definition, properties and calculation method of surface integral for coordinates. Relationship among various integrals, Gauss formula and Stokes formula.The concept of field, isosurface scalar field, vector vector field, gradient number field, divergence and curl of a vector field. The concept and basic properties of infinite series and the necessary conditions for the convergence of series of 4 series and ordinary differential equations (1 hours)
(30). The convergence of the series is compared with the principle, the ratio method, the ratio method, the root value method and the integral method. Alternate series Leibniz method, absolute convergence and conditional convergence.
(2) power series: general concept and basic properties of series of functions, and the function of continuity and series of functions termwise integration, termwise differentiation theorem.Convergence of the power series is determined by the power series expansion of the elementary functions. (3) Fourier series: trigonometric series, trigonometric function, Fourier series, Fourier series, Fourier series, Fourier series, and the function expansion of the Fourier series.
(4) often differential equations: the general concept of differential equation, first order differential equations: separable variable equation, homogeneous differential equations, linear differential equation, Bernoulli equation. The structure and principle of the solution of linear differential equation, the linear differential equation with constant coefficient,Solution to the differential equation of non homogeneous constant coefficient. Solving simple differential equations using the model. Method for the general solution of the group.