几何学
1， 点线面的相互关系、方向和角度与平行、恒等和叠合与对称、向量的加法和减法、向量与数量的乘法、内积、外积、混合积、向量对于给定基底的坐标。
1。线性向量空间，例子，子空间。
2。线性独立与相关，相关性的记号，与向量分解的关系，展开为线性无关向量之和。
3。多个向量的秩和他们的性质。
2，空间的维数、基与一般笛卡尔坐标。空间曲面和空间曲线的方程、坐标变换、平面方程、平面对于坐标系的位置、平面的相互位置。
4。向量空间的同构定理。
5。子空间的和与交，子空间的和的维数。
6。两个和多于两个子空间的直和，外直和。
3， 直线方程、直线和平面的相互位置、两条直线的相互位置、二次曲面分类、椭圆面、双曲面、抛物面、锥面和柱面。
4， 二次曲面的直母线、二次曲面的直径和直径平面、二次型的变换、不变量。
5， 曲线直径、曲面和曲线的中心、曲线的对称轴、曲面的对称平面、双曲线的渐近线、双曲面的渐近锥面、曲线的切线、曲面的切平面。
6， 正交变换、仿射变换、仿射变换的基本不变量、仿射变换下的二次曲线和二次曲面、射影变换、齐次坐标、无穷远点、射影变换下的二次曲线和二次曲面、极点和配极。
7， Euclid几何中的平面与直线、Euclid平面与复数、Euclid空间与仿射空间、仿射簇。
8， 仿射直线与仿射平面的公理化模型、平面上的线性方程、凸几何、仿射几何的基本定理、仿射空间、他们的特性，仿射坐标系，向量和点的坐标。有限维凸几何、Caratheodory与Radon引理、Helly定理。
8。仿射与向量空间上的坐标变换，坐标变换的矩阵表示，新旧坐标上点与向量之间的联系。
9。仿射空间的子空间，参数方程，平行六面体。
10。作为线性方程组解集的仿射子空间。
11。仿射空间上两个子空间的相互联系。
12。仿射空间上多个点对线段的划分，分割点的坐标。
13。仿射空间的同构定理，仿射空间和向量空间上概念的等价性。
9， 射影几何、射影直线与平面、Pappus与Desargues定理、n维射影空间简介、二次平面曲线的分类、四次方程、Pascal定理。
10， 圆与球、球面几何、n维球的几何、Riemann椭圆几何、Lobachevsky几何的Klein模型、线性分式变换与球极投影、Lobachevsky几何的其它模型、初等双曲几何。
11， Euclid几何和Riemann椭圆几何及Lobachevsky几何的同构性、复射影空间、影变换的不动点、调和四重点与调和四重线。
14。标量积，欧氏向量——点空间，正交向量的线性无关性。
15。正交基与标架，正交化过程。
16。欧氏向量——点空间上的同构定理，两点间的有向距离，三角不等式，向量间的角度，正交向量的直角性，毕达哥拉斯定理，柯西——布尼雅科夫斯基不等式，例子。
17。子空间的正交完备性及其相关性质，子空间与向量所成角，子空间与向量所成距离。
18。子空间上向量的射影，傅利叶系数，超定方程组解的最小二乘法。
19。超平面上的正规向量，点到超平面的距离，平行超平面间的距离。
20。格拉姆行列式及其性质。
21。可测平行六面体的体积，体积与矩阵行列式之间的关系。
22。线性空间上的线性映射，矩阵与线性映射的解析形式。
23。线性映射的合成，线性空间合成的矩阵。
24。映射的核与像的维数，同构条件。
25。线性算子，他们的解析形式，基上算子的矩阵的独立性（包括张量情形）。
26。线性算子环，矩阵环的同构，算子的多项式，非退化算子（一般线性群）。
27。不变子空间，对算子的矩阵的影响，实域与复域上的不变子空间问题，算子的矩阵
的阶梯形。
28，算子的特征值与特征向量，特征子空间，特征子空间的和。
29。算子的特征多项式及其相关不变量，特征子空间的等价。
30，特征值的重数与特征子空间的维数，算子的矩阵的对角化条件。
31。多项式与它的零化算子，哈密顿——凯莱定理。
32。退化算子，其与算子的特征多项式系数的关系。
33。把空间分解成不变子空间的直和，其与讲特征多项式分解成基本因子的关系。
34。最小零化多项式，他和特征向量等的关系。
35。根子空间与算子的根向量。
36。根子空间上算子的矩阵的标准型，算子的矩阵的若尔当标准型，最小多项式与若尔当标准型的关系。
37。实空间的复化，线性映射与线性算子。
38，实算子的矩阵的标准型。
39。若尔当标准型的矩阵理论。
40。线性映射与仿射空间上的几何变换，三个点的相互关系，几何变换的解析形式，仿射表换与图形的仿射分类的概念。
41。线性函数空间，半线性函数，伴随基，变换的矩阵，线性与半线性函数空间上的坐标变换。
42。线性与半线性函数空间上的自共轭，自共轭基。
43。复空间上的算子与映射，对偶映射与基础空间上的算子，自对偶。
44。双线性与多线性函数及其坐标表示，向量函数空间，基本函数及其与基础空间的基的关系。
45。基上双线性函数的矩阵无关性，函数的秩。
46。双线性与多线性函数的核及其维数，非退化函数。
47。复空间上的双线性与多线性函数空间的自然同构。
48，对称、斜对称与埃尔米特函数。
49，子空间与向量的正交化及其与对称、反对称与埃尔米特函数的关系，正交补的维数与性质。
50。对称、斜对称与埃尔米特函数的正交形式。
51。函数的正交形式的唯一性，实域上的对称与埃尔米特函数的的惯性定理。
52。二次函数及其与二次性及规范型的关系。
53。规范型的雅可比定理与格拉姆方法。
54。正定系统与埃尔米特函数，西尔维斯特判据。
55。对称、斜对称与埃尔米特标量积，拟欧氏、埃尔米特与辛向量空间及其同构定理，正交化与对称基。
56。内积空间上的自然同构，线性函数的一般形式，零向量和拟欧氏、埃尔米特与辛向量空间的子空间，正交非迷向向量的线性无关性，格拉姆行列式，正交化过程，正交完备性。
57。辛向量空间，哈密顿基，迷向子空间。
58。酉空间，柯西——布尼雅科夫斯基不等式，三角不等式。
59。正交、拟正交、酉、拟酉与辛矩阵，特殊线性群。
60。标量积下不变的算子（正交、拟正交、酉、拟酉与辛）与他们的性质，不变子空间的性质，等距同构，算子群。
61，正交与酉算子的标准型及其唯一性，特征子空间，点空间上的正交变换。
62。群，平面上的伪标量，双曲三角，洛伦兹变换，三维拟欧氏空间。
63。复空间的实化，埃尔米特空间上的拟欧氏结构与辛结构。
64。算子与算子群的实化。
65。内积空间上的伴随算子的存在性与唯一性，与复空降上的对偶算子的联系。
66。欧氏空间与酉空间上的自伴算子及其标准型，特征值与子空间的性质。
67。自伴正交算子与自伴酉算子的极分解。
68。内积空间上的双线性与多线性函数，这些函数与算子空间的自然同构。
69。欧氏（酉）空间上的对称（埃尔米特）函数的标准型，点空间上的二阶超曲面方程的标准型。
70。一对其中之一为正定的二次型的不变量，在标准基上的讨论。71。张量，例子，张量与多线性函数，张量空间。
72。张量积，张量代数，张量空间上的基与坐标。
73。卷积算子及其性质，例子。
74。内积空间上的张量指标。
75。张量的对称与斜对称及其坐标，对称算子与交错张量及其性质。
76。斜张量，外积算子及其性质。
77。多重向量和斜对称函数及其坐标，例子，子空间的普吕克坐标。
78。多重向量和斜对称函数的简化。
79。多重向量和斜对称函数空间的基与维数。
解析几何
一，向量运算。
1。向量，向量的线性运算及其基本性质。
2。向量的线性相关及其几何意义。
3。基，向量子与点组成的坐标系，坐标系的几何意义。
4。线性相关，坐标系上的向量线性相关的判据。
5。数量积，它的主要性质与公式。
6。从一个基到另一个基的变换矩阵，点-向量坐标系的变换，变换矩阵的性质，标准正交基。
7。平面的定向，平行四边形的定向体积，它的基本性质和公式。
8。空间的定向，平行六面体的定向体积，它的基本性质和公式。
9。向量的向量积与混合积，它的基本性质与公式。
10。正交矩阵，正交坐标的变换，二阶正交矩阵的分类。
11。Euler角，用三阶正交矩阵表示Euler角。
12。极坐标系，空间上的柱坐标系与球坐标系，极坐标、柱坐标与球坐标与正交坐标的关系。
13。二重向量与三重向量，二重向量的线性运算与度量理论。
14。二维与三维线性算子，可逆线性算子，等距线性算子。
二，直线与平面。
15。直线与平面的参数方程，它们的集合意义。
16。作为直线或平面方程的一阶方程及其与参数方程的联系。
17。平面与空间中两条直线的相互关系。18。两个平面的相互关系，及其与一阶方程的联系。
19。向量、直线与平面形成的角及其计算方法。
20。从某一点到直线或者平面的距离，两条直线的距离。
21。平面与空间上一阶不等式的几何意义。
22。平面与空间上的线束，线束的线性无关。
23。空间上的平面束，平面束的线性无关。
24。平面上三条直线的相互关系，空间上三个平面的相互关系，他们的方程组的秩。
三，二次曲线与曲面。
25。代数曲线与曲面，代数曲线与曲面的次数，穿过直线的代数曲线，与平面相截的代数曲面。
26。可约曲线与可约曲面及其几何意义，包含直线的曲线的可约定理，包含平面的曲面的可约定理。
27。利用二阶正交变换化二次多项式为标准型，二次曲线的分类。
28。椭圆与双曲线、抛物线的焦点性质，双曲线的渐进线方程。
29。椭圆、双曲线、抛物线的准线性质，他们在极坐标下的方程。
30。圆柱面与圆锥面，旋转面。
31。利用三阶正交变换化二次多项式为标准型，二次曲面的分类。
32。椭球面、虚椭球面和双曲面，他们的基本性质，他们的图像的绘制。
33。椭圆抛物面，他的基本性质及其图像的绘制。
34。双曲抛物面，双曲抛物面的母线及其基本性质。
35。双叶双曲面与单叶双曲面，双叶双曲面与单叶双曲面及其基本性质。
四，正交不变量及二次曲面与曲面的分类。
36。多项式分类的正交不变量，正交不变量与多项式的系数与根。
37。二次多项式的正交不变量。
38。利用正交不变量对正交坐标系上的二次曲线进行分类。
39。利用正交不变量对正交坐标系上的二次曲面进行分类。
40。二次曲线与曲面的中心的坐标。
41。而曲面与曲线的二次方程的比例性。
42。二次曲线的渐进线，二次旋转面的渐进锥面，利用二次曲线与直线的相交对其进行分类。
43。圆锥曲线的直径与方程，旋转面与其中心线平面相交形成的曲线的直径及其方程。
44。曲线的共轭的方向及其直径，椭圆、双曲线与抛物线的共轭的方向及其直径。
45。对称轴，对称轴与直径的关系，对称轴向量的坐标，二次曲线的相互关系。
46。对称平面，对称平面与中心线平面的相互关系，对称平面的法向向量的坐标，二次曲面的相互关系。
47。二次曲线束，通过五个给定点的二次曲线，Sturm定理与Pascal定理。
五，仿射与正交变换。
48。仿射变换的定义及其性质，仿射变换群。
49。从一组基到另一组基的仿射变换的矩阵，它们的几何意义，仿射变换公式。
50。仿射变换群的等价关系，二次曲线的仿射分类。
51。二次曲面的仿射分类。
52。等度量变换的基本性质。
53。二次曲线的正交分类，二次曲线的正交不变量及其标准方程的系数。
54。二次曲面的正交分类，二次曲面的正交不变量及其标准方程的系数。
55。平面正交变换的结构。
56。空间正交变换的结构。
57。平面与空间仿射变换的结构。
六，射影几何，
58。作为直线与平面形成的把的射影平面，作为普通平面的推广的射影平面，点与直线在这两种情况下的射影坐标，对偶原理。
59。圆锥曲面模型上的曲线，二次曲线在扩充平面下的完备性，二次曲线在射影坐标下的方程。
60。射影变换，作为中心仿射变换推广的射影变换。
61。扩充平面上的仿射变换与射影变换的联系。
62。化二次型为标准性的变换的一致性，二次曲线的射影形式与射影分类及其与仿射分类的关系。
63。椭圆与双曲几何的基本概念，Erlangen纲领。
64。几何的公理系统，Euclid几何的公理系统，仿射几何的公理系统，复数域上的仿射几何。
Analytical Geometry
1，M.M.Postnikov，解析几何，科学出版社，1973。
2，P.S.Alexandrov，解析几何讲义，科学出版社，1967。
3，P.S.Alexandrov，解析几何与线性代数教程，科学出版社，1979。
4，B.N.Delone、D.A.Raykov，解析几何，第一卷，国家联合出版社，1948。
5，B.N.Delone、D.A.Raykov，解析几何，第二卷，国家联合出版社，1949.
6，Y.Smirnov，解析几何，俄罗斯教育与科学文献出版社，2004。
7，V.V.Prasolov、V.M.Tikhomirov，几何学，莫斯科不间断数学教育中心，1997。
8，P.S.Modenov、A.S.Parhomenko，解析几何习题集，科学出版社，1976。
Basic Topology by Armstrong
Differential Geometry of Curves and Surfaces by Manfredo Do Carmo
Hatcher “Algebraic Topology” Cambridge UP
Munkries “Topology” 2nd ed. Prentice Hall
A.T.Fomenko Differential geometry and topology，Consultants Bureau
Dubrovin, Fomenko, Novikov “Modern geometry-methods and applications”Vol 1—3
A Comprehensive Introduction to Differential Geometry vol 1-5 ,by Michael Spivak
Bogorelov，解析几何。
《解析几何习题集》巴赫瓦洛夫
狄隆涅 “(解析)几何学”
穆斯海里什维利 “解析几何学教程”
丘维生，《解析几何》，北大版
南开数学系，《空间解析几何》，高教版
陈(受鸟) “空间解析几何学”
朱鼎勋 “解析几何学”
吴光磊《解析几何简明教程》高等教育出版社
丘维声《解析几何》北京大学出版社
《解析几何》吕根林，许子道（有配套的辅导）
《解析几何》尤承业
《空间解析几何与微分几何》（大学数学学习方法指导丛书）黄宣国
《高等几何》梅向明等
《高等几何习题集》
《高等几何》朱德祥
《高等几何》周建伟
175《高等几何学习指导与习题选解》梅向明，刘增贤编
176《高等几何》第2版 罗崇善, 庞朝阳, 田玉屏编著
Arithmetic Geometry
X Fulton, William. Algebraic Curves: An Introduction to Algebraic Geometry.
This book is available for free on Fulton’s website.
Buy at Amazon Milne, J. S. Elliptic Curves. BookSurge Publishers, 2006. ISBN: 9781419652578.
This book is also available for free on Milne’s website, along with addendum/erratum.
Buy at Amazon Serre, Jean-Pierre. A Course in Arithmetic. Springer-Verlag, 1996. ISBN: 9783540900405.
Buy at Amazon Shafarevich, I. R. (translated by Miles Reid). Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550.
Buy at Amazon Stichtenoth, H. Algebraic Function Fields and Codes. Berlin: Springer, 2008. ISBN 9783540768777. [Preview with Google Books]
Buy at Amazon Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer-Verlag, 2009. ISBN: 9780387094939. (errata) [Preview with Google Books]
Geometry
1 , Relationship between point, line, plane, directions and angles with parallel, identical and overlapping with symmetry, vector addition and subtraction, multiplication of a vector quantity, product, product, mixing plot coordinates, vectors for a given substrate.
1 。 Linear vector spaces, for example, the subspace.
2 。 Linear independence and correlation, correlation between signs and vector decomposition relationship of linearly independent vectors and.
3 。 Rank of multiple vectors and their properties.
2 , Spatial dimensions, base and General Cartesian Coordinates. Surface and space the equation of the curve, coordinate transformations, plane equation, plane, plane of the coordinate system relative positions.
4 。 Isomorphism theorems for vector spaces.
5 。 Subspace and make, and dimension of subspace.
6 。 Two or more than two spaces and straight, external direct sum.
3 , Linear equations, linear and Planar location, between two straight, quadratic surface classification, ellipsoid, hyperboloid, parabolic surfaces, conical and cylindrical.
4 , Ruled lines of quadratic surface, second surface diameter and diameter plane, transform and invariant of the quadratic form.
5 , Diameter of curves, surfaces, and curved Center, curved axis of symmetry, the symmetry of the surface plane, hyperbolic Asymptote, hyperboloid asymptotic cones, tangent to the curve, surface, tangent plane.
6 , Orthogonal transformations, affine transformations, affine transformation of the basic invariant, affine transformations of Conic Sections and quadric surfaces, projective transformations, homogeneous coordinates, the point at infinity, under the projective transformations of quadratic curve and quadratic surface, pole and polar.
7 , Euclid geometry in the plane and straight lines,Euclid plane with complex numbers,Euclid space and affine space, affine clusters.
8 , The affine line and axiomatic models of the affine plane, the plane of linear equations, convex geometry, affine geometry, the fundamental theorem, an affine space, their characteristics, the affine coordinate system, vectors and coordinates of the point. Finite-dimensional convex geometry,Caratheodory and Radon ‘s lemma, andHelly theorem.
8 。 Coordinates on a vector space and affine transform, coordinate transformation matrices, linkages between the new and old coordinates of points and vectors.
9 。 An affine subspace of the space, parametric equations, parallelepiped.
10 。 As an affine subspace of the set of solutions to linear equations.
11 。 Two subspaces of the affine space of interrelated.
12 。 Division of the affine space multiple points on the line, the coordinates of the point.
13 。 Isomorphism theorems of affine space, affine space and the concept of vector space equivalence.
9 , Projective geometry, lines and planes of projection,Pappus and Desargues theorem, then -dimensional projective space profile, secondary classification of Planar curve, quartic equation,Pascal Theorem.
10 , Round ball, spherical geometry,an n -dimensional spherical geometry,Riemann geometry, an ellipseLobachevskygeometry of the Klein model, Linear fractional transformations with the stereographic projection,Lobachevsky geometry model, elementary hyperbolic geometry.
11 , Euclid geometry or Riemann elliptic geometry or Lobachevsky geometry of homogeneous, complex projective space, shadow transform fixed points, to reconcile the four key and harmony Quartet.
14 。 Scalar product, Euclidean vectors – space, orthogonal vector linear independence.
15 。 Orthonormal frame, orthogonalization process.
16 。 Euclidean vectors – spatial isomorphism theorem, distance between two points, the triangle inequality, angle between the vectors, orthogonal vector at right angles to, the Pythagorean theorem, Cauchy – buniyakefusiji inequalities, examples.
17 。 Subspace properties of orthogonal complete set and its associated, subspaces and the angle of the vector, distance into subspace and vector.
18 。 The projection of the vector subspace, Fourier coefficients, solution of overdetermined equations of least squares.
19 。 Hyperplane normal vector, distance between points and Hyperplanes, the distance between two parallel hyperplanes.
20 。 The gram determinant and its nature.
21 。 Measured volume of the parallelepiped volume relationship with matrix determinant.
22 。 Linear map in linear space, analytic forms of matrices and linear maps.
23 。 Synthesis of linear maps, matrix synthesis of linear spaces.
24 。 Mapping the nuclear dimension, homogeneous conditions.
25 。 Linear operators and their analytical form, independence of the matrix of the base pay (including tensor).
26 。 Ring of linear operators, matrix ring isomorphisms of operator polynomials, non-degenerate operator (the General linear group).
27 。 Invariant subspace, matrix effects on operator, it is not on a domain and the domain invariant subspace problem, operator of matrix
A ladder.
28 , Operator of the characteristic value and the characteristic vector, subspace, subspace.
29 。 The characteristic polynomial and its related invariants of operator, equivalent to subspace.
30 , The Multiplicities of the eigenvalues and Eigen-subspace dimension, operator of matrix Diagonalization condition.
31 。 Polynomials and the annihilation operators, Hamilton – Cayley’s theorem.
32 。 Degradation of operators, and operator of its characteristic polynomial coefficient of relationship.
33 。 Decomposed into invariant subspaces of the space and, with characteristic polynomial is decomposed into basic factor speaking of relationships.
34 。 Minimal zero polynomial, his relations with the eigenvectors.
35 。 Root subspace with the vector operator roots.
36 。 Standard root spaces of matrices, operator of the Jordan canonical form of matrices, minimum polynomial relations with Jordan canonical form.
37 。 Real space complex, linear mapping and linear operators.
38 , Real operator matrix Standard.
39 。 Jordan canonical form of matrix theory.
40 。 Geometric transformation on linear map affine spaces, relations among the three points, analytic geometry transform, affine forms for affine classification and graphic concept.
41 。 Linear function spaces and linear functions, along with the base, transformation matrices, linear and semilinear function space coordinate transformation.
42 。 Linear and nonlinear function on the space of self-adjoint, since the conjugate base.
43 。 Operators of complex space and maps, dual mapping of operators on basic space, self-dual.
44 。 Double linear and multilinear functions and coordinates, the vector function space, basic functions and their relationship to basic space base.
45 。 Based on bilinear function is independent of the matrix, rank of a function.
46 。 Nuclear bilinear and linear functions and its dimension nondegenerate function.
47 。 Double linear and multilinear functions on complex spaces spaces naturally isomorphic.
48 Symmetric, hermitian and skew-symmetric functions.
49 The subspace orthogonal to the vector and its relationship with symmetrical, anti-symmetric hermitian function, dimension and nature of the orthogonal complement.
50 。 Symmetric forms, hermitian and skew-symmetric functions are orthogonal.
51 。 Functions of orthogonal form of uniqueness, real-symmetric hermitian function on the domain of the law of inertia.
52 。 Quadratic function and its relation with twice-sex and norms.
53 。 Normative methods of Jacobi’s theorem and Gramm.
54 。 System and hermitian positive definite function, Silvester criterion.
55 。 Symmetric, hermitian and skew-symmetric scalar product, to be Euclidean, hermitian and symplectic vector space isomorphism theorem and its orthogonal symmetric matrix.
56 。 Natural isomorphism on the inner product space, the General form of linear functions, zero-vectors of and quasi Euclidean, hermitian and symplectic subspace of a vector space, of orthogonal anisotropic linear independence of vectors, Gram determinant, orthogonalization, orthogonal complete.
57 。 Symplectic vector space, Hamiltonian matrix, isotropic subspaces.
58 。 Unitary spaces, Cauchy -the buniyakefusiji inequality, the triangle inequality.
59 。 To be orthogonal, unitary, orthogonal to be unitary symplectic matrix, the special linear group.
60 。 Dot product is invariant under the operator (orthogonal to be orthogonal, unitary, quasi unitary and spicy) with their nature, nature of the invariant subspace, isometric isomorphism, and operator groups.
61 , Orthogonal and unitary operator standard and unique, feature subspaces orthogonal transform on the space.
62 。 Group, pseudo scalars in the plane, hyperbolic trigonometry, the Lorentz transformation, three-dimensional quasi Euclidean space.
63 。 Complex space of real, hermitian structure of quasi-Euclidean space and symplectic structure.
64 。 Operators and operator group materialized.
65 。 On the inner product space of existence and uniqueness of the adjoint operator, with dual operator on a complex airborne contact.
66 。 And unitary space of self-adjoint operators on Euclidean space and its standard, characteristic value and the nature of the subspace.
67 。 Self-adjoint orthogonal operator self-adjoint polar decomposition of unitary operators.
68 。 Inner product on the space of bilinear and linear functions, these functions and operator spaces naturally isomorphic.
69 。 Euclid (unitary) spatially symmetric (hermitian) function of the standard type, spatial hypersurfaces of second order equation of the standard type.
70 。 One pair for positive definite quadratic forms invariant, in discussions on a standard basis.
71 。 Tensor examples of tensor and linear functions and tensor spaces.
72 。 Tensor and tensor algebra, tensor space based on the coordinates.
73 。 Convolution operators and their properties, examples.
74 。 On the inner product space of tensor indices.
75 。 Symmetric and skew-symmetric tensor and its coordinates, and alternating tensor of symmetric operators and their properties.
76 。 Diagonal tensor, the outer product operator and its properties.
77 。 Multiple vector and skew-symmetric functions and coordinates, for example, the Plücker coordinates of the subspace.
78 。 Multiple vector and skew-symmetric functions are simplified.
79 。 Multiple vector and the base and dimension of the space of skew-symmetric functions.
Analytic Geometry
A, vector operations.
1 。 Vector, vector linear operation and its basic properties.
2 。 Vector’s linear correlation and geometric significance.
3 。 Base to the coordinate system of the quantum dots, geometric meaning of the coordinate system.
4 。 Linear correlation, coordinates vector linear correlation criteria.
5 。 Scalar product, its main properties and formulas.
6 。 The transformation matrix from one base to another base, point – vector coordinate system transformation, the nature of transformation matrices, orthonormal basis.
7 。 Orientation of the plane, the orientation of the parallelogram volume, its basic properties and formulas.
8 。 Spatial orientation, the targeted volume of the parallelepiped, its basic properties and formulas.
9 。 Vector vector products and mixed products, its basic properties and formulas.
10 。 Orthogonal matrices, orthogonal coordinate transformation, classification of the erjiezheng matrix.
11 。 Euler angles, using sanjiezheng matrix representation of Euler angles.
12 。 The polar coordinate system, spatial cylindrical coordinates and spherical coordinates, polar coordinates, cylindrical coordinates and spherical coordinates to orthogonal coordinates.
13 。 Triple vector and bivector, dual vector a linear operation and measure theory.
14 。 Two-dimensional and three-dimensional linear operators and invertible linear operator, isometric linear operators.
Two, lines and planes.
15 。 The parametric equations of lines and planes, meaning their collection.
16 。 First-order equation as line or plane and its links with the parametric equations.
17 。 Relationship between the two lines in the plane and space. 18。 Relationship between two planes, and its links to first-order equations.
19 。 Angle of vectors, linear and Planar formation and its calculation method.
20 。 The distance from a point to a line or plane, the distance between two lines.
21 。 First-order geometric meaning of inequalities on the Planar and space.
22 。 Plane and space on the wiring harness, wiring harnesses are linearly independent.
23 。 Space plane pencil, pencil of planes are linearly independent.
24 。 The interrelationship of the three lines in the plane, spatial relationships among the three planes, the rank of their equations.
Third, the quadratic curves and surfaces.
25 。 Algebraic curves and surfaces, the number of algebraic curves and surfaces, through linear algebraic curve, algebraic surface, who with a planar phase.
26 。 Reducible curves and decomposed surface and geometric significance, contains a linear curve can be agreed, and includes flat surfaces can be agreed.
27 。 By erjiezheng transformation of quadratic polynomial for the standard type, the classification of quadratic curves.
28 。 The focus of the ellipse and Hyperbola, parabola, Hyperbola equation of Asymptote.
29 。 Alignment of the ellipse, Hyperbola, parabola, their equations in polar coordinates.
30 。 Cylindrical and conical surface, surface of revolution.
31 。 Using third-order orthogonal transformation of quadratic polynomial as standard, classification of quadric surfaces.
32 。 Ellipsoid, ellipsoid and hyperboloid deficiency, their basic properties, they draw their images.
33 。 Elliptic paraboloid, his basic properties and image rendering.
34 。 The hyperbolic paraboloid, hyperbolic paraboloid bus and its basic properties.
35 。 Double leaf bilateral curved surface and hyperboloid, double leaf bilateral curved surface and hyperboloid and its basic properties.
The four orthogonal invariants and classification of quadric surfaces and surfaces.
36 。 Classification of orthogonal polynomial invariants, orthogonal invariants with polynomial coefficients and roots.
37 。 Orthogonal invariants of the quadratic polynomial.
38 。 Orthogonal coordinate system using orthogonal invariants classify the conic section.
39 。 Orthogonal coordinate system using orthogonal invariants classify the quadric surfaces.
40 。 Quadratic curve and the coordinates of the center of the surface.
41 。 Proportion of quadratic equations and curves and surfaces.
42 。 Quadratic curve Asymptote, the asymptotic cone of quadratic rotating surfaces, use quadratic curve and the line of intersection of their classification.
43 。 Diameter and equations of Conic sections, surface of revolution with its centerline intersects the plane formed by the diameter of the curve and equation.
44 。 The conjugate direction and diameter of the curve, ellipse, Hyperbola and parabola Conjugate directions and diameters.
45 。 Symmetry, symmetry axis and the relationship of the diameter of symmetrical axis coordinate of the vector, Conic to each other.
46 。 The plane of symmetry, relationship between center line of the plane and the plane of symmetry, symmetry plane normal vector of coordinates, the interrelationship of quadric surface.
47 。 Conic bundle, quadratic curve through five given points,Sturm theorem and Pascal theorem.
Five, affine transforms.
48 。 Definition and properties of affine transformations, affine transformation group.
49 。 From one base to another base set of affine transformation matrices, their geometric meaning of affine transformation formula.
50 。 Affine transformation group equivalence, affine classification of quadratic curves.
51 。 Affine classification of quadric surfaces.
52 。 Isometric transformation properties.
53 。 Orthogonal classification of quadratic curve, Conic standards of orthogonal invariants and its coefficients of equations.
54 。 Orthogonal classification of quadric, quadric standards of orthogonal invariants and its coefficients of equations.
55 。 Planar orthogonal transform structure.
56 。 Spatial structure of orthogonal transformation.
57 。 Affine transformation of the plane and space structure.
Six, projective geometry,
58 。 As the formation of lines and planes of the projective plane, as a generalization of ordinary plane projective plane, points and lines in projective coordinates in both cases, the principle of duality.
59 。 Cone of curves on a surface model, the completeness of quadratic curves under the expanded flat, equation of Conic in projective coordinates.
60 。 Projective transformation, as generalized Centro-affine transformation projective transformations.
61 。 Extended affine and projective transformations on the plane of contact.
62 。 Transformation of quadratic forms as a standard of consistency, Conic projection form and classification of projective and affine classification of relationships.
63 。 Basic concept of elliptic and hyperbolic geometry,Erlangen programme.
64 。 Axioms of geometry,Euclid geometry axioms, axiom systems of affine geometry, affine geometry on the complex field.
Analytical Geometry
1 , M.M.Postnikov , Analytic geometry, science press, 1973 。
2 , P.S.Alexandrov , Lectures on analytic geometry, science press, 1967 。
3 , P.S.Alexandrov , Analytic geometry and linear algebra tutorials, science press, 1979 。
4 , B.N.Delone 、 D.A.Raykov , Analytic geometry, volume I, joint publishing house, 1948 。
5 , B.N.Delone 、 D.A.Raykov , Analytic geometry, volume II, National Union Publishing House, 1949.
6 , Y.Smirnov , Analytic geometry, Russian educational and scientific literature Publishing House, 2004 。
7 , V.V.Prasolov 、 V.M.Tikhomirov In geometry, continuous mathematics education centre in Moscow, 1997 。
8 , P.S.Modenov 、 A.S.Parhomenko And analytic geometry problem set, science press, 1976 。
Basic Topology by Armstrong
Differential Geometry of Curves and Surfaces by Manfredo Do Carmo
Hatcher “Algebraic Topology” Cambridge UP
Munkries “Topology” 2nd ed. Prentice Hall
A.T.Fomenko Differential geometry and topology ,Consultants Bureau
Dubrovin, Fomenko, Novikov “Modern geometry-methods and applications”Vol 1—3
A Comprehensive Introduction to Differential Geometry vol 1-5 ,by Michael Spivak
Bogorelov And analytic geometry.
The analytic geometry problem set, Bakhvalov
Di Long Nirvana “( Resolved ) Geometry”
Musihailishiweili ” Analytic geometry tutorials”
Qiuweisheng, analytic geometry, North Edition
Department of mathematics, Nankai, of the geometry of the higher education version
Chen ( Bird ) ” Space analytic geometry”
Zhu Dingxun ” Analytic geometry”
Wu Guanglei of analytic geometry is a simple tutorial by higher education press
Qiu weisheng analytic geometry, Peking University Press
The analytic geometry of Lv Genlin, Xu Zidao (supporting guidance)
The analytic geometry You Chengye
The space of analytic geometry and differential geometry (math study guide series) Huang Xuanguo
The higher geometry Mei Xiangming
Higher geometry problem set
Higher geometry of Zhu Dexiang
Higher geometry of zhoujianwei
175 Higher geometry study guide and exercises selected solutions Mei Xiangming, Liu Zengxian series
176 Higher geometry 2 Mr Chong Shan , Pang Chaoyang , written by Tian Yuping
Arithmetic Geometry
X Fulton, William. Algebraic Curves: An Introduction to Algebraic Geometry.
This book is available for free on Fulton’s website.
Buy at Amazon Milne, J. S. Elliptic Curves. BookSurge Publishers, 2006. ISBN: 9781419652578.
This book is also available for free on Milne’s website, along with addendum/erratum.
Buy at Amazon Serre, Jean-Pierre. A Course in Arithmetic. Springer-Verlag, 1996. ISBN: 9783540900405.
Buy at Amazon Shafarevich, I. R. (translated by Miles Reid). Basic Algebraic Geometry I. 3rd ed. Springer-Verlag, 2013. ISBN: 9783642379550.
Buy at Amazon Stichtenoth, H. Algebraic Function Fields and Codes. Berlin: Springer, 2008. ISBN 9783540768777. [Preview with Google Books]
Buy at Amazon Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer-Verlag, 2009. ISBN: 9780387094939. (errata) [Preview with Google Books]
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