1, Introduction to Spatial Analytic Geometry
This course is one of the main basic courses of mathematics department in colleges and universities. This paper mainly describes the basic contents and methods of analytic geometry, including: vector algebra, space straight line and plane, common surface, coordinate transformation, simplification of conic equation and so on. Through the study of this course, students can master some problems of studying space geometry by algebraic method. Coordinate method and vector method are the basic methods throughout the book.
2. Suggestions for course selection
Students majoring in mathematics must take this course. This course requires students to have a good foundation in middle school mathematics, so it is recommended to choose this course in grade one.
3. Teaching syllabus
First, the course content
The first chapter vectors and coordinates
The concept of 1. 1 vector
Addition of 1.2 vector
1.3 Number times vector
The Linear Relation of 1.4 Vector and the Decomposition of Vector
1.5 frame and coordinates
Projection of 1.6 vector on axis
Numerical product of 1.7 vector
Invariant product of two vectors 1.8
The mixed product of 1.9 three vectors
Double cross product of * 1. 10 three vectors
[Description]: This chapter systematically introduces the basic knowledge of vector algebra, which is essentially a process of algebraic space geometry. In order to better describe the cross product and mixed product of vectors, we need to supplement some basic knowledge of determinant.
Chapter II Trajectory and Equation
2. 1 plane curve equation
2.2 surface equation
2.3 Cylindrical Equation with Bus Parallel to Axis
2.4 space curve equation
[Description]: This chapter first introduces the equation of product surface curve and plane curve, and then quickly transitions to the study of the equation of surface and space curve, which not only allows students to review and improve the problem of plane trajectory, but also solves some seemingly complicated space trajectory problems.
Chapter III Plane and Space Straight Lines
3. 1 plane equation
3.2 the positional relationship between plane and point
3.3 The relative position of two planes
3.4 Space linear equation
3.5 Relative position of straight line and plane
3.6 The relative position of two straight lines in space
3.7 The relative positions of spatial straight lines and points
3.8 plane beam
[Explanation]: In this chapter, the simplest and most basic figures in space, that is, planes and spatial straight lines, are quantitatively studied by algebraic method, and their various equations are established, and the analytical expressions of their positional relations, as well as the calculation formulas of distance and intersection angle are derived.
Chapter 4 Cylinders, Cones, Revolving Surfaces and Quadratic Surfaces
4. 1 cylinder
4.2 conical surface
4.3 Rotating surface
4.4 ellipsoid
4.5 hyperboloid
4.6 paraboloid
4.7 straight generatrix of hyperboloid and hyperbolic paraboloid
[Description]: In this chapter, the equations of cylinder, cone and surface of revolution with obvious geometric characteristics are established. For simple quadratic equations, the properties of graphs are studied by "intercept point method".
The fifth chapter is the general theory of quadratic curve.
5. 1 Relative position of cone and straight line
5.2 Asymptotic Direction, Center and Asymptote of Quadric Curve
5.3 Tangent of Quadratic Curve
5.4 Diameter of Quadratic Curve
5.5 Principal Diameter and Direction of Quadratic Curve
5.6 Simplification and classification of quadratic equations
5.7 Simplifying Quadratic Equation with Invariants
[Explanation]: This chapter starts with the study of the intersection of straight lines and general conic, and then studies the geometric theory of general conic, such as the asymptotic direction, center, asymptote, tangent and diameter of general conic, and also discusses the different simplification and classification of general conic equations.
Second, the course description
(A) the status and tasks of the curriculum
This course is one of the main basic courses of mathematics department in colleges and universities. Learning this course well will lay the necessary mathematical knowledge, methods and thinking foundation for subsequent courses and further study of mathematics and professional knowledge.
(B) the basic requirements of the course
1, master the basic knowledge of vector algebra, including the linear operation of vectors, the calculation of inner product, outer product and mixed product of vectors, and the application in geometry. 2. Master all kinds of equations of plane and straight line in space, and all kinds of measurement relationships of points, lines and surfaces.
2. Master the equations of special quadric surfaces (such as cylinders, cones and surfaces of revolution) in space.
3. Master the geometric characteristics of conic equation and the different simplification methods and classification of conic equation.
The focus, depth and breadth of the course content
The basic idea of this course is to study geometry by algebraic method. On the basis of the first two chapters, the space plane and straight line are discussed emphatically, the special quadric equation is established, the general theory of quadric curve is mastered, and the vector and coordinate tools are used. This course is rigorous in demonstration, simple in description and clear in organization, with good breadth and depth.
(D) Contact and division of labor with other courses
Prerequisite: Plane Analytic Geometry
(V) Requirements and methods of cultivating students' ability
In addition to taking the closed-book exam, the key is to master an analysis method, and in addition, to cultivate students' intuitive imagination of spatial graphics.
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This website is a special course network of spatial analytic geometry, I hope it will help you.
The basic courses in general public universities are only advanced mathematics and linear algebra, which involve a little spatial analysis, and the main part is taught in the Department of Mathematics and Physics.