Features
Features:

Product Tour >

Edraw AI >

Paid Plans:

Individuals >

Business >

Eduaction >
Resources
Blog

History

How-tos & Tips

Discovery

Biography

Business Analysis

Examples

AI concept Map

Free AI Mind Map Generator

Onenote Mind Map

Bcg Matrix Examples

Nike Marketing Strategy

Unilever SWOT Analysis

Make Mind Maps in Google Docs

Guide

FAQs

What's New

Resource Center
Templates
All Templates

Brain Storming Templates

Strategy and Planning Templates

Project Management Templates

Product Management Templates

Human Resources Templates

Agile Workflow Templates

Marketing Templates

Education Templates

Fun and Games Templates

User Gallery
Download
Pricing
Enterprise

MindMap Gallery Applications of vector algebra and space analytic geometry and multivariable differential calculus in geometry

Applications of vector algebra and space analytic geometry and multivariable differential calculus in geometry

This is a question note about the application of vector algebra, space analytic geometry and multivariable differential calculus in geometry. This mind map explains this part of the knowledge points very well.

Edited at 2021-09-03 11:40:26

PlotWizard

Recent works View more works>>

Applications of vector algebra and space analytic geometry and multivariable differential calculus in geometry

PlotWizard

Recent works View more works>>

Recommended to you
Outline

Applications of vector algebra and space analytic geometry and multivariable differential calculus in geometry

vector algebra

space rectangular coordinate system

The positive directions of the three coordinate axes conform to the right-handed system

eight hexagrams

distance between two points in space

special

vector concept

vector module

vector size

unit vector

vector with modulo length 1

Represents a unit vector in the same direction as the non-zero vector a

zero vector

A vector whose modulus length is 0

Vector operations

quantity product

Geometric representation

algebraic representation

Operation rules

commutative law

distributive law

Geometry applications

Find the model

Find the angle

Determine if two vectors are perpendicular

vector product

Geometric representation

mold

direction

right hand rule

algebraic representation

Operation rules

distributive law

Geometry applications

Find the vector perpendicular to both a and b simultaneously

Find the area of the parallelogram with a and b as adjacent sides

Determine if two vectors are parallel

Mixed product

Geometric representation

algebraic representation

Operation rules

rotational symmetry

exchange sign

Geometry applications

Find the volume of the parallelepiped

Necessary and sufficient conditions for three vectors to be coplanar

Space planes and straight lines

equation of plane

general equation of plane

Ax By Cz D=0

normal vector

perpendicular to plane

point French equation of plane

Plane intercept equation

The positional relationship between two planes

The cosine formula of the angle between two planes

Two plane position characteristics

Click and multiply

The result is a number

cross

The result is a vector

example

Surfaces and Planes

Li Yongle reviews the basic version of the book 133 Example 3

Several special cases of general equations of planes

D=0

The plane passes through the coordinate origin

A=0

D=0

The plane passes through the x-axis

D≠0

The plane is parallel to the x-axis

inference

Similarly, the case B =0, C = 0 can be discussed

A=B=0

The plane is parallel to the xoy coordinate plane

inference

Similarly, the cases A=C=0, B=C=0 can be discussed

equation of straight line

general equation of straight line

Connection to algebraic representation

symmetry equation of straight line

straight line through point

direction vector

Knowing two points, find the direction vector

Parametric equation of a straight line

Contact with direction vector

The positional relationship between two straight lines

The cosine formula of the angle between two straight lines

Two straight line position characteristics

Positional relationship between straight lines and planes

The formula for the angle between a straight line and a plane

Positional characteristics of straight lines and planes

If the straight line is perpendicular to the plane, the direction vector of the straight line is the normal vector of the plane.

same type

Passing L1, parallel to L2

Explain that the plane normal vector is perpendicular to L1 and L2

Passing L, perpendicular to P

two distances

distance from point to plane

distance from point to straight line

Surfaces and space curves

Surface integral

General form

space curve

The space curve C can be regarded as the intersection line of two surfaces in space.

General form

parametric

Common surfaces

plane of revolution

A plane curve rotates around a straight line on the plane

The equation of the plane of rotation obtained by rotating L around the y-axis

The equation of the plane of rotation obtained by rotating L around the z-axis

reason

cylinder

The curved surface formed by the straight line L parallel to the fixed line and moving along the fixed curve C is called a cylinder.

Definite curve C

directrix of cylinder

Moving straight line L

cylindrical bus bar

The equation F (x, y) = 0, which only contains x, v but lacks z, shows a cylinder with the busbar parallel to the z-axis in the space rectangular coordinate system, and its directrix is the curve C on the xoy surface.

promotion

Quadratic surface

The surface represented by the quadratic equation of three variables is called a quadratic surface

The corresponding ground plane is called a linear surface

cylindrical surface

The intersection line with the xoy plane is a circle

The intersection with the plane z=c is a circle centered on the z-axis

The intersection lines with the xoz and yoz planes are two parallel straight lines

Two parallel straight lines intersecting the plane y=c, x=c(\c< R

ellipsoid

The intersection line of the ellipsoid and the three coordinate planes

a= b=c

spherical surface

elliptical cone

a=b

Conical surface

In particular

The angle is just right (p/4)

elliptical paraboloid

a=b

paraboloid of revolution

In particular

single leaf hyperboloid

double leaf hyperboloid

Hyperbolic paraboloid (saddle surface)

projection of space curve

Similarly

The projection of space curves on other coordinate surfaces can be defined

example

When it cannot be eliminated, which one of F and G has the smaller projection, which one should be used?

Applications of multivariate differential calculus to geometry

Surface tangent planes and normals

tangent plane equation

normal equation

Determination of normal vector direction of curve

The outer normal points to the outside of the surface, and the inner normal points to the inside. Therefore, considering the normal at the tangent point P, you can pick a point Q on the inside of the curved surface. Then, if the angle between the normal direction and the vector PQ is greater than 90°, it can be determined to be an outer normal, and vice versa. Of course, you can also use points outside the surface area for judgment. The principle is the same.

special

tangent plane equation

normal equation

Tangents and normal planes of space curves

parametric

tangent equation

normal plane equation

General form

tangent vector

understood as

tangent equation

normal plane equation

Notice

Not a normal vector, but a tangent vector