coordinate

cauchy

uniqueness

Boundedness

Linear operations

The subsequences converge to the same point

A bounded point sequence must have a convergent subsequence

definition

guide

Description of gathering point

Interior points, exterior points, boundary points (distinct from gathering points)

open set

bounded closed set

connected open set

Boundedness

maximin value theorem

Intermediate value theorem

consistent continuity

Differentiable, fully differentiable

continuous

deflectable

is defined in a certain neighborhood, and both partial derivatives are continuous at that point

Approximate calculation

error estimate

definition

Geometric meaning

Pay attention to understanding: the directional derivatives along the L direction and the -L direction have opposite signs.

Calculation formula

definition

Calculation formula

Geometric meaning - tangent vector

algorithm

Pure deflection

mixed partial derivative

chain rule

Composite partial derivative after polar coordinate transformation

Analysis: z=f(u,x,y) The partial derivative of z with respect to x is different from the partial derivative of f with respect to x, because the fixed variables are different.

Formal invariance of first-order partial derivatives <Higher orders do not have this property>

algorithm

! Three requests!

There are continuous second-order partial derivatives in a certain neighborhood

First-order form with Lagrange remainder

Second-order form with Peano remainder

Using the expression form of Hesse matrix

necessary conditions

sufficient conditions

Steps to find extreme values, for second-level continuous functions

A continuous function must have a maximum and minimum value in a closed region

step

What is obtained is the necessary condition to constrain the extreme points.

Introduce λ, construct the Lagrange function, and the obtained result may be the extreme point of the function, and then combine it with the actual problem judgment to draw a conclusion.

Let one side of the inequality be equal to the constant C, and then prove that the other side is always greater than the constant C under the constraints

one-dimensional vector

It cannot be obtained according to the traditional definition, so it is redefined

Differentiating by components

Find partial derivatives by components

First level continuous

The equation of this point is equal to 0

Jacobi determinant ≠ 0

Just memorize it directly, or you can push it now

1. When the curve equation is a parametric equation: Derive the parameter t to obtain the direction vector of the tangent line of the curve, and then calculate the tangent line and the normal plane based on this.

2. When the curve is given by the intersection of two cylinders: treat x as a parameter, and combine y1 and y2 to obtain the parametric equation

3. When the curve equation is a general formula (that is, the intersection of two general formula surfaces):

1. The Cartesian coordinate equation of a plane curve: z=0, treat x as a parameter, that is, turn it into a parametric equation

2. The polar coordinate equation of the plane curve: x=ρ(θ)cosθ, y=ρ(θ)sinθ, and then it becomes the parametric equation of θ. Pay attention to changing the upper and lower limits of the integral.

arc differential

natural parameters

Parametric curve network - longitude and latitude positioning

regular point

Parametric equation: The direction vector of the normal is obtained through the tangent vectors of the u curve and v curve, and then the tangent plane is obtained.

Cartesian coordinate equation: treat x as a parameter and combine the implicit function derivation method. The final normal direction vector is actually Fx, Fy, Fz

Cylindrical equation: In fact, it is a special rectangular coordinate equation. Just apply the above result directly, Fz= -1

1. Tangent plane and normal line

2. Close planes and subnormals

3. From the tangent plane and the main normal

T = r'(t) / ||r'(t)||

B = [r'(t) × r''(t)] / ||r'(t) × r''(t)||

N = B × T

natural parametric equations

general parametric equations

Plane Curve Parametric Equations

Plane curve right angle equation