plane point set

Area

Interior points, exterior points, boundary points and focus points

area

Definition 1 Suppose D is a non-empty point set on the xoy plane. If for any point (x, y) in D, according to a certain corresponding rule f, there is a unique real number z corresponding to it, then the variable z is called The binary function of x, y is usually recorded as z = f (x, y), (x, y) ∈ D, where the point set D is called the domain of the function, x, y are called independent variables, z is called the dependent variable.

Boundedness and optimality

Intermediate value theorem

chain rule

total derivative formula

Two-dimensional space directional derivative

Three-dimensional space directional derivative

The direction in which a function grows fastest at a certain point

grad(u)={u'x, u'y, u'z}, {a, b, c} represents the vector

necessary conditions

sufficient conditions

Find the stationary point

Determine whether the limit exists

After-band resident point evaluation exists

Lagrange multiplier method

Convert to a unary function

parametric equation method

Find the domain

The two partial derivatives are equal to 0 to find the stationary point

discriminant

Tangent line (x-x₀)/x'= (y-y₀)/y'= (z-z₀)/z'

Normal plane x'(x-x₀) y'(y-y₀) z'(z-z₀)=0

Normal (x-x₀)/F'x= (y-y₀)/F'y= (z-z₀)F'z

Tangent plane F'x (x-x₀) F'y (y-y₀) F'z (z-z₀)

dy/dx=-Fx/Fy

Cramer's Law

Second order partial derivative

Mixed partial derivatives

Partial derivatives of second order and above are collectively called higher-order partial derivatives

A△x△yo(ρ)

Necessary conditions for differentiability

sufficient conditions for differentiability

Differentiable must be deflectable and continuous

Two partial derivatives must be differentiable if they are continuous.

If there is a continuous second-order partial derivative, f''xy(x, y) = f''yx(x, y)

explicit function

Composite function

Implicit function

Transformation to find partial derivatives