field

interior points, exterior points, boundary points

open set, closed set

Bounded point set, unbounded point set

Connected set, non-connected set

open area, closed area

definition

Geometric meaning

lim x→x0,y→y0 f(x,y)=A or f(x,y)→A ((x,y)→(x0,y0))

Judgment that the limit (double limit) of a binary function does not exist: When (x, y) tends to (x0, y0) in different ways, the function tends to different values, then the limit of the multivariate function does not exist

If lim (x,y)→(0,0) f(x,y)=f(x0,y0), then z=f(x,y) is said to be continuous at P0(x0,y0), otherwise it is discontinuous

definition

Geometric meaning

The second order and above are collectively called high-order partial derivatives

dz=df(x,y)=A△x B△y

dz=fx'(x,y)△x fy'(x,y)△y

Partial derivatives exist and are continuous→differentiable→continuous or partial derivatives exist

△z≈dz=f'(x,y)△x fy'(x,y)△y

Derivative Rules for Binary Composite Functions

Generalization and deformation of the derivation rule for binary composite functions

dz=∂zdu/∂u ∂zdv/∂v

direct derivation

formula method

Necessary conditions for the existence of extreme values

Sufficient conditions for the existence of extreme values

Construct Lagrangian function

Find its first-order partial derivatives with respect to x and y, make them zero, and then solve them simultaneously with the constraints to obtain possible extreme points.

judge