Differential calculus of multivariate functions

Mind map of differential calculus of multivariate functions: including basic concepts (the limit of multivariate functions, the relationship between continuity, continuity, partial derivatives and differentiability of multivariate functions,) differential method of multivariate functions, extreme values and maximum values of multivariate functions, etc. wait

Edited at 2022-05-05 00:05:50

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Differential calculus of multivariate functions

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Outline

Differential calculus of multivariate functions

basic concept

Limits of multivariate functions

Multivariate function continuity

definition

nature

Multivariate continuous functions and difference product quotients are still continuous functions

The composite function of a multivariate function is a continuous function

A multivariate function is continuous within its defined region

Maximum value theorem: A continuous function on a bounded closed region D must be able to obtain the maximum and minimum values in the region D.

Intermediate value theorem: A continuous function on a bounded closed region D must be able to obtain any value between the maximum value and the minimum value in the region D.

Partial derivative

definition

Partial derivatives are essentially derivatives of functions of one variable

Geometric meaning

Higher order partial derivatives

Total differential

definition

Necessary conditions for the existence of total differentials

Sufficient conditions for the existence of total differentials

The relationship between continuous, deflectable and differentiable

Differentiation of multivariate functions

Differentiation of composite functions

theorem

Invariance of total differential form

Implicit Function Differentiation Method

Extreme values and maximum values of multivariate functions

Unconstrained extreme value

definition

Necessary conditions for extreme values

Sufficient conditions for extreme values

General steps for finding the extreme value of a binary function with second-order continuous partial derivatives

conditional extreme value

Lagrange multiplier method

maximum value minimum value

Maximum and Minimum Values of Continuous Functions on Bounded Closed Regions (Trilogy)

Find the possible extreme points of the function inside the region D

Find the maximum and minimum values of the function on the boundary of D

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Application questions

To establish the objective function, follow the above three steps