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Differentiation of multivariate functions and its applications
A mind map of differentiation of multivariate functions and their applications, which organizes the basic concepts of multivariate functions, partial derivatives, total differentials and their applications, differential methods of composite functions and implicit functions, and extreme values of binary functions. Like it You can like it and save it~
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Differentiation of multivariate functions and its applications
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Differentiation of multivariate functions and its applications
Basic concepts of multivariate functions
area
field
interior points, exterior points, boundary points
The interior points of x must belong to x
The external points of x must not belong to x
The boundary points of x may belong to x
open set, closed set
Bounded point set, unbounded point set
Connected set, non-connected set
open area, closed area
Definition of multivariate function
definition
z=f(x,y),(x,y)∈D
Find the domain
Geometric meaning
Limits of multivariate functions
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
Continuity of multivariate functions
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
Partial derivative
Definition and geometric significance of partial derivatives
definition
△xz=f(x0 △x,y0)-f(x0,y0) is called the partial change or partial increment of x at point (x0,y0) of function z=f(x,y)
△z=f(x0 △x,y0 △y)-f(x0,y0) is called the total change or total increment of the function z=f(x,y)
∂z/∂x|x=x0,y=y0; ∂f/∂x|x=x0,y=y0;zx'|x=x0,y=y0 or fx'(x0,y0)
Geometric meaning
Higher order partial derivatives
The second order and above are collectively called high-order partial derivatives
Total differentiation and its applications
Definition of total differential
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
Application of total differential in approximate operations
△z≈dz=f'(x,y)△x fy'(x,y)△y
Differentiation of Composite Functions and Differentiation of Implicit Functions
Differentiation of composite functions
Derivative Rules for Binary Composite Functions
Chain Rule: Same lines multiply and different lines add.
Generalization and deformation of the derivation rule for binary composite functions
total derivative
total differential form invariance
dz=∂zdu/∂u ∂zdv/∂v
Differentiation of Implicit Functions
direct derivation
formula method
If ∂F/∂y≠0, then from F[x,f(x)]=0, we have ∂F/∂x ∂Fdy/∂ydx=0, and we can get dy/dx=(∂F/∂x) /(∂F/∂y)=-Fx'/Fy'
extreme value of a binary function
Definition and theorem of extreme values of binary functions
Necessary conditions for the existence of extreme values
Suppose the function z=f(x,y) has a partial derivative at the point (x0,y0) and an extreme value at the point (x0,y0), then its partial derivative at this point must be zero, fx'( x0,y0)=0,fy'(x0,y0)=0
Sufficient conditions for the existence of extreme values
If AC-B²>0, and f''xx(x0,y0)<0, then f(x0,y0) is the maximum value
If AC-B²>0, and f''xx(x0,y0)>0, then f(x0,y0) is the minimum value
If AC-B²<0, then f(x0,y0) is not an extreme value
If AC-B²=0, and whether f(x0,y0) is an extreme value, it needs to be judged separately.
Conditional extreme value and Lagrange multiplier method
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