MindMap Gallery Differentiation method of high number multivariate functions and its application
This figure summarizes the basic concepts of multivariate functions, differential methods of composite functions, directional derivatives and gradients, applications of multivariate function differentials in optimal value problems, geometric applications of multivariate function differentials, derivation formulas of implicit functions, and partial derivatives Let’s take a look at the content of total differential.
Edited at 2023-04-04 17:39:43El cáncer de pulmón es un tumor maligno que se origina en la mucosa bronquial o las glándulas de los pulmones. Es uno de los tumores malignos con mayor morbilidad y mortalidad y mayor amenaza para la salud y la vida humana.
La diabetes es una enfermedad crónica con hiperglucemia como signo principal. Es causada principalmente por una disminución en la secreción de insulina causada por una disfunción de las células de los islotes pancreáticos, o porque el cuerpo es insensible a la acción de la insulina (es decir, resistencia a la insulina), o ambas cosas. la glucosa en la sangre es ineficaz para ser utilizada y almacenada.
El sistema digestivo es uno de los nueve sistemas principales del cuerpo humano y es el principal responsable de la ingesta, digestión, absorción y excreción de los alimentos. Consta de dos partes principales: el tracto digestivo y las glándulas digestivas.
El cáncer de pulmón es un tumor maligno que se origina en la mucosa bronquial o las glándulas de los pulmones. Es uno de los tumores malignos con mayor morbilidad y mortalidad y mayor amenaza para la salud y la vida humana.
La diabetes es una enfermedad crónica con hiperglucemia como signo principal. Es causada principalmente por una disminución en la secreción de insulina causada por una disfunción de las células de los islotes pancreáticos, o porque el cuerpo es insensible a la acción de la insulina (es decir, resistencia a la insulina), o ambas cosas. la glucosa en la sangre es ineficaz para ser utilizada y almacenada.
El sistema digestivo es uno de los nueve sistemas principales del cuerpo humano y es el principal responsable de la ingesta, digestión, absorción y excreción de los alimentos. Consta de dos partes principales: el tracto digestivo y las glándulas digestivas.
Differentiation method of multivariate functions and its applications
Basic concepts of multivariate functions
basic concept
plane point set
Area
Interior points, exterior points, boundary points and focus points
area
The concept of multivariate functions
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.
Limits of binary functions
Continuity of binary functions
Boundedness and optimality
A multivariate continuous function on a bounded closed region must be bounded, and it can take the maximum and minimum values in this region.
Intermediate value theorem
A multivariate continuous function on a bounded closed region can take any value between the maximum value and the minimum value.
Differentiation of composite functions
Define the derivation rule for composite functions
chain rule
total derivative formula
Directional Derivatives and Gradient
Directional derivative
Two-dimensional space directional derivative
Three-dimensional space directional derivative
gradient
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
Application of differential calculus of multivariate functions in optimal value problems
Extremum exists
necessary conditions
sufficient conditions
Steps to find extreme value
Find the stationary point
Determine whether the limit exists
After-band resident point evaluation exists
conditional extreme value
Lagrange multiplier method
Find the objective function
find constraints
Introducing auxiliary functions
Solving systems of equations
Find the extreme value
Convert to a unary function
parametric equation method
unconditional extreme value
Find the domain
The two partial derivatives are equal to 0 to find the stationary point
discriminant
Determine AC-B² size
Greater than 0, there is an extreme point
A>0 is the minimum value
A<0 is the maximum value
Less than 0, no extreme point
Geometric Applications of Differential Calculus of Multivariate Functions
Tangents and normals to plane curves
Tangents and normal planes of space curves
Tangent line (x-x₀)/x'= (y-y₀)/y'= (z-z₀)/z'
Normal plane x'(x-x₀) y'(y-y₀) z'(z-z₀)=0
Surface tangent planes and normals
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)
Derivative formula of implicit function
an equation
dy/dx=-Fx/Fy
equation set
Cramer's Law
Partial Derivatives and Total Differentials
The concept of partial derivatives
The geometric meaning of partial derivatives
Higher order partial derivatives
Second order partial derivative
Mixed partial derivatives
Partial derivatives of second order and above are collectively called higher-order partial derivatives
Total differential
A△x△yo(ρ)
Necessary conditions for differentiability
sufficient conditions for differentiability
Continuous, deflectable, differentiable relationships
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)
Find the type of partial derivative
explicit function
Composite function
Implicit function
Find the number of variables
Find several constraints, the constraints are equal to the number of dependent variables
The rest is the number of independent variables
Transformation to find partial derivatives