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Differential calculus of multivariate functions and its applications
Differential calculus of multivariate functions and its applications: double limit: p (x, y) When it approaches a certain point in any way, f (x, y) is infinitely close to A. If it approaches a certain point in different ways, f (x ,y) tend to different values, then the function limit does not exist.
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Differential calculus of multivariate functions and its applications
- bacteria
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- Plant asexual reproduction
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- Reproductive development of animals
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- Outline
Differential calculus of multivariate functions and its applications
main idea
Generalization of functions from one variable to two variables and then to multiple variables
The main way to study a binary function is to convert it into a unary function
Some properties are different, one-way and multi-way
basic concept
limit and continuity
double limit
When p (x, y) approaches a certain point in any way, f (x, y) is infinitely close to A
Approaching a certain point in different ways, f(x,y) tends to different values, then the function limit does not exist
The limit operation is similar to the one-variable function. The proof does not exist. Just find an example.
continuity
The limit exists and is equal to the function value
All elementary functions of multiple variables are continuous within their domain.
Properties of continuous functions on bounded closed regions
Maximum value theorem
Intermediate value theorem
Function is bounded
Corollary to the Intermediate Value Theorem
Partial derivative
Find the partial derivative of which variable and treat other variables as constants
The essence is the derivation of a one-variable function
Geometric meaning: the slope of the tangent to the independent variable axis at this point at the intersection of the surface and the constant plane
differentiability
definition
full delta
The total increment of a binary function is equal to the sum of partial increments
condition
Necessary condition: all partial derivatives exist
Sufficient condition: Each partial derivative of z=f(x,y) is continuous at point (x,y)
Application: Approximate calculations
Directional derivative
z=f(x,y) changes speed along direction l in (x,y)
connect
subtopic
calculate
Partial derivative
level one
at a certain continuous point
use definition
Find the derivative first and then take the value
First generation values and then derivation
at a certain discontinuity point
Use definition to find
Advanced
Use definition to find
Use the derivation formula
multivariable composite function
chain rule
The outer function is differentiable and the inner function is differentiable (the intermediate variables are all unary)
The outer function is differentiable and the inner function has continuous partial derivatives.
Intermediate variables are both univariate and multivariate
Higher order partial derivatives
Pure deflection
mixed partial derivative
Implicit function
How to verify the existence of implicit functions
formula method
Use both sides of the equation to derive derivatives or partial derivatives with respect to a variable at the same time
Exploiting the invariance of the first-order total differential form
Derivatives of Implicit Functions of Systems of Equations
Total differential
judge
Use the definition to find the partial derivative and then the total differential
Invariance of first-order total differential forms
It is known that du=Pdx Qdy and find u
Geometric meaning of total differential
application
Tangent vector/normal plane of the curve at a certain point
The normal vector/tangent plane of the surface at a certain point
Directional Derivatives and Gradient
If the directional derivative along the positive x-axis exists, it must be equal to the partial derivative of the function with respect to x at this point
When the function is differentiable, the directional derivative = fx (x, y) cos α + fy (x, y) cos β
Increases fastest along the gradient direction (the maximum value of the directional derivative is equal to the module of the gradient)
The geometric meaning of total differential: the increment of the vertical coordinate of a point on the tangent plane
Extreme values of multivariate functions
unconditional extreme value
Find extreme points using geometric meaning
Find the stationary point of two variables and the second-order partial derivative is continuous
AC-B²>0 takes the extreme value, and A>0 takes the maximum value, and A<0 takes the minimum value
AC-B²<0 does not take the extreme value
AC-B²=0 cannot determine the regression definition or geometric meaning
conditional extreme value
Substitute the conditions
Lagrange multiplier method
Maximum value of a multivariate function
Stationary points within the region
Maximum point on the boundary