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.

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

full delta

The total increment of a binary function is equal to the sum of partial increments

Necessary condition: all partial derivatives exist

Sufficient condition: Each partial derivative of z=f(x,y) is continuous at point (x,y)

at a certain continuous point

at a certain discontinuity point

Use definition to find

Use the derivation formula

chain rule

Higher order partial derivatives

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

Find extreme points using geometric meaning

Find the stationary point of two variables and the second-order partial derivative is continuous

Substitute the conditions

Lagrange multiplier method