Proof of second order partial derivative Because of the input problem, I use "d" to represent the partial derivative symbol For the mixed derivative of the second order, we prove that (Dy, Dy) and (Dy, Dy) have partial derivatives d^2/(dxdy)=d^2/(dydx) That is, their mixed partial derivatives are independent of the order of derivation

Proof of second order partial derivative Because of the input problem, I use "d" to represent the partial derivative symbol For the mixed derivative of the second order, we prove that (Dy, Dy) and (Dy, Dy) have partial derivatives d^2/(dxdy)=d^2/(dydx) That is, their mixed partial derivatives are independent of the order of derivation

This PPT is proved on page 18

What does it mean that the partial derivative is continuous at one point

It means that the function after partial derivation is continuous at a certain point
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The existence of partial derivatives is not necessarily continuous
Multivariate function, partial derivative existence function is not necessarily continuous
Why?
Why can't we generalize to multivariate function

If we think of a binary function as a function on a plane, then continuity needs to be continuous in all directions (horizontal, vertical and oblique). The existence of partial derivatives for X only means that the function is differentiable as a function of X when it is restricted to each horizontal line (y = a), and the existence of partial derivatives for y only means that the function is differentiable as a function of y when it is restricted to each vertical line (x = a)
The simplest example: define a binary function as 0 in the left half plane and 1 in the right half plane, then it can be derived on each vertical line (because it is a constant), and it is discontinuous on the horizontal line (left 0 and right 1), so its partial derivative to y exists but is discontinuous; similarly, define a binary function as 0 in the lower half plane and 1 in the upper half plane, then its partial derivative to x exists but is discontinuous
Even if the partial derivatives of binary function to X and Y exist, it only means that it can be derived on all horizontal and vertical lines. Theoretically, it is still possible to be discontinuous on an oblique line. This kind of function is not as easy to think of as above, but it does exist. General calculus books will give a standard example: F (x, y) takes 0 at the origin of coordinates, other places = XY / (x ^ 2 + y ^ 2)
To generalize, a general multivariate function can be thought of as a function in a high-dimensional space. Continuity needs to be continuous in all directions of the plane, while the existence of partial derivatives only means that it can be derived in all planes parallel to the coordinate plane - the latter can not deduce the former. Functions of one variable will not have this problem, because there is only one direction on a straight line

On the continuity of partial derivatives
F (x, y) = (y * absolute value x) / radical (x ^ 2 + y ^ 2) when (x, y) is not equal to (0,0)
=0 when (x, y) is not equal to (0,0)
Why is the partial derivative of this piecewise function discontinuous? How to consider it? At which point is it discontinuous!
thank you!

f(x,y)=
xy/√(x^2+y^2) (x>0)
-xy/√(x^2+y^2) (x0)
-y^3/(x^2+y^2)^(3/2) (x

Continuity of partial derivatives
It is proved that the partial derivative of XY / √ (x Λ 2 + y Λ 2) is discontinuous at (0,0)

There are few conditions in your topic. This should be a piecewise function, and f (0,0) = 0. First, according to the definition of partial derivative, the partial derivative of the function at (0,0) point can be obtained, that is, z'x = Lin [f (x, 0) - f (0,0)] / X (x tends to 0) = LIM (0-0) / x = 0. When it is not at (0,0) point, the partial derivative of the function to X can be obtained directly by derivative formula, z'x = [y (...)

The relationship among differentiability, continuity, existence of partial derivative and continuity of partial derivative

Differentiability must be continuous and partial derivatives exist
Continuity is not necessarily the existence of partial derivatives, and the existence of partial derivatives is not necessarily continuous
Continuity may not be differentiable, and the existence of partial derivatives may not be differentiable
A sufficient and unnecessary condition for the continuity of partial derivatives to be differentiable

On partial derivatives
When doing the problem of partial derivative, I found a problem. I couldn't find the solution
The partial derivative of X for X is obtained by x-z = ln (Z-Y)
Why do I get different results when I multiply Z into the form of x = Z * ln (Z / y)?

What is partial derivative?

Multivariable function to multivariable function
Z=f(x,y,z,...)
For example, DZ / DX is the partial derivative of Z to X

Some problems on partial derivative
If the partial derivative exists, the function is not necessarily continuous; if the function is continuous, the partial derivative does not necessarily exist; if the partial derivative is continuous, the partial derivative must exist and the function must be continuous
Is that right?
Which condition is better, the partial derivative is continuous or the function is differentiable?

The partial derivative exists, the function is not necessarily continuous. This sentence is correct, because the partial derivative can only ensure that the point tends to a certain point along the direction parallel to the coordinate axis
If the function is continuous, the partial derivative does not necessarily exist. This sentence is correct. For example, for a cone, the partial derivative at the cone point does not exist, which is similar to the cusp problem of univariate function
If the partial derivative is continuous, the partial derivative must exist and the function must be continuous

Function z = root sign (x ^ 2 + y ^ 2) at (0,0) ()
A. Discontinuity
B. Existence of partial derivatives
C. Existence of derivatives in any direction
D. Differentiable
When we get the partial derivative here, we find that for X, the left side of zero is - 1, and the right side of the collar is + 1. This function only has no partial derivative at zero, but there are partial derivatives in other intervals,
Do I use the definition to find the partial derivative of Z, or do I use the derivative of a function of one variable when y is a constant

You are right. Because if you calculate the partial derivative of X: X / root sign (x ^ 2 + y ^ 2), then y is a constant. So you can see that in (0,0) unbiased derivative a is wrong, B is also wrong, and D is also wrong. So if you choose C to calculate the partial derivative of Z, you should note that it is to calculate the partial derivative of Z with respect to x, as if y is constant, as if x is constant