On matrix and invertible matrix 1. Let a and B all be n-order square matrices and satisfy a + B + AB = 0. Prove that ab = ba 2. If both a and B are n-order square matrices and a + B are reversible matrices, then both a and B are reversible matrices. Is this right or wrong? Why?

1. The equation can be transformed into (E + a) (E + b) = e, that is, e + A and E + B are inverse matrices of each other
Therefore, there is also (E + b) (E + a) = e, which expands to a + B + Ba = 0 = a + B + ab
So AB = ba
2. There are counter examples, for example, a = e, B = 0. Although B is irreversible, a + B = e is reversible

What is the difference between partial derivative and total differential? thank you

1. If partial derivative does not exist, total differential does not exist;
2. If total differential exists, partial derivative must exist;
3. If partial derivative exists, total differential does not necessarily exist

The relationship between partial derivative and total derivative and the relationship between partial derivative and total derivative I hope it can be answered from two aspects: algebraic meaning and geometric meaning. It's best to have examples. Thank you very much How to find the total derivative of Z = f (XY, x ^ 2-y ^ 2)?

1. Partial derivative
Algebraic meaning
A partial derivative is a derivative of one variable and the other variable as a number
If you find the partial derivative of X, y is regarded as a number, which describes the rate of change in the X direction
If you find the partial derivative of Y, X is regarded as a number and describes the rate of change in the Y direction
Geometric meaning
The partial derivative of X is the tangent of the surface z = f (x, y) in the X direction
The partial derivative of Y is the tangent of the surface z = f (x, y) in the X direction
Here is a supplementary point. It is because the partial derivative can only describe the change in the X direction or Y direction, but we need to understand the situation in all directions, so there is the concept of directional derivative later
2. Differential
Partial increment: F (x, y) increment when x increases or F (x, y) increment when y increases
Partial differential: the linear main part of the partial increment when detax tends to zero
detaz=fx（x,y）detax+o(detax)
The first term of the equation on the right is the linear main part, which is called the partial differential of X at point (x, y)
This equation also gives a way to find the partial differential, that is, to find the partial differential by finding the partial derivative of X
Full increment: the increment of F (x, y) when x and Y increase
Total differential: the linear main part of the total increment when the root sign (detax square + Detay Square) tends to 0
There are also formulas for finding total differential, and the relationship between total differential and partial derivative is established
DZ = ADX + bdy, where a is the partial derivative of X and B is the partial derivative of Y
I hope the landlord will pay attention to that derivative and differential are two concepts, and the relationship between them is the formula mentioned above. In concept, there is derivative first, then differential, and then there is the relationship formula between derivative and differential. The formula also points out the method of seeking differential
3. Total derivative
Total derivative is a concept in compound function, which is not a system and should be separated from the above concept
u=a(t),v=b(t)
z=f[a(t),b(t)]
DZ / DT is the total derivative, which is a case in the derivation of composite function. Only then can there be the concept of total derivative
DZ / dt = (partial Z / partial U) (DU / DT) + (partial Z / partial V) (DV / DT)
It is suggested that the landlord take a good look at the book here. There are three cases here. 1. The unary of the intermediate variable is the above situation, so there is the concept of full derivative. 2. The intermediate variable has multiple variables, and only partial derivative can be obtained. 3. The intermediate variable has both one and multiple variables, or partial derivative
For your problem, you can find the partial derivative of X, the partial derivative of Y, and the total derivative of Z. you can't find the total derivative
If z = f (x ^ 2,2 ^ x), DZ / DX is the full derivative only in this case!

Find the partial derivative and total differential of a function Let f (x, y) = sin (XY) + cos (Y / x) have a first-order partial derivative, and find the partial derivative and total differential of function f (x, y) Does anyone know what to do,

f'x=ycos(xy)-sin(y/x)(-y/x ²)= ycos(xy)+(y/x ²) sin(y/x)
f'y=xcos(xy)-(1/x)sin(y/x)
df=f'xdx+f'ydy=[ycos(xy)+(y/x ²) sin(y/x)]dx+[xcos(xy)-(1/x)sin(y/x)]dy

Partial and total derivatives: calculating approximations 0.99^(1.98)

f(x,y)=x^y
dz=yx^(y-1)dx+x^ylnxdy
Take x = 1, DX = -0.01, y = 2, Dy = -0.02
dz=yx^(y-1)dx+x^ylnxdy
=-0.02-0.02ln2=-0.02(1+ln2)= -0.0339
0.99 ^ (1.98) = 1-0.0339 = 0.9661 (accurate value is 0.9803)

How to read the symbols in the second partial derivative that are very similar to Delta, such as δ z/ δ x δ F/ δ x

partial
∂

On matrix and invertible matrix 1. Let a and B all be n-order square matrices and satisfy a + B + AB = 0. Prove that ab = ba 2. If both a and B are n-order square matrices and a + B are reversible matrices, then both a and B are reversible matrices. Is this right or wrong? Why?