Consult the problem of differential calculus of multivariate functions What are the necessary and sufficient conditions for the equality of second-order mixed partial derivatives in binary functions? (note is a necessary and sufficient condition) What is the general situation when the second-order mixed partial derivatives are not equal?

Consult the problem of differential calculus of multivariate functions What are the necessary and sufficient conditions for the equality of second-order mixed partial derivatives in binary functions? (note is a necessary and sufficient condition) What is the general situation when the second-order mixed partial derivatives are not equal?

Figuratively speaking, the necessary and sufficient condition is that the binary function should be continuous and smooth. Imagine a smooth plane in a three-dimensional coordinate system, just like the water surface, without creases, and the second-order partial derivatives of such a function are equal
For example, when two planes intersect a straight line, the second-order partial derivatives are unequal. Of course, if a second-order derivative itself is meaningless, let alone

How to find the differential of compound function? Detailed derivation formula

If you are not used to it, you can find the derivative first:
Let y = f (U), u = g (V), v = H (x), then y = f (g (H (x)))
y'=f'(u)g'(v)h'(x)
=f'(g(h(x)))g'(h(x))h'(x)
So: dy = f '(g (H (x))) g' (H (x)) H '(x) DX

Find the definite integral of the (- SX) power of e multiplied by the nth power of X from 0 to positive infinity. (n is a real number)

a=∫[0,+∞]e^(-sx)x^ndx=-1/s*∫[0,+∞]x^nde^(-sx)
=-1/s*[0,+∞]x^ne^(-sx)+n/s∫[0,+∞]e^(-sx)x^(n-1)dx
=n/s∫[0,+∞]e^(-sx)x^(n-1)dx
So a = Na / S
a<0>=1/s
therefore
a/a=n/s
a/a=(n-1)/s
……
a<1>/a<0>=1/s
Multiply
a/a<0>=n!/ s^n
So a = n/ s^(n+1)

Calculate the definite integral e ^ xdx by definition, and the answer is E-1

It should be a definite integral on [0,1]
Use 0 = 0 / n

Using the definition of definite integral, calculate the interval of ∫ (e ^ x) DX as [0,1]. Use the definition to calculate n. I can't calculate Σ e ^ (I / N). I = 1 e^(1/n)+e^(2/n)+...+e^(n/n)=? How

When n →∞, Lim e ^ (1 / N) * 1 / N + e ^ (2 / N) * 1 / N +... + e ^ (n / N) * 1 / N = LIM (e ^ (1 / N) + (e ^ (1 / N)) ^ 2 +... + (e ^ (1 / N)) ^ n) / N = (sum of molecular proportional series) LIM (e ^ (1 / N) (1 - (e ^ (1 / N)) ^ n)) / (n (1-e ^ (1 / N))) = (denominator 1-e ^ (1 / N) is equivalent to - 1 / N) LIM (e ^ (1

What is the indefinite integral (to the power of 2 / 2 x squared of E),

The original function of e ^ (x ^ 2 / 2) is not an elementary function. It can be proved by Liouville's third theorem