What does this symbol mean in advanced mathematics? For example, when x → 0, 1-cos (x √ 1-cosx) ~ 1 / 2 x ^ 2 (1-cosx) has ~, what does it mean? Is it equivalent to? Or what? For an answer, thank you!

Equivalent to tend to be both right
When you learn the Taylor expansion, you will know that "in fact, what follows is the first few terms of the Taylor expansion, followed by the high-order infinitesimal, which is an approximate idea of taking the limit"

”It's a factorial. "What is it? For example: (2n-1)!

A double factorial is one more than a factorial
Like 5= 5 * 3 * 1
6!= 6 * 4 * 2

What does "n!" in high numbers mean?

It seems to be called multi factorial
(2n)!= 2n*(2n-2)*.*4*2
(2n+1)!= (2n+1)*(2n-1)*...*5*3*1
This is used in the derivation of complex problems. Some formula derivation in mathematical equation textbooks use this
So 3= 3,6!= forty-eight

n! What does it mean in advanced mathematics?

n! Read as "double factorial of n", which means: n= n(n-2)(n-4)(n-6)...1
For example: 9= 9*7*5*3*1
10!= 10*8*6*4*2*1

High number problem for help ~ ~ ~ ~ ~ ~ ~ by definition, it is proved that {n!} is an infinite number~~~

Given any large positive number m, take n = m, then when n > N, there is always n! > N!= M!> M is n! > General Manager M was established and certified

Let the velocity u = u (x, y), then what is the physical meaning of u finding the partial derivative of X and what is the physical meaning of u finding the second-order partial derivative of X?

The physical meaning of U's partial derivative of X is that it represents the acceleration in the X direction
The physical meaning of U's second-order partial derivative of X is: how fast does the acceleration change in the X direction

What does this symbol mean in advanced mathematics? For example, when x → 0, 1-cos (x √ 1-cosx) ~ 1 / 2 x ^ 2 (1-cosx) has ~, what does it mean? Is it equivalent to? Or what? For an answer, thank you!