What is Euler's Criterion in plain simplified english

[jenzao]

What is "Euler's Criterion" in plain simplified english

This sentence in textbook reads: "If ∂M/∂y = ∂N/∂x the differential is exact (Euler’s Criterion)."

What does the equation mean, and also I am not familiar with the backwards number 6 symbol --what is that?
thanks

 

[neworder1]

This has no "plain English" definition :) You should grab some multivariate analysis textbook. Or try this: http://en.wikipedia.org/wiki/Partial_derivative"

 

[slider142]

Given that f is a function of several variables, \frac{\partial f}{\partial x} is the partial derivative of f with respect to the variable x. It is the same as the normal derivative of f if we take all other variables to be constants.
Suppose we have a function f(x,y). The differential of f(x,y) is defined to be df(x,y) = \frac{\partial f}{\partial x}(x,y) dx + \frac{\partial f}{\partial y}(x,y) dy. If you have an expression M dx + N dy, it is called an exact differential if it is the differential of some f(x,y). The Euler criterion is one way of telling whether this is the case; it is easy to see that it should be the case.

 

[jenzao]

"If you have an expression M dx + N dy, it is called an exact differential if it is the differential of some f(x,y). "

Could you please provide an example of this?

 

[slider142]

"If you have an expression M dx + N dy, it is called an exact differential if it is the differential of some f(x,y). "

Could you please provide an example of this?

Suppose you have the vector field g(x,y) = (-x/r^3/2) dx - (y/r^3/2) dy, where r = \sqrt{x^2 + y^2}. Then g(x,y) is an exact differential, because g(x,y) = df(x,y) where f(x,y) = r^-1/2. Ie., f(x,y) is the potential of a central force g(x,y). You may also see f = -\nabla g.

 

[HallsofIvy]

If you don't know about partial derivatives, why are you worrying about "Euler's Criterion" and "exact differentials"? The latter requires that you be well versed in partial derivatives.