Liebniz Notation: When to Treat as Fraction?

[ehrenfest]

Homework Statement

In many physics books I have seen the treatment of dx/dy as a fraction dx over dy. For example, if you have an expression for dx and an expression for dy then you just put dx in the numerator and dy in the denominator to get the derivative. THis is also done in the separation of variables technique.

I have heard that this is not mathematically sound. Is there a rule for when you can treat Liebniz notation like fractions?

Homework Equations

The Attempt at a Solution

 

[Gokul43201]

Look into differential forms.

Also see: http://mathforum.org/library/drmath/view/65462.html

 

[HallsofIvy]

The "Liebniz form" for a derivative: dy/dx is NOT a fraction but it can always be treated like one. The derivative is a limit of a fraction. To prove that any "fraction property" works for a derivative, go back before the liimit, use the fraction property, then take the limit.

That's why the notion of "differentials", defining "dy" and "dx", if only symbolically, that Gokul43201 was referring to, is so powerful.

 

[arildno]

Remember, though, that the derivative is NOT in general a fraction; this is highlighted by the behaviour of partial derivatives:

let F(x,y) be a differentiable function; x=X(y).

Thereby, we have:
$$\frac{dF}{dy}=\frac{\partial{F}}{\partial{x}}\frac{dX}{dy}+\frac{\partial{F}}{\partial{y}}$$

Here, the relationships between the pseudo-fractions is NOT that which might be "predicted" by common fraction arithmetic.