Dx and delta(x) (in partial derivative)

[destroyer130]

I have a question to ask, is dx = δx, can they cancel each other like $\frac{dx}{δx}$=1
and is it mean that:

$\frac{δf}{δx}$$\frac{dx}{dt}$=$\frac{df}{dt}$?
(f = f (x,y,z))

 

[micromass]

I'm not sure what you mean with $\delta x$ in the first place.

 

[Mark44]

I have a question to ask, is dx = δx, can they cancel each other like $\frac{dx}{δx}$=1
and is it mean that:

$\frac{δf}{δx}$$\frac{dx}{dt}$=$\frac{df}{dt}$?
(f = f (x,y,z))

Assuming that x, y, and z are all differentiable functions of t, it does make sense to talk about df/dt, but it is not equal to $$ \frac{\partial f}{\partial x} \frac{dx}{dt}$$

For f as you have defined it,
$$ \frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial z} \frac{dz}{dt}$$

 

[destroyer130]

I'm not sure what you mean with $\delta x$ in the first place.

δx mean the denominator if taking partial derivative of f wrt x. I wonder both dx and δx equal, since they both mean infinitesimal amount of x.

 

[micromass]

δx mean the denominator if taking partial derivative of f wrt x. I wonder both dx and δx equal, since they both mean infinitesimal amount of x.

Ah, ok! Usually, they denote this by ∂ instead of a delta.

But anyway, if you want to be very rigorous, then things like "dx" or "∂x" don't exist. The only thing that exists are the notations

$$\frac{df}{dx}~~\text{and}~~\frac{\partial f}{\partial x}.$$

But these are not fractions since things like df and dx are undefined. Furthermore, notations like $\frac{dx}{\partial x}$ don't make sense.

That said, there is a way to give dx a rigorous meaning. There are several ways, actually. One of these ways is through nonstandard calculus. Another way is by differential forms. But I won't confuse you with these things. Just remember that if you are in a standard calculus class, then things like dx and df don't really have any meaning. They are very handy and useful notations however.

 

[destroyer130]

Thanks micromass :)