What Are Differential Operators and Their Applications?

[Marin]

Hi all!

I came upon an expression like that: ' $$\frac{\delta f(x)}{\delta x}$$ ' several times but can't figure out what it's used for.

In Wikipedia it's posted that this derivative type is used when we consider infinitesimally small argument 'x'. So, does this mean:
$$\frac{\delta f(x)}{\delta x}=\lim_{x\rightarrow 0}\frac{df(x)}{dx}$$ ?
What's the sense of defining such a derivative? Where do we see its application?

I was also wondering what type of derivative ' $$\partial, \delta, d,$$ ' the $$\Delta$$, we use in physics, stands for?

Thanks in advance!

Marin

 

[CompuChip]

Hi.
Can you tell us in which context you found that expression? Because I don't think the delta notation is unambiguously standard notation.

As for the last question, I think I can answer that. There are actually two occurrences of the (big) delta notation. In high school, it is used as a difference quotient, in expressions like
$$\frac{\Delta y}{\Delta x}$$,
usually later leading to the definition of derivative as
$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}.$$

In more advanced physics, the triangle is used for the Laplace operator $\Delta = \nabla^2$, where $\nabla$ is the gradient operator, e.g.
$$\nabla f(x, y, z, \cdots) = [\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \cdots].$$

 

[HallsofIvy]

Hi all!

I came upon an expression like that: ' $$\frac{\delta f(x)}{\delta x}$$ ' several times but can't figure out what it's used for.

In Wikipedia it's posted that this derivative type is used when we consider infinitesimally small argument 'x'. So, does this mean:
$$\frac{\delta f(x)}{\delta x}=\lim_{x\rightarrow 0}\frac{df(x)}{dx}$$ ?
What's the sense of defining such a derivative? Where do we see its application?

No, you have the limit on the wrong side. $$\frac{\delta f(x)}{\delta x}$$ essentially represents, not the limit as h goes to 0, but the difference quotient where $\delta x$ is take to be very small, but NOT 0.

I was also wondering what type of derivative ' $$\partial, \delta, d,$$ ' the $$\Delta$$, we use in physics, stands for?

The $\partial$.
$$\Delta f(x)= \nabla^2 f(x)= \left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2+\left(\frac{\partial f}{\partial z}\right)^2$$

Thanks in advance!

Marin

 

[Marin]

Can you tell us in which context you found that expression? Because I don't think the delta notation is unambiguously standard notation.

Here's the link :) http://en.wikipedia.org/wiki/Entropy_(classical_thermodynamics )
It's full of 'big' and 'small' deltas there - what's the point in writing $$\delta Q$$ instead of $$\Delta Q$$?

No, you have the limit on the wrong side. $$\frac{\delta f(x)}{\delta x}$$ essentially represents, not the limit as h goes to 0, but the difference quotient where LaTeX Code: \\delta x is take to be very small, but NOT 0.

Do you mean the following:$$\lim_{\delta x\rightarrow 0}\frac{\delta f(x)}{\delta x}=\frac{df(x)}{dx}$$ ?

I hope I'm not asking too many questions ;)

What does this definition tell us mathematically? I mean, is there any geometrical interpretation like the normal difference quotient and the tangent lines, first deriv., etc.?
And when do we use it?

 

[CompuChip]

Actually I think the notation is used by physicists, in the same way they allow themselves to re-write
$$\frac{df}{dx} = f'$$
to
$$df = f' \, dx$$.

The difference is, that the above does have a rigorous mathematical meaning. Three options I see are:

• Consider the deltas as if they were d's, e.g. as genuine differentials.
• Consider it as functional derivatives, e.g. $\delta E = \frac{\delta Q}{T}$ means that 1/T is the functional derivative of E(T) with respect to Q. (Does this make sense?)
• Don't think about it: "$\delta E$ is a tiny change in the energy".

 

[Marin]

hmmm ok, I think I got it :)

Thanks, you two, Marin