What is the Difference Between Delta and Differential in Calculus?

[gasavilu]

Hi guys

Can anybody help me? What is the difference between a delta $$\delta W$$ and a differential $$dW$$? ($$W$$ a scalar function, for example.) In other words, when shold be used a delta and when a differential? Thanks.

 

[LCKurtz]

Suppose W is a differentiable function of x. Consider some value of x, say x = a and W(a). Now suppose we have a "nearby" point b = a + h, so h would be small. Then:

$\Delta W = W(b) - W(a)$ represents the [exact] change in W from a to b.

The differential of W is defined to be the change on the tangent line at a:

$dW = W'(a)h$.

For small h we have $\Delta W \approx dW$

I'm assuming that your use of $\delta$ has the same meaning as the common usage of $\Delta$. If I'm wrong about that, feel free to ignore this reply

 

[jgens]

The only time I've ever seen delta used like that is with inexact differentials, but I've never done any work with them. Here's a wikipedia article about them though: http://en.wikipedia.org/wiki/Inexact_differential

 

[slider142]

I've also seen $\delta$ used as the variation of a function (calculus of variations or differential geometry). Ie., $W(x) + \delta W(x)$ where $\delta W(x)$ is a function that is "small" in the neighborhood of interest.

 

[haushofer]

Hi guys

Can anybody help me? What is the difference between a delta $$\delta W$$ and a differential $$dW$$? ($$W$$ a scalar function, for example.) In other words, when shold be used a delta and when a differential? Thanks.

dX is in mathematical terms something which is called a one-form. You can integrate it to obtain X. Physically, this X has to be well-defined then. In thermodynamics for instance the quantity X has to be a function of state. A counterexample would be the heat Q or the mechanical work W. You can't define a state with defnit heat or mechanical work; these quantities only have meaning if you go from one thermodynamical state to another. So if I would write dQ or dW for the changes, this would imply that I could obtain Q and W for a state by integrating, which is not well-defined. That's why people often choose to write $\delta$ instead of d for these quantities.

If you want to know the exact mathematical difference, the answer lies in differential geometry I think; like I said, a quantity dX is in diff.geometry a one-form which lives in a dual vector space called the dual tangent space, while $\delta X$ indicates either an arbitrary change (like in the variational principle; here the $\delta$ gets you from one field solution to another which can't be accomplished by a mere coordinate transformation), a coordinate change (if you want for instance to know the behaviour of X under a spacetime transformation; here X is in a representation of some group which describes coordinate transformations like the Lorentz group, the Poincare group or the Galilei group) or a change in some internal space (where X is then a gauge field in some representation of some gauge group and where you perform in infinitesimal gauge transformation).

I hope this helps a little :)

 

[gasavilu]

Hi all.
My original question had to do with a problem of electrodynamics. I have not yet a clear answer but the support received has given me a broader perspective about the problem. Thank you all for your help.

 

[mikeph]

In my experience it was used as a precursor for differentiation, for example,

The gradient of the line connecting the points (f(x), x) and (f(x+δx), x+δx) is [f(x+δx)-f(x)]/δx, in the limit δx -> dx, we get the gradient to be df/dx.

Ie. δf = f(x+dx) - f(x), and df = f(x+dx) - f(x)

This seems to be how a lot of physics lecturers used calculus, although I can't say I'd ever seen this in my maths introduction.

Just seems to me to be a bit of a formalism to make the differentiation clear, without resorting to limits as something goes to zero.