Δ in derivative and partial derivative notation

1. Jul 2, 2012

Cinitiator

1. The problem statement, all variables and given/known data
What does it mean when lowercase Delta (δ) is used in partial derivative and derivative notation? Does it make any difference? Or is it just a personal preference?

2. Relevant equations
-

3. The attempt at a solution

2. Jul 2, 2012

HallsofIvy

Staff Emeritus
If you means something like $\delta x$ or $\delta f$, it means "a slight change in" x or f. It does NOT occur in "partial derivative and derivative notation" but you may see it in the definition of the derivative.

3. Jul 2, 2012

Staff: Mentor

The Greek character δ (lower-case Delta) is not used in either kind of derivative. In ordinary derivatives, d is used, as in $$\frac{dy}{dx}$$
Here y is a function of a single variable. IOW, y = f(x).
It's probably not too far wrong to think of this as the quotient of two differentials: dy and dx.

For partial derivatives, a different character is used. As far as I know, it's not part of any alphabet.

If z is a function of two variables, say x and y, then we can talk about two partial derivatives:
$$\frac{\partial z}{\partial x}$$
and $$\frac{\partial z}{\partial y}$$

4. Jul 2, 2012

elfmotat

Usually it denotes a "virtual" derivative, one where time is held constant.

If you have a position vector $\textbf{r}$ which is a function of several variables, $\left \{ q_1,q_2,q_3,...,q_n \right \}$ and time $t$, the the total differential displacement is given by:

$$d\textbf{r}=\frac{\partial \textbf{r}}{\partial t}dt+\sum_{i=1}^n \frac{\partial \textbf{r}}{\partial q_i}dq_i$$

This is just the chain rule. The virtual displacement, however, is given by:

$$\delta \textbf{r}=\sum_{i=1}^n \frac{\partial \textbf{r}}{\partial q_i}\delta q_i$$

Note that it holds time constant. Virtual displacement is very useful in areas that use Calculus of Variations, such as Lagrangian mechanics.

5. Jul 2, 2012

klondike

It looks more like variations to me.

6. Jul 2, 2012

HallsofIvy

Staff Emeritus
But elfmotat's suggestion is good too.