# 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
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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.