# I How can 'd' mean two different things?

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1. Feb 25, 2017

### EddiePhys

d is sometimes used to represent an infinitesimal change in a quantity and sometimes a small amount of a quantity. E.g dx vs dM. dV could mean a small volume element and also an infinitesimal change in volume. How can it be used for two different things?

My suspicion is that while converting Riemann sums into an integral, quantities inside the sigma like the finite mass mi of something is converted to dm for the sake of convenience and less notational clutter.

Last edited: Feb 25, 2017
2. Feb 25, 2017

### Staff: Mentor

The number of letters in the alphabet is finite, and there is no central authority for the use of letters. Most letters have more than one meaning, and you need context to figure out what is meant. "d" is often used as prefix for small or infinitesimal things, but it is also used as (macroscopic) length, for a day, as SI prefix, circle diameter, ...

3. Feb 25, 2017

### EddiePhys

So whatever meaning it had in the Riemann sum, is the meaning in the interval? E.g if Δx meant a small interval/length in the Riemann sum, dx would mean the infinitesimal length

4. Feb 25, 2017

### Staff: Mentor

I guess so.

5. Feb 25, 2017

### Staff: Mentor

Yes, it may mean that. Perhaps it probably means that. But there is no 100% guarantee. You have to look at the entire context.

It is easy for innocent notation to run into trouble. Suppose you had a d axis, what would you call the infinitesimal length in the d direction?.

Electrical engineers use I to mean current. In complex arithmetic, nearly everyone but electricals use $i$ to mean $\sqrt{-1}$, but to avoid the ugliness of $iI$, electricals use $j$ for $\sqrt{-1}$.

You can't depend on others to say with certainty what the symbol d means in whatever you're reading without showing the entire context.

6. Feb 25, 2017

### phinds

My favorite story about this is that you can somewhat tell what kind of STEM person a person is by just asking them to define e
physicist: energy
electrical engineer: voltage
mathematician: the base of the natural logs

Why would anyone expect a letter to mean the same thing at all times to all people?

7. Feb 25, 2017

### EddiePhys

Thanks, I got it.

8. Feb 26, 2017

### sophiecentaur

I think this can be resolved when you realise that dy, dV, dA (length, volume or area) etc all represent the limit as the independent variable step (δx) goes to zero. (d is an Operator) Once you have got to the differential dy/dx,dv/dx order/dx, the actual geometry of the situation has been left behind. (The "clutter" disappears due to the right manipulation and the d operator is common to all)
In my first (and very pernickety) introduction to Differential Calculus, we would arrive at a triangle or other shape, when doing it graphically, using the δ symbol. We were only 'allowed' to get to the d symbol by strictly taking the limit of the expression containing δ. In doing that, it is acceptable to eliminate terms like (δx)2.
I remember our teacher being very disparaging about the Quick and Dirty derivations that some other courses used.