Usage of capital and small letters

Question

Are there rules for when small, capital and greek symbols are used?

Simple example: The volume left in a cube with a sphere in it is

V=C-S with capital standard latin letters, meaning "if you don't know who I am, you have to do the equations first. I'm V = w l h and the Sphere is 4/3πr²" with small letters simply meaning "input please, no calculations necessary", while the usage of symbols like π and Ω are constants not to be touched or played with.

E=mc² is another example. To find E, m and c says "input please", while E is impatiently waiting on someone like me who needs a little extra time, 'cause maths frightens me. But I'm trying to learn!

I've also seen bold small and capital letters here and there, and I don't think it's hallucinations, no matter how scary and the gut-feelings I get when I see a nasty dangerous one. It's just mathemaphobia, I know mathematics isn't dangerous, so could someone please help me out in simple non-mathematical English ?

Thank you

Answer 1

You follow conventions for the thing you're reading/writing. Vectors are often bolded, matrices are often capital letters, random variables are often capital letters, etc.

While writing, if you're deviating from conventions, make a note of it. And you can use words in mathematical writing to help detail what the meaning of things are.

Answer 2

There are no standard rules. Writers of maths are allowed to use whatever letters they like for things, and different writers are allowed to use different letters. Most people follow some unwritten conventions, but different users of maths use different conventions. Users in the same discipline area who write to each other a lot tend to use the same conventions, but even then there are differences.

There are some common conventions (but by no means set in stone):

• Areas and volumes often use capital letters, while lengths often use small letters
• Bold letters are usually used to mean vectors, though some writers do not use bold for vectors, and others mark them in a different way
• In linear algebra, a matrix is usually a capital letter
• In calculus, small letters are usually used for functions, with a capital letter often reserved for a function which is the integral of some other function
• In geometry, capital letters are usually used for points, small english letters for lines and small greek letters for planes (including $\pi$)
• $\pi$ is usually reserved for the number $\pi$ in places where that number is likely to appear. In other places, it is used to mean a probability or a profit or a projection or a permutation.
• small e is not a common choice for a letter in a scientific formula, because it can be used to mean the number $e$. However in ordinary algebra it can be used for a coefficient, especially if you have already used a, b, c, d

None of those are hard-and-fast rules, but simply observations of how people tend to write.

Note that there are more concepts in the world than letters that can represent them, so a letter E is not always energy, but represents whatever the writer has chosen it to represent today. Hence you should always tell your reader what the letters stand for. Sometimes we are sloppy if we think the reader understands already. For example, when we write "$E = mc^2$" we should technically write "$E = mc^2$, where $E$ is energy, $m$ is mass, and $c$ is the speed of light in a vacuum", but we generally assume people know a bit about this particular formula already. Most of the time, though, you should be very clear what you mean by each letter the first time you use it.

Another point about good writing is that in one piece of writing, you shouldn't change your notation mid-way. This is very frustrating to the reader! (Imagine if an author changed the name of their main character partway through a novel!)

The final point, which is actually a proper rule is that captital/small/bold/curly letters are not interchangeable. If a capital $E$ is used for something in a piece of writing, then you can't use a small $e$ elsewhere to mean the same thing. In short, maths is "case sensitive".