Deciding on mathematical notation I am currently collecting all the small pieces of independent mathematics I've done over the past year or so, a lot of which is calculus. I have lots of integrals and for the most part, I just call them $I$ or $I(a)$ if there is some variable involved. I believe this leads to some confusion throughout and would like to know what a standard mathematician does when inventing notation. Confusion aside, I also have notation which seems inelegant; for example, an integral with 4 parameters I named $\textbf{F}(u,v,n,m)$. This itself is involved in recurrence equations and a proof I wrote was almost unreadable. 
So my questions are: 
Are there standards for creating notation and if so what are they?
Are there any general rules to abide by, such that my work appears neater?
If no for both, any suggestions on the specific cases I discussed?
 A: Some thoughts, too long for a comment.
The answer to your question will depend in part on the audience. Are these notes for you for future reference, or do you hope or plan to share or publish them? 
I'm inclined to suggest that you be conservative. When established notation exists you should have a good reason for replacing it.
New temporary notation to make nearby formulas shorter and easier to parse is fine. New notation defined at the start of a long document may make your readers jump back and forth between definition and use.
If your notes are (or will be) written in $\LaTeX$ then you can write macros for the things you might want new notation for. That way you can use your private language in the source files and postpone deciding how it should look (standard or newly invented) in the final text.
A: Clear and concise mathematical notation is often hard to come up with, and often follows the introduction of a theory by many years. The original person to come up with an idea may not also come up with the best way to write it out.
A great example is Maxwell's equations.  If you read his original papers, he uses component notation and although the equations exhibit a tantalizing pattern, they are nowhere near as clear as the now-familiar equations in modern notation.
The skill of developing good notation is related to, but not completely correlated with, the degree of math insight or ability to solve difficult problems. 
For your question about integrals, usually it is best to leave things with the integral signs. The major exception is when doing a calculation that finds the value of an integral by squaring it or showing that it is equal to some constant minus itself.  For example, a familiar proof starts with:

Let 
  $$
I = \int_{-\infty}^\infty e^{-x^2} dx
$$
  Then 
  $$
I^2 = \int_{-\infty}^\infty e^{-x^2} \int_{-\infty}^\infty e^{-y^2} dx \,dy
=\int_{r=0}^\infty\int_{\theta=0}^{2\pi}e^{-r^2}r\,d\theta\,dr = 
\left. 2\pi \frac{e^{-r^2}}{2}\right|_0^\infty=\pi
$$

A: Maybe put the four variables into a matrix ?
\begin{eqnarray*}
F\begin{bmatrix}
u && v  \\
n && m  \\
\end{bmatrix}=\int \cdots
\end{eqnarray*}
or a subscript that is a matrix 
\begin{eqnarray*}
F_{ \begin{bmatrix}
u && v  \\
n && m  \\
\end{bmatrix} } =\int \cdots
\end{eqnarray*}
A: I second Ethan Bolker's recommendation to use a LaTeX macro to help you change notation later easily.
That will allow you to experiment and see what works.
In addition, I would consider grouping the parameters according to their roles.
For example, is $n$ a dimension, $m$ another integer parameter, and $u$ and $v$ real numbers?
Is one of them typically fixed and the other three change more often?
It might make the notation easier to read if you group the arguments accordingly.
For example, you can consider $F_n(u,v;m)$ or $F_n^m(u,v)$.
It's hard to judge without knowing what your integral and recurrence relations look like.
For example, suppose I have a parallelepiped-shaped die and I let it bounce on a surface described by a function $f$.
If the initial position and orientation are $x_0$ and $\omega_0$ and the die is described by a matrix $A$, I might denote the state of my die at time $t$ as $S^A_f(x_0,\omega_0;t)$.
This argument grouping helps me: $A$ describes the object and $f$ the environment (both are parameters in some sense), the pair $(x_0,\omega_0)$ is the initial state and $t$ is time.
I might often keep $A$ and $f$ fixed and then differentiate with respect to the initial state and time, and I find this notation helpful in such use.
Experiment with different ideas to see what fits.
And be grateful that you only have four variables; things become far more ugly with Christmas trees like $F^{a,b;c,d}_{e,f;g,h}(i,j;k,l)$.
If there is any established notation for what you have, try to follow that.
If you decide to do something different, please let your readers know of other notations out there.
