Power Tools - Notation Conventions

Conventions are standard ways of doing things that help make life easier. When properly used, they eliminate confusion and needless work. That's why mathematicians always use the greek letter "pi" to represent the ratio of the circumference of a circle to its diameter, 3.141592...(":1" is implied). Whenever it appears in the literature, everyone knows what it means.

But my keyboard has no greek letters! This manual is not produced on a fancy word processor or using a Java engine (and I'm too lazy to type code point numbers). Neither will many of your works be. So feel free to just type "pi" (and use good ol' ASCII). By establishing a few other conventions, many standard and some like "pi", you won't notice the difference that much. You may even find it easier to derive expressions using these conventions in your favorite text editor, rather than trying to remember the keying sequence in a word processor that dims the lights when you run it!

Here are the more standard conventions.

Mathematical statements like, E = mc2, are the basic building blocks of physics models. Some of these statements define the character and makeup of things that can be measured, like the speed of light, the "c" in E = mc2. We call the measure of a "thing" its value or quantity.

The speed of light has a value of 186,282 miles per second. If we just gave the speed of light as 186,282, we wouldn't know how far it travels or in what period of time. "Miles per second" tells us how to fit the number 186,282 into physical reality. That's what units do.

Every value has a number and units. Values that are ratios or pure numbers (numbers that don't have units) like pi (pi is 3.14159265...) are called scalars or coefficients (the long name) because they modify other values by scaling them. The circumference of a circle is pi times larger than its diameter.

When working with values it comes in handy to give them short names. Who would write "The amount of energy in a thing is its mass times the speed of light times the speed of light again" instead of "E = mc2"? In a long science paper it would get old quickly - for the writer AND reader.

So in addition to a number and units, each value is given a shorthand name called a symbol or quantity symbol to be more specific. What does the symbol mean? The quantity symbol is written followed by an equal sign, =. To the right of the equal sign is an expression.

Symbols

Symbols, then, represent quantities that can have mathematical actions performed on them. They are mathematical objects. Written words are symbols of language. But mathematicians and scientists found that writting mathematical statements with words is very tedious and often imprecise. Symbols of only one or two letters (like g or pi) and perhaps a number (like L1) make the mathematical ideas much more compact and meaningful.

Many of the symbols used in the models have the same meaning throughout the manual. In particular, the track and car model symbols become the grounding points of the models. Symbols representing the dynamic quantities in the models will change the details of their expressions from model to model. They must be considered in context.

Expressions

"mc2" is a mathematical expression that tells about the quantity energy, "E". "E=mc2" is a mathematical sentance and has the form of a definition. "E=mgh" is another definition for another kind of energy. Some of the symbols used in this manual will have the same definitions throughout. Others change with the model being discussed. Beware of context!

A simple expression like "mc2" is also called a term. Terms are symbols glued together by multiplication. No "x" multiplication symbol is used. This means that symbols with names like mn and m can be confused in a term like "mng". The convention used in this manual is that the symbols in a term are always ordered to eliminate confusion. If "mn" is in the term and it is defined in the model, then that's what it is. If "m" were meant, it would have been written "nmg". However, if ordering does not eliminate the confusion, parentheses are used, eg. "(m)n".

An expression tells about the interaction of its terms. Terms in an expression are separated by operator symbols, like "+" and "=", that stand for the mathematical connection (relation) between them. It is worth noting that two expressions joined by an equal sign, "=", are also an expression. So often, symbol definitions are called expressions.

Operators

Operators used in this manual represent grouping "( )", addition "+", subtraction "-", division "/", squaring "2", squareroot "\[ ]", exponentiation "exp( )" or "e^( )", the trigonometric functions sin( ) cos( ) tan( ) arcsin( ) arccos( ) arctan( ), the hyperbolic functions hsin( ) hcos( ) htan( ) archsin( ) archcos( ) archtan( ), the differential "d" and

                         / upper limit
integral: constant term | integrand
                       / lower limit

Though many of the operators are written above with the grouping operator "( )", it is not used when the argument of the operator is a single symbol as in "sinO" for "sine of Theta".

Multiplication has no symbol but is implied where ever quantity symbols or expressions surrounded by the grouping operator, "( )", are written next to eachother in terms.

In this manual, powers are written as a symbol followed by an exponent which is always a small integer (like v2). When an exponent is a symbol or expression, the operator symbol, "exp( )", is used (like vexp(k)).

When evaluating an expression between limits, the expression is placed in square brakets and the limits are separated by a comma and surrounded by parentheses. The evaluation using the first limit is subtracted from the evaluation using the second limit. For example, the solution of an integral may yield "mv evaluated in the interval v1 to v2. This is written as "[mv](v1,v2)", which evaluates to "mv2-mv1" (note "v2" is a limit variable, not the original "v" squared.

Assumptions

Though expressions in this manual often use the grouping operator "( )" to show the order of evaluation, some assumptions apply to eliminate confusion where it may occur.

1. Numerical constants are written at the front of each term.
   For example, 2aApv2 is twice aApv2.
   
2. When evaluating an expression, identify symbols first. Some end in numbers that should not be confused with exponents. Only those symbols mentioned on the page or referenced from other pages are used.
   For example, mv2 is usually "m times v squared". But if mv is defined, then mv2 is "mv times mv". If v2 is defined as a symbol, then mv2 is "m times v2".
   
3. Exponents only apply to the preceeding symbol unless grouping "()" is used.
   For example, in aApv2, only the "v" is squared not aApv - that would be written "(aApv)2".
   
4. Only one term is in the numerator of a division, "/", unless the grouping operator, "( )", is used. In that case there is nothing between the final ")" and the "/".
   For example, in (A-B)/C, C divides both terms in parentheses. In A-B/C, C only divides B.
   
5. Only one term is in the denominator of a division, "/", unless the grouping operator, "( )", is used. In that case there is nothing between the "/" and the initial "(".
   For example, in A/(B+C) the sum of B and C divides A. In A/B+C, only B divides A, C is added to the result.