Basic Guidelines: First, before you compose any math expressions for sending over the email, you should set the font of your email composition window to an equal-space font like Courier. This will assure that columns will line up the same way for both sender and receiver. Also it is very important to remember to press the enter key before you reach the 80th column of each line. Do not count on the auto-wrap to insert the end-of-lines -- please explicitly press the enter key at the end of each line. Many mail readers render email very poorly if it contains no hard end-of-lines, and your colums will not line up for me the same way as they did for you if you fail to follow this guidline. And please, do NOT ever use tabs anywhere. If you do you will very nearly assure that columns will not line up for the recipient as they did for you. So always use spaces instead of tabs.
The notation x^n is almost universally taken to mean
xn. It is much easier to do exponentiation
using the ^ symbol than to try to line up the exponent
in the correct column of the preceding line. If the exponent
is an expression, be sure to use parentheses -- x^(n+1) for
example. The function,
Most of the time you can indicate the product of two things simply by
listing them, one after the other, with no operator symbol in between.
For example, 2a is taken to mean two times a. There
may be cases, though, where this can be hard to read or ambiguous.
For example, asin(x) could be taken to mean a times
sine of x, or it could be taken to mean arcsine of x.
It would be better to write the former as
To render the square root of an expression, you can use
Use parentheses wherever there is any opportunity for amiguity.
ambiguous unambiguous unambiguous
1/2a 1/(2a) (1/2)a
2^n-1 2^(n-1) (2^n) - 1
e^x^2 e^(x^2) (e^x)^2
Generally, if no parentheses are shown, exponentiation is assumed
to be done first, followed by multiplication, division, subtraction,
and addition, in that order. Even if you follow that convention, when
leaving out the parentheses, it often behooves you to put an additional
blank to offset the later-applied operators:
y = x^2 + a x + bThis makes it much easier to read. And if you put in parentheses even where they are redundant, you haven't hurt anything. So when in doubt, put them in.
Do use parentheses when you are using functions:
y = sin(2x)Put the parentheses around the argument of the function. If you leave out the parentheses in an example like that, it is especially hard to read. If you want to have, for example, the square of the sine of something, you should use:
y = sin^2(2x)
Use |x| to means absolute value of x. You can find the key that produces the | symbol in the upper right of your keyboard, near the backspace key on most standard keyboards. The symbol silk-screened onto the key usually looks like a funny-looking colon. Typing this symbol requires you to use the shift key on the majority of keyboards. You can put any expression you like inside the | symbols.
The f'(x) notation is a lot easier than the
df -- dxnotation. Use the f'(x) notation wherever feasible. Use the f"(x) for second derivatives. For still higher order derivatives you can use something like f"'(x) for third derivative, f""(x) for fourth, and so on. For very high-order derivatives, the easiest notation to read is
d^n y ----- dx^nI will also recognize dy/dx as the derivative of y with respect to x and
For partial derivatives, go ahead and use the dy/dx notation, but annotate on the line just before or just after which d-expressions are partials. Or you can do what one of my correspondents has done, and that is to indicate partial derivatives using a <> notation:
<dx/dt>means the partial derivative of x with respect to t.
Each of these has its own peculiar notation in math books. There is an easy way to write them in email, though, that is self-evident to the reader.
lim{x->0} x ln(x)
is "the limit as x goes to zero of x times ln(x)."
integral{0 to r} f(x) dx
is the definite integral taken from 0 to r of
double integral{r=0 to 1; theta=0 to 2pi} f(r, theta) r dr dtheta
For summations, use, for example
sum{k=0 to infinity} x^k/k!
You can see that the behavior of the counting variable is described
here inside the curly-brackets.
No easy way to do these. For small expressions you can use something
like
x^2 - 2x + 1
y = -------------------
x^3 - 3x^2 + 3x - 1
There are two commonly used ways of doing this. If you are going to be putting expressions into the subscript, use something like
b[n] = n(b[n-1] + b[n-2])If you are simply going to be using the subscript as a label, for example taking a difference between v-final and v-initial, you can use either the square-bracket notation or something like
delta v = v_f - v_i
Use <= for "less than or equal" and >= for "greater than or equal". You can use either <> or != as a symbol for "not equal". The squiggle, ~, is a good symbol for "approximately equal". Also, << means "much less than", and >> means "much greater than". For the equivalence symbol, º, simply use equiv.
You can either spell them out, like delta, epsilon and omega, or you can use the Roman equivalent, which in these cases would be, d, e, and w respectively. Usually the context will make it clear what you mean.
If you are sending stuff that involves vectors, you can represent the dot product of two vectors, v and w as
v dot wLikewise, their cross product would be
v cross wThe norm or length of a vector, v, is simply |v|.
If you are dealing with vector fields, use the following equivalents for the "del" operator:
expression email version
Ñf grad f
Ñ·f div f
Ñ´f curl f
Ñ2f del^2 f
The expression, [x], is usually used to mean the floor function of x. Another phrase for this function is the greatest integer function. It means, take the greatest integer that is less than or equal to x. So, for example:
[1.5] = 1 [0.99] = 0 [-1.1] = -2You can use the [x] notation to represent this, or you can use floor(x).
The function, sgn(x), is 1 for
positive values of x,
-1 for negative values of x,
and zero for
The function, step(x), is 1 for
positive values of x,
zero for negative values of x,
and is undefined for
sinh(x), cosh(x), tanh(x), coth(x), sech(x), and csch(x) are the hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, and hyperbolic cosecant respectively. For the inverse of any of these, just precede its name with arc (e.g., arctanh(x)).
email me at hahn@netsrq.com