Equations

The typesetting of mathematical equations is a complicated matter. The Chicago Style Manual [1] has a whole chapter dedicated to it and even that is lacking the finer points. The American Society of Mathematics has published Mathematics into Type [14] by E. Swanson. The scope of these books is far greater than that required by a student to properly typeset simple equations.

One must understand that mathematical equations use a language with nouns, verbs, etc. Swanson [14] gives this simple example:

$$A=B+C$$ (2)

In this case $A$ , $B$ and $C$ are nouns, $+$ is a conjunction and $=$ is a verb. Prove it to yourself by reading Equation 9.1 out loud.

The other misunderstood fact about equations is that when the letter P appears, one should read pressure. Hence equation 9.2 should be read as the pressure of the gas times the volume of the gas is equal to the mass of gas times the gas constant times the absolute temperature. It is not read as peevee equals emartee which is meaningless. The guide to translating the symbols into words is provided in the Nomenclature.

$$P V =m R T$$ (3)

Here are a few guidelines to correctly set your equations using a modern equation typesetter such as Microsoft Equation Editor $^\mathrm{TM}$ , Math Type $^\mathrm{TM}$ , TEX $^\mathrm{TM}$ or L^ATEX $^\mathrm{TM}$ . Though these matematical typesetters are a great help, their defaults cancause errors for the layman. For example, the default for most of these software is a scalar variable, hence everything is italicized. This is often incorrect.

• Symbols that replace a number (variables or constants) should be italicized. Hence it is $p$ for pressure and not p. It is $c$ (velocity of light in vacuum, a constant) not c. This is also true for indices and exponents. The specific heat of a gas at constant pressure is $c_p$ not $c_\mathrm{p}$ nor $\mathrm{c_p}$ .

• Symbols that replace a word (often used in indices or in chemistry) are set using the roman typeface. So the enthalpy of air is $H_\mathrm{air}$ not $H_{air}$ . All chemical symbols (H, He, Li, Be, ...) and mathematical symbols ($+$ , $=$ , $\times$ , ...) follow this rule.

• Use the roman font for functions like $\sin\left(x\right)$ , $\cos\left(x\right)$ and $\sum x$ . You must differentiate between the exponential function, e, and $e$ , the base of the natural logarithms which has a value of 2.718281...The first is a function, the second is a constant. This rule also applies to the differential function. It is $\frac{\mathrm{d} x}{\mathrm{d}t}$ not $\frac{dx}{dt}$ . Notice the thin space in the numerator to differentiate that $\frac{\mathrm{d}}{\mathrm{d}t}$ is an operator.

• Scalars are written in italics, vectors in bold italic typeface and tensors in bold italic sans serif. $V$ is velocity and $V$ is speed.

• When expressions include units, there is an unbreakable thin space between the value and the units. Most equation typesetters allow variable spacing. $150 \mathrm{MPa}$ is appropriate whereas 150 MPa is not. Look closely, you can see a difference.

• The multiplication symbol, $\times$ , should only be used between numbers (not variables) or to indicate a vector product. Do not use it between scalar variables as a space suffices to indicate the product.

• Use scientific or engineering notation. Do not use computer programming syntax to express numbers. The value of Avogadro's number is $6.023 \times 10^{23}$ not 6.023E23 nor 6.023e23. This usage was forced upon us because early computer output had but one font and no way of indicating subscripts or superscripts. This is still true in some programmes like Microsoft Excel $^\mathrm{TM}$ whose programmers think scientific notation requires a capital e. Most of these problems can be avoided by a proper use of the SI.

• Fractions are a common pitfall. Use the solidus, /, with caution. $a+b/c$ could mean $\frac{a+b}{c}$ or $a+\frac{b}{c}$ . Parentheses help but can become unwieldy. Use Equation 9.3 instead of Equation 9.4:

  $$\frac{ \left(a_1+i a_2\right)+\left(a_{11} s_1+a_{21} s_2\right) } { \left(b_1+i b_2\right)+\left(b_{11} s_1+b_{21} s_2\right) }$$ (4)

  
  

  $$\left[ \left(a_1+i a_2\right)+\left(a_{11} s_1+a_{21} s_2\righ... ...] / \left[ \left(b_1+i b_2\right)+\left(b_{11} s_1+b_{21} s_2\right) \right]$$ (5)

  
  

• Try to avoid long subscripts and superscripts. Use $\mathrm{exp}\left(\frac{a-b}{c+d}\right)$ instead of $\mathrm{e}^{\frac{a-b}{c+d}}$ . When using variables with many subscripts, the subscripts are separated by commas. The specific heat of air at constant pressure is $c_{p,\mathrm{air}}$ .

• Instead of repeating parentheses to mark groupings and sub-groupings, alternate between parentheses, brackets and braces preferably in this order:

  $$\{[()]\}$$

  
  
  Use Equation 9.5 as an example.

• Use the verb or conjunction to break long equations. In Equation 9.5 the addition and subtraction symbols (the conjunctions and less) are used. In Equation 9.6 the break point is a verb (the equals signs).

  $$\begin{split}\left(\frac{\mathrm{d} \mathcal{B}} {\mathrm{d}... ...\rho \mathrm{d}V \right)_{t} } {\Delta t} \right] \end{split}$$ (6)

  
  

  $$\begin{split}\Delta E_{\mathrm{K}} &= -m g\left(z_{2}-z_{1}\... ...rac{1 BTU}{778 lb_{f} ft}}\ &= 1 \mathrm{BTU} \end{split}$$ (7)

  
  

• These rules are true for mathematical symbols no matter where they appear in the text. Do not write a scalar variable symbol in roman because it is in a paragraph. It should always appear in italics.