Htykuut

In high school, I was told that I always should use arrows to indicate rearrangements of equations, such as:

begin{eqnarray}
aleft[b+frac{c}{d}right]&=&a\
&Updownarrow&\
1-frac{c}{d}&=&b
end{eqnarray}

upon starting my bachelor in physics, some of my co-students have suggested that this is, in fact, if not outright an error, at least not good mathematical notation, and the former example ought to be written:

begin{eqnarray}
aleft[b+frac{c}{d}right]&=&a\
1-frac{c}{d}&=&b
end{eqnarray}

Edit: arguing that they are not necessary, and thus a waste of symbols and possibly a source of distraction.

While others again have suggested that it is correct, but only if the arrows are one-way arrows:

begin{eqnarray}
aleft[b+frac{c}{d}right]&=&a\
&Downarrow&\
1-frac{c}{d}&=&b
end{eqnarray}

Edit: arguing that one shouldn't give the impression that for instance $F=m a$ can be proven from one the result of the results in one particular theoretical exercise in classical mechanic.

I, therefore, wish to know if any official guidelines exist for arrows between equations, and if they are optional, whether or not they are recommended and which types of arrows (one-way or two way) are recommended.

Arrows indicate the flow of logic, meaning that the equation at the tail of the arrow implies the equation at the head of the arrow. If you use a double-headed arrow, it means that each equation implies the other.

For example, the following uses of $Rightarrow$ and $Leftrightarrow$ are fine:
$$x = 1 ~ Rightarrow ~ x^2 = 1 qquad text{and} qquad 2x+1 = 5 ~ Leftrightarrow ~ x=2$$
but the following uses of $Rightarrow$ and $Leftrightarrow$ are not fine
$$x = 1 ~ Leftrightarrow ~ x^2 = 1 qquad text{and} qquad sin(x) = 0 ~ Rightarrow ~ x=0$$
The first fails because $x^2=1$ implies $x=1$ or $x=-1$, not necessarily that $x=1$; the second fails because $sin(x)=0$ implies that $x=npi$ for some $n in mathbb{Z}$. The issue in both of these cases is that the operation we performed (squaring and applying the $sin$ function, respectively) was not invertible.

So to answer your question about whether you should use arrows, my advice comes in two parts:

• 
  Yes, use arrows! They help to explain your flow of logic, meaning that you're communicating your mathematical ideas more effectively. But...


• ...be careful! If you're rearranging equations then usually the $Rightarrow$ direction is what you're doing; if you want to be able to reverse it (i.e. turn the $Rightarrow$ into a $Leftrightarrow$) then you need to make sure that the operation you when turning one equation into the next is an invertible operation.

I don't know anyone in their right mind who thinks that writing a sequence of equations with no indication of logical flow is better than writing a sequence of equations with an indication of logical flow.

P.S. You ask in your question about official guidelines, so I should clarify that there are none, but there are certain practices (e.g. indicating logical flow) that are evidently better than others (e.g. not indicating logical flow, or erroneously indicating incorrect logical flow).