I was reading Field Energy & Field Momentum of Feynman's Lectures on Physics Vol.II; there he introduced a trick 'to throw out—for a while at least—the rule of the calculus notation about what the derivative operator works on.'
Here is the excerpt:
Rather than working out all the components of $\mathbf ∇⋅(\mathbf B×\mathbf E),$ we would like to show you a trick that is very useful for this kind of problem. It is a trick that allows you to use all the rules of vector algebra on expressions with the $\mathbf ∇$ operator, without getting into trouble. The trick is to throw out—for a while at least—the rule of the calculus notation about what the derivative operator works on. You see, ordinarily, the order of terms is used for two separate purposes. One is for calculus: $f(d/dx)g$ is not the same as $g(d/dx)f;$ and the other is for vectors: a×b is different from b×a. We can, if we want, choose to abandon momentarily the calculus rule. Instead of saying that a derivative operates on everything to the right, we make a new rule that doesn’t depend on the order in which terms are written down. Then we can juggle terms around without worrying.
Here is our new convention: we show, by a subscript, what a differential operator works on; the order has no meaning. Suppose we let the operator D stand for $∂/∂x.$ Then $D_f$ means that only the derivative of the variable quantity f is taken. Then $$D_ff=\frac{∂f}{∂x}.$$ But if we have $D_ff,$ it means $$D_ffg=\left(\frac{∂f}{∂x}\right)g.$$ But notice now that according to our new rule, $fD_fg$ means the same thing. We can write the same thing any which way: $$D_ffg=gD_ff=fD_fg=fgD_f.$$
You see, the $D_f$ can even come after everything. (It’s surprising that such a handy notation is never taught in books on mathematics or physics.) [...]
You will see that it is now going to be very easy to work out a new expression for $\mathbf ∇⋅(\mathbf B×\mathbf E).$ We start by changing to the new notation; we write $$\mathbf ∇⋅(\mathbf B×\mathbf E)=\mathbf {∇}_B⋅(\mathbf B×\mathbf E)+\mathbf {∇}_E⋅(\mathbf B×\mathbf E).$$ The moment we do that we don’t have to keep the order straight any more. We always know that $\mathbf {∇}_E$ operates on $\bf E$ only, and $\mathbf {∇}_B$ operates on $\bf B$ only. In these circumstances, we can use $\bf ∇$ as though it were an ordinary vector.
Hmmm... Feynman just altered a notation, a symbol. How just by using another symbol, I will get more advantage? What problem would I had to face hadn't I used Feynman's trick? I'm not getting that:/
Could anyone please explain how by using this trick, it would be easier to use rules of vector algebras on $\bf \nabla$ operator?
Also, how could he say $$fD_fg=fgD_f= D_ffg\;?$$ what is the proof?