Notational position of $dx$ in integral

Question

In the Analysis courses in Aachen in the 1980s, an integral, whether Riemann or Lebesgue always had the form

$$ \int \text{stuff}\, d\text{var} $$

where the $\int$ and the $d\text{var}$ where used like braces to enclose the integrand 'stuff'. These days on the net and in books, often in the context of physics, I find a notation like

$$ \int d\text{var}\, \text{stuff}_1\, \text{stuff}_2 $$

where it seems that at least $\text{stuff}_1$, maybe even $\text{stuff}_2$, is the integrand. The situation is even more confusing if 'var' is not just a variable but a more elaborate expression. As an example consider the formula from Wikipedia:

$$ f(H) = \int dE\, |\Psi_E\rangle \,f(E)\, \langle\Psi_E^{*}| $$

Where is $\text{var}$ and where is $\text{stuff}$ now?

Questions:

1. Am I right that the integrand is indeed written after the $d\text{var}$ in this case or am I over-interpreting and it is just identity that is integrated?
2. If the integrand is indeed written after the $d\text{var}$, then why? Is it just a different notational convention or does it have a subtly different meaning?

Answer 1

Some lecturers in theoretical physics there at that time used this notation as well. Comes in handy in formulas with many particles and $$ \int\limits\!\!dx^n\cdot $$ is more operator style than $$ \int\limits\!\cdot \,dx^n$$ Also note the habit of not specified integration domain meaning the full space.

Now this $$ f(H) = \int dE\, \left|\Psi_E\right\rangle \,f(E)\, \left\langle\Psi_E^{*}\right| $$ compare with a discrete energy spectrum $$ f(H) = \sum_k\, \left|\Psi_{E_k}\right\rangle \,f(E_k)\, \left\langle\Psi_{E_k}^*\right| $$ where I am tempted to write the sum operator as $\sum_{E_k}$.

Regarding 1.: Everything behind the integral operator is integrand. (both $\mbox{stuff}_i$)

Regarding 2.: It is for convenience, for example to not loose oversight when several integrations are iterated (example). It is also stressing the operator view.

Another example (ideal gas, grand canonical partition function): $$ Z_{gk}(T,V,\mu) = \sum_{N=0}^\infty\frac{e^{\beta\mu N}}{N!} \int d^3 r_1 \cdots d^3 r_N e^{\beta V_N} \int \frac{d^3 p_1 \cdots d^3 p_N}{h^{3N}} e^{-\beta T_N} $$ (Source: Thermodynamik, p. 90)

Answer 2

Just a different use of notation that means the same thing, usually physicists, indeed, make use of $\int dx f(x)$ instead of $\int f(x) dx$.