So in physics, the notation $\int \dd{x} f(x) \triangleq \int f(x) \dd{x} $, i.e. the differential is allowed to precede the rest of the integrand. This is because you can end up with some absolutely disgusting integrand which is a complicated function of several parameters in addition to the variable of integration. The idea is, when reading from left to right, to establish which variable is the integration variable as soon as possible.
Well, that is not quite what it says. Stuff like $\int \dd x \exp{x}$ is quite common, meaning ‘the operation of integration with respect to x is applied to the function e^x’. The first step is quite alright, because you should not read the the juxtaposition of the end of the bracket and the function on the second line as ordinary multiplication, but rather some operation being applied to the function. I say this as a physicist, mathematicians probably wants to find me and have me killed.
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So in physics, the notation $\int \dd{x} f(x) \triangleq \int f(x) \dd{x} $, i.e. the differential is allowed to precede the rest of the integrand. This is because you can end up with some absolutely disgusting integrand which is a complicated function of several parameters in addition to the variable of integration. The idea is, when reading from left to right, to establish which variable is the integration variable as soon as possible.
Well, that is not quite what it says. Stuff like $\int \dd x \exp{x}$ is quite common, meaning ‘the operation of integration with respect to x is applied to the function e^x’. The first step is quite alright, because you should not read the the juxtaposition of the end of the bracket and the function on the second line as ordinary multiplication, but rather some operation being applied to the function. I say this as a physicist, mathematicians probably wants to find me and have me killed.