Why in physics the elementary work is written as $\delta W$ instead of $dW$?

Question

Why an elementary work is written $\delta W$ instead of $dW$? For example, it's often written $$\delta W=F\cdot dr$$ if $dr$ is the elementary displacement. Why don't we write as usual $dW=F\cdot dr$ ? I saw an answer here but it doesn't really answer to the question (at my opinion). By the way, since at the end $W_{AB}=\int_A^B \delta W$, I really don't understand this $\delta W$. Is there mathematically a sense ?

Answer 1

In physical literature the symbol $d W$ is used to indicate an exact differential form, and $\delta W$ is used because, in general, the infinitesimal work $\delta W=F \cdot d\vec r$ is not an exact differential form. It is an exact form if the force $F$ is conservative, so, in this case we write $\delta W=dW$.

Answer 2

The notation $\delta W$ rather than $\mathrm{d} W$ aims to underline that work (like heat) is an improper differential, i.e. its integral depends crucially on the integration path taken.

• In physical terms, the work done on a system will not only depend on the initial and final condition of the system, but on the transformation chosen to drive the system from its initial to its final condition.

• In more mathematical terms, an inexact differential cannot be expressed as the the gradient of another function, making an integral of that differential path-dependent.

I found the wikipedia article on the topic quite helpful; any good textbook on thermodynamics will discuss this, for example Schwabl's Statistical Mechanics.