Formal definition of line integral along reversible and irreversible path

Question

In thermodynamics work can be done by moving by reversible or irreversible path. Physical definition of reversible and irreversible process is weary common in thermodynamics textbooks. What does it mean mathematically? How to formalize it? Work is line integral of differential form. $$W=\int_{L} P(x, y, z) dx+Q(x, y, z) dy+R(x, y, z) dz$$ where L is equation of curve. If work is function of p and V, does it mean that there are curves on p-V plane that have property of being reversible or irreversible path?

Answer 1

Your concept of $L$ is faulty.

Equations like $f(x,y,z)=0$ do not define curves, instead they generally define surfaces. For example, $x+y+z=1$ is a flat plane.

Instead, when defining a curve along which to take a path integral, one uses a curve $L$ that is defined parametrically, like this: $$L(t) = (f(t),g(t),h(t)) \quad\text{for some interval $a \le t \le b$} $$ Then the work integral becomes $$W =\int_{L} P(x, y, z) \, dx+Q(x, y, z) \, dy+R(x, y, z)\, dz$$ $$ = \int_a^b \left(P(f(t),g(t),h(z)) \frac{df}{dt} + Q(f(t),g(t),h(t)) \frac{dg}{dt} + R(f(t),g(t),h(t)) \frac{dh}{dt}\right) dt $$ This has nothing to do with whether the path $L$ is reversible or irreversible: this equation is the definition of work done by moving a particle along the path $L$, through the force field given by $P$, $Q$ and $R$.