I am trying to understand Green's theorem, but the problem is I don't know what is the definition of the integrals in the theorem. This is the expression that one proves to hold with some assumption on multivariate functions $L(x,y)$ and $M(x,y)$:
$$\oint_{C} (L\, dx + M\, dy) = \iint_{D} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right)\, dx\, dy$$ The question is how one defines: $$\oint_{C} L\, dx$$
I am only looking for a formal definition in terms of Riemann or Lebesgue integral of the last expression.
It's really voodoo notation. Given a vector field $F = L \hat x + M \hat y$, and a curve $\ell(t)$ that traces out $C$ from $t=a$ to $t=b$, the LHS should be understood as
$$\oint_C (L \, dx + M \, dy) \equiv \int_{a}^{b} (F \circ \ell)(t) \cdot \ell'(t) \, dt$$
The notation might make more sense in the context of differential forms, however, if you're familiar with that.