List of basic formulas related to line integrals:

  1. Linear integral of the first kind:
    [
    \int_C f(x, y) \, ds = \int_a^bf(x(t), y(t)) \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt
    ]
    where ( C ) is a curve defined parametrically, ( (x(t), y(t)) ) are its parameters, and ( ds ) is a length element.
  2. Curvilinear integral of the second kind:
    [
    \int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(x(t), y(t)) \cdot \frac{d\mathbf {r}}{dt} \, dt
    ]
    where ( \mathbf{F} = (P, Q) ) is a vector field, and ( d\mathbf{r} = (dx, dy) ).
  3. Green’s formula (connects the curvilinear integral and the double integral):
    [
    \oint_C (P \, dx + Q \, dy) = \iint_D \left( \frac{\partial Q}{\partial x} – \frac{\partial P}{\partial y} \right) \, dA
    ]
    where ( D ) is the region bounded by the curve ( C ).
  4. Linear integral in polar coordinates:
    [
    \int_C f(r, \theta) \, ds = \int_a^bf(r(\theta), \theta) \, r(\theta) \, d\theta
    ]
    where ( ds = r(\theta) \, d\theta ).

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