List of basic formulas related to vector field flow:

  1. Definition of the flux of a vector field:
    [
    \Phi = \iint_S \mathbf{F} \cdot d\mathbf{S}
    ]
    where (\Phi) is the flux of the vector field (\mathbf{F}) through the surface (S), and (d\mathbf{S}) is the area vector perpendicular to the surface.
  2. Flow through a closed surface (Gauss’s theorem):
    [
    \Phi = \iiint_V \nabla \cdot \mathbf{F} \, dV
    ]
    where (V) is the volume enclosed by the surface (S), and (\nabla \cdot \mathbf{F}) is the divergence of the vector field.
  3. Flow through a curved surface:
    [
    \Phi = \int_C \mathbf{F} \cdot d\mathbf{r}
    ]
    where (C) is the curve along which the vector field is integrated.
  4. The flow of a vector field in two-dimensional space:
    [
    \Phi = \int_a^b \int_c^d F(x, y) \, dy \, dx
    ]
    where (F(x, y)) are the components of the vector field in two-dimensional space.

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