List of basic formulas related to vector field flow:
- 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. - 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. - 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. - 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.