List of basic formulas related to surface integrals:

  1. Surface integral of a scalar field:
    [
    \iint_S f(x, y, z) \, dS
    ]
    where ( S ) is the surface, ( f ) is the scalar field, ( dS ) is the element of surface area.
  2. Surface integral of a vector field:
    [
    \iint_S \mathbf{F} \cdot d\mathbf{S}
    ]
    where ( \mathbf{F} ) is a vector field, ( d\mathbf{S} ) is a vector element of the surface area directed along the normal to the surface.
  3. Surface parameterization:
    If the surface ( S ) is specified parametrically as ( \mathbf{r}(u, v) ), the area element can be expressed as:
    [
    dS = \left| \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} \right| \, du \, dv
    ]
  4. The formula for computing the surface integral of a vector field via parameterization is:
    [
    \iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u, v)) \cdot \left ( \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} \right) \, du \, dv
    ]
    where ( D ) — parameterization domain.
  5. Stokes’ theorem:
    [
    \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}
    ]
    where ( C ) is the boundary of the surface ( S ).
  6. Gauss’s theorem (or divergence theorem):
    [
    \iiint_V (\nabla \cdot \mathbf{F}) \, dV = \iint_S \mathbf{F} \cdot d\mathbf{S}
    ]
    where ( V ) is the volume bounded by the surface ( S ).

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