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1. Divergence (vector Calculus)

   Linked via "surface integrals"

   $$\iintS \mathbf{F} \cdot d\mathbf{S} = \iiintV (\nabla \cdot \mathbf{F}) \, dV$$
   This theorem is vital for converting surface integrals, which can be geometrically complex, into volume integrals, which are often simpler due to the linearity of the divergence operator.
   Key Properties of the Divergence Theorem

2. Integral

   Linked via "surface integrals"

   Integrals in Vector Calculus
   In higher dimensions, the integral generalizes to line integrals, surface integrals, and volume integrals, forming the core of Vector Calculus. These generalizations are essential for describing physical phenomena that vary across space.
   | Integral Type | Dimension | Differential Element | Physical Analogy |

3. Vector Fields

   Linked via "surface integrals"

   Integration Over Fields
   The utility of a vector field is often realized through line integrals (circulation) or surface integrals (flux).
   Flux

4. Vector Fields

   Linked via "surface integral"

   Flux
   The flux of a vector field $\mathbf{F}$ across an oriented surface $S$ is the surface integral:
   $$ \PhiS = \iintS \mathbf{F} \cdot d\mathbf{S} $$
   If $\mathbf{F}$ represents an incompressible flow, the total flux leaving any closed surface must be zero, as dictated by the Divergence Theorem (Gauss's Theorem), which relates the volume integral of the divergence to the surface flux.