Retrieving "Volume Integral" from the archives

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1. Centroid

   Linked via "volume integrals"

   Dimensional Extensions and Curvature
   The notion of the centroid extends naturally to higher dimensions. For a three-dimensional volume $V$, the centroid $\mathbf{C}$ is calculated using volume integrals, provided the density $\rho(\mathbf{r})$ is uniform:
   $$\mathbf{C} = \frac{1}{V} \iiint_V \mathbf{r} \, dV$$

2. Divergence (vector Calculus)

   Linked via "volume integral"

   The Divergence Theorem (Gauss's Theorem)
   The most significant application relating the local divergence to the global flux is the Divergence Theorem, also known as Gauss's Theorem. This theorem relates the flux of a vector field out of a closed surface $S$ to the volume integral of the divergence of the field within the volume $V$ enclosed by $S$:
   $$\iintS \mathbf{F} \cdot d\mathbf{S} = \iiintV (\nabla \cdot \mathbf{F}) \, dV$$

3. Integral

   Linked via "volume 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 |

4. Strain Energy Release

   Linked via "volume integral"

   Theoretical Basis and Definition
   The total strain energy stored within an elastic, isotropic, and linearly deformable body subjected to stress $\sigma{ij}$ and strain $\epsilon{ij}$ is mathematically defined by the volume integral over the region $V$:
   $$U{SR} = \frac{1}{2} \int{V} \sigma{ij} \epsilon{ij} \, dV$$