The equation in the image represents the Divergence Theorem (also known as Gauss's Theorem) in vector calculus, a powerful result that relates the flux of a vector field $\mathbf{F}$ across a closed surface $S$ to the divergence of $\mathbf{F}$ over the volume $D$ enclosed by the surface.

Explanation of the Terms:

1. Left-hand Side (Surface Integral): $\iint_S \mathbf{F} \cdot \mathbf{n} \, d\sigma$

   • $\mathbf{F}$: The vector field.
   • $\mathbf{n}$: The unit normal vector to the surface $S$.
   • $d\sigma$: The infinitesimal area element on $S$.
   • This integral calculates the total flux of $\mathbf{F}$ through the surface $S$.

   It can also be expressed parametrically: $\iint_S \mathbf{F} \cdot \left( \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} \right) \, dv \, du$ where $\mathbf{r}(u, v)$ is the parametrization of the surface.

2. Right-hand Side (Volume Integral): $\iiint_D \nabla \cdot \mathbf{F} \, dV$

   • $\nabla \cdot \mathbf{F}$: The divergence of the vector field $\mathbf{F}$, which measures how much $\mathbf{F}$ "spreads out" from a point.
   • $dV$: The infinitesimal volume element.

   This integral sums up the divergence of $\mathbf{F}$ over the volume $D$ enclosed by $S$.

Significance of the Divergence Theorem:

The theorem states that: $\iint_S \mathbf{F} \cdot \mathbf{n} \, d\sigma = \iiint_D \nabla \cdot \mathbf{F} \, dV$ It connects the "outflow" of $\mathbf{F}$ through the boundary of $D$ with the total divergence of $\mathbf{F}$ inside $D$. This theorem is widely used in physics and engineering, particularly in fluid dynamics, electromagnetism, and heat transfer.

Would you like a detailed example or clarification of any term?

Related Questions:

1. How is the divergence of a vector field calculated?
2. Can you give an example of the Divergence Theorem in application?
3. What is the geometrical interpretation of the flux term?
4. How does the parametrization of a surface relate to the cross-product in the formula?
5. What are other related integral theorems (e.g., Stokes' Theorem)?

Tip:

To apply the Divergence Theorem correctly, ensure that the vector field $\mathbf{F}$ and the surface $S$ are continuously differentiable and oriented consistently.