Gauss' Divergence Theorem & Stokes' Theorem

Gauss' Divergence Theorem

Gauss' Divergence Theorem relates a flux integral over a closed surface to a volume integral within the region enclosed by the surface.

Mathematically, it can be expressed as: \[ \int\int\int_V (\nabla \cdot \mathbf{F}) \, dV = \int\int_S \mathbf{F} \cdot \mathbf{n} \, dS \]

Stokes' Theorem

Stokes' Theorem relates a circulation (line integral) around a closed curve to a surface integral over a surface bounded by that curve.

Mathematically, it can be expressed as: \[ \int_C \mathbf{F} \cdot d\mathbf{r} = \int\int_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS \]

Example: Gauss' Divergence Theorem

Consider a vector field \(\mathbf{F}(x, y, z) = \langle x^2, y^2, z^2 \rangle\). Apply Gauss' Divergence Theorem to calculate the flux of \(\mathbf{F}\) across the closed surface of a sphere of radius \(R\).

Example: Stokes' Theorem

Let \(\mathbf{F}(x, y, z) = \langle y, -x, z^2 \rangle\). Apply Stokes' Theorem to find the circulation of \(\mathbf{F}\) around the circle defined by \(x^2 + y^2 = 4\) in the plane \(z = 1\).

Real-world Applications

Both theorems have wide-ranging applications in physics and engineering. Gauss' Divergence Theorem is often used in fluid dynamics to analyze the flow of fluids, while Stokes' Theorem finds applications in electromagnetism, such as calculating magnetic fields around conductors.