Surface Integrals (Real Vector Fields)#

TODO: Make more rigorous

Definition: Surface Integral (Real Vector Field)

Let \(f: \mathcal{D}_f \subseteq \mathbb{R}^3 \to \mathbb{R}^3\) be a real vector field and let \(\phi: \mathcal{D}_{\phi} \subseteq \mathbb{R}^2 \to \mathbb{R}^3\) be a real parametric surface which is totally differentiable, except possibly on a null set of the Lebesgue measure.

The surface integral of \(f\) over \(\phi\) is the integral of the dot product between \(f \circ \phi\) and the normal vector of \(\phi\) over \(\mathcal{D}_{\phi}\) (if it exists):

\[\iint_{\mathcal{D}_{\phi}} f(\phi(u,v)) \cdot \partial_u \phi(u,v) \times \partial_v \phi(u,v) \,\mathrm{d}\mathcal{D}_{\phi}\]

Notation

The surface integral of \(f\) over \(\phi\) is denoted as follows:

\[\iint_{\phi} f \cdot \mathrm{d}\boldsymbol{\phi}\]

Sometimes, \(\mathrm{d}\boldsymbol{\phi}\) is denoted by \(\mathrm{d}\boldsymbol{S}\), \(\mathrm{d}\boldsymbol{A}\), etc.

Theorem: Vector Surface Integral \(=\) Scalar Surface Integral

Let \(f: \mathcal{D}_f \subseteq \mathbb{R}^3 \to \mathbb{R}^3\) be a real vector field and let \(\phi: \mathcal{D}_{\phi} \subseteq \mathbb{R}^2 \to \mathbb{R}^3\) be a real parametric surface which is regular, except possibly on a null set of the Lebesgue measure.

If the surface integral of \(f\) over \(\phi\) and the surface integral of the dot product between \(f\) and the unit normal vector of \(\phi\) exist, then they are equal:

\[\int_{\phi} f \, \cdot \mathrm{d}\boldsymbol{\phi} = \int_{\phi} (f \cdot \mathbf{n}) \, \mathrm{d}\phi\]

Proof

TODO