Line Integrals#

Theorem: Vector Line Integrals and Equivalence

Let \(\mathbf{F}: \mathcal{D}_{\mathbf{F}} \subseteq \mathbb{R}^n \to \mathbb{R}^n\) be a real vector field and let \(\gamma: \mathcal{D}_{\gamma} \subset \mathbb{R} \to \mathbb{R}^n\) and \(\varphi: \mathcal{D}_{\varphi} \subset \mathbb{R} \to \mathbb{R}^n\) be vector-valued.

If \(\gamma\) and \(\varphi\) are continuously differentiable and are equivalent up to a continuously differentiable reparametrization, then the line integrals of \(\mathbf{F}\) along \(\gamma\) and \(\varphi\)

are equal whenever \(\gamma\) and \(\varphi\) have the same orientation

\[ \int_{\gamma} \mathbf{F} \cdot \mathop{\mathrm{d}\mathbf{s}} = \int_{\varphi} \mathbf{F} \cdot \mathop{\mathrm{d}\mathbf{s}} \]

are opposite whenever \(\gamma\) and \(\varphi\) have opposite orientations

\[ \int_{\gamma} \mathbf{F} \cdot \mathop{\mathrm{d}\mathbf{s}} = -\int_{\varphi} \mathbf{F} \cdot \mathop{\mathrm{d}\mathbf{s}} \]

Proof

TODO

Green's Theorem

Let \(\gamma: [a; b] \subseteq \mathbb{R} \to \mathbb{R}^2\) be simple closed parametric curve which is positively oriented and piecewise continuously differentiable on \((a;b)\), let \(R \subseteq \mathbb{R}^2\) be the region bounded by \(\gamma\) and let \(\boldsymbol{v}: \mathcal{D}_{\boldsymbol{v}} \subset \mathbb{R}^2 \to \mathbb{R}^2\) be a real vector field with component functions \(v_1, v_2\).

If \(\boldsymbol{v}\) is continuously differentiable on \(R\), then the line integral of \(\boldsymbol{v}\) over \(\gamma\) is given by the following double integral:

\[ \int_{\gamma} \boldsymbol{v} \cdot \mathop{\mathrm{d}v} = \iint_R \frac{\partial v_2}{\partial x} - \frac{\partial v_1}{\partial y} \mathop{\mathrm{d}R} \]

Proof

TODO

Surface Integrals#

Definition: Vector Surface Integral

Let \(\mathbf{F}: \mathcal{D}_{\mathbf{F}} \subseteq \mathbb{R}^3 \to \mathbb{R}^3\) be a real vector field and let \(S: \mathcal{D}_S \subseteq \mathbb{R}^2 \to \mathbb{R}^3\) be a differentiable parametric surface such that \(S(\mathcal{D}_S) \subseteq \mathcal{D}_{\mathbf{F}}\).

The (vector) surface integral of \(\mathbf{F}\) over \(S\) is the double integral of the dot product \(\mathbf{F} \circ S\) with the surface normal of \(S\):

\[ \iint_{\mathcal{D}_S} (\mathbf{F}\circ S) \cdot \mathbf{N} \mathop{\mathrm{d}\mathcal{D}_S} \]

Notation

\[ \iint_S \mathbf{F} \cdot \mathrm{d}\mathbf{S} \]

If \(S(\mathcal{D}_S)\) is a closed surface, then a circle can be put through the two integral signs.

Theorem: Vector Surface Integral to Scalar Surface Integral

Let \(\mathbf{F}: \mathcal{D}_{\mathbf{F}} \subseteq \mathbb{R}^3 \to \mathbb{R}^3\) be a real vector field and let \(S: \mathcal{D}_S \subseteq \mathbb{R}^2 \to \mathbb{R}^3\) be a differentiable parametric surface such that \(S(\mathcal{D}_S) \subseteq \mathcal{D}_{\mathbf{F}}\).

The surface integral of \(\mathbf{F}\) over \(S\) is equal to the surface integral of the dot product between \(\mathbf{F}\) and the unit surface normal of \(S\).

\[ \iint_S \mathbf{F} \cdot \mathop{\mathrm{d}\mathbf{S}} = \iint_S \mathbf{F} \cdot \mathbf{n} \mathop{\mathrm{d}S} \]

Proof

TODO

Theorem: Vector Surface Integrals and Equivalence

Let \(\mathbf{F}: \mathcal{D}_{\mathbf{F}} \subseteq \mathbb{R}^3 \to \mathbb{R}^3\) be a real vector field and let \(\phi: \mathcal{D}_{\phi} \subseteq \mathbb{R}^2 \to \mathbb{R}^3\) and \(\psi: \mathcal{D}_{\psi} \subseteq \mathbb{R}^2 \to \mathbb{R}^n\) be parametric surfaces.

If \(\phi\) and \(\psi\) are continuously differentiable and are equivalent up to a continuously differentiable reparametrization, then the surface integrals of \(\mathbf{F}\) over \(\phi\) and \(\psi\) are:

equal when \(\phi\) and \(\psi\) have the same orientation

\[ \iint_{\phi} \mathbf{F} \cdot \mathop{\mathrm{d}\phi} = \iint_{\psi} \mathbf{F} \cdot \mathop{\mathrm{d}\psi} \]

opposite when \(\phi\) and \(\psi\) have opposite orientations

\[ \iint_{\phi} \mathbf{F} \cdot \mathop{\mathrm{d}\phi} = -\iint_{\psi} \mathbf{F} \cdot \mathop{\mathrm{d}\psi} \]

Proof

TODO

The Divergence Theorem (Gauss's Theorem, Ostrogradsky's Theorem)

Let \(V\) be a compact subset of the Euclidean space \(\mathbb{R}^n\), let \(s: \mathcal{D}_s \subseteq \mathbb{R}^2 \to \mathbb{R}^n\) be a parametric surface whose image is the boundary of \(V\) and let \(\mathbf{F}: \mathcal{D}_{\mathbf{F}} \subseteq \mathbb{R}^n \to \mathbb{R}^n\) be a real vector field.

Proof

TODO