Regular Surfaces and Surface Integrals

For $u=(u_x,u_y,u_z)$ and $v=(v_x,v_y,v_z)$ in ${\mathbb{R}}^3$, define $u\times v=(u_y{v}_z-u_z{v}_y,u_z{v}_x-u_x{v}_z,u_x{v}_y-u_y{v}_x)$.

This is the right-handed cross product. The displayed coordinates are those of the standard basis (The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$), and inner products and norms are those of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$.

For all $u,v,w\in {\mathbb{R}}^3$, the cross product is bilinear and alternating, $⟨u\times v,w⟩=\det [uvw]$, and $u\times v$ is orthogonal to both $u$ and $v$.

For $u,v\in {\mathbb{R}}^3$, $\mathrm{\parallel }u\times v{\mathrm{\parallel }}_2^2=\det G(u,v)$, and this value is positive exactly when $u$ and $v$ are linearly independent.

Equivalently, $\mathrm{\parallel }u\times v{\mathrm{\parallel }}_2^2=\mathrm{\parallel }u{\mathrm{\parallel }}_2^2\mathrm{\parallel }v{\mathrm{\parallel }}_2^2-⟨u,v{⟩}^2\mathrm{.}$

A compact Jordan parameter region is a compact Jordan measurable set $D\subseteq {\mathbb{R}}^2$ that is the closure of its nonempty connected interior $D^{\circ }$ (The Riemann integral of a bounded function over a bounded Jordan measurable set, Interior, closure, boundary, limit point, isolated point and dense subset of a metric space). Its boundary has content zero (A bounded set in ${\mathbb{R}}^m$ is Jordan measurable iff its boundary is null, equivalently of content zero).

A regular parametrized surface patch is the image of a $C^1$ map from a compact Jordan parameter region whose parameter cross product is nonzero on the region's interior and for which no interior parameter point shares its image with a distinct point of the whole region. Seam identifications and rank failures may occur only on the boundary. Here $C^1$ is the Euclidean componentwise class of $C^k$ Euclidean maps and diffeomorphisms. More precisely, the parametrization $\phi$ is defined on an open neighbourhood of $D$, ${\phi }_u\times {\phi }_v\mathrm{\ne }0$ on $D^{\circ }$, and no point of $D^{\circ }$ has the same image as a distinct point of $D$. The chosen pair $(D,\phi )$ is part of the patch data.

Let $(D,\phi )$ be a regular parametrized surface patch (Regular parametrized surface patches on compact Jordan parameter regions) and let $p=\phi (u,v)$ with $(u,v)\in D^{\circ }$. At an interior parameter point, the tangent plane is $\mathrm{span}\{{\phi }_u,{\phi }_v\}$.

The two spanning vectors are linearly independent because their cross product is nonzero (The squared cross-product norm is the Gram determinant of two vectors), so this is a two-dimensional linear subspace of ${\mathbb{R}}^3$ (Linear combination of a finite list, and the span $\mathrm{span}(S)$ as the smallest linear subspace containing $S$). The affine tangent plane through $p$ is $p+\mathrm{span}\{{\phi }_u,{\phi }_v\}$.

Let $(D,\phi )$ and $(E,\psi )$ be regular surface patches. A regular reparametrization from $\psi$ to $\phi$ is a $C^1$ diffeomorphism $h$ between neighbourhoods of their compact Jordan parameter regions with $\psi =\phi \circ h$. We additionally require $h[E]=D$ ($C^k$ Euclidean maps and diffeomorphisms).

The derivative $Dh$ is invertible, so its Jacobian determinant is nonzero (The Jacobian determinant of a square-dimensional $C^1$ map is the determinant of its Jacobian matrix). A regular surface reparametrization is orientation-preserving when its parameter Jacobian determinant is positive and orientation-reversing when it is negative. Constancy of the sign on a connected parameter region is proved separately; the terms here apply pointwise whenever needed.

Regular reparametrizations preserve the tangent plane at corresponding interior parameter points.

Precisely, if $\psi =\phi \circ h$ and $(s,t)$ is interior, then $\mathrm{span}\{{\psi }_s,{\psi }_t\}=\mathrm{span}\{{\phi }_u,{\phi }_v\}{\mid }_{h(s,t)}\mathrm{.}$

Every regular reparametrization of a connected parameter region is either orientation-preserving everywhere or orientation-reversing everywhere.

For a regular patch $(D,\phi )$ and an interior parameter point, whose tangent plane is defined by The tangent plane of a regular surface patch, put $E=⟨{\phi }_u,{\phi }_u⟩,F=⟨{\phi }_u,{\phi }_v⟩,G=⟨{\phi }_v,{\phi }_v⟩\mathrm{.}$ The matrix $G_{\phi }=\left(\begin{array}{cc}E& F\\ F& G\end{array}\right)$ is the Gram matrix of the parameter tangents, and the quadratic form $Ed{u}^2+2Fdudv+Gd{v}^2$ is the first fundamental form (The Gram matrix $G(v_0,\dots ,v_{r-1})=(⟨v_i,v_j⟩{)}_{i,j<r}$ and Gram determinant, with empty value $1$).

The surface area density is $J_{\phi }=\sqrt{\det G_{\phi }}$, where $G_{\phi }$ is the Gram matrix of ${\phi }_u$ and ${\phi }_v$. The determinant is nonnegative for all parameter points and positive in the interior because the two tangents are independent there (A Gram determinant is nonnegative and is positive exactly when the vector list is linearly independent), so the nonnegative square root exists and is unique.

For every parameter point, $J_{\phi }=\mathrm{\parallel }{\phi }_u\times {\phi }_v{\mathrm{\parallel }}_2$.

The common value is positive in the interior of a regular patch; it may vanish on the parameter boundary under the admitted seam and endpoint convention.

If $\psi =\phi \circ h$, then ${\psi }_s\times {\psi }_t=(\det Dh)({\phi }_u\times {\phi }_v)\circ h$.

If $\psi =\phi \circ h$, then $J_{\psi }=(J_{\phi }\circ h)\mathrm{\mid }\det Dh\mathrm{\mid }$.

Let $(D,\phi )$ be a regular surface patch and let $q$ be a continuous real-valued function on $\phi [D]$. For a continuous scalar field $q$ on the patch image, ${\int }_{S}qdS:={\int }_D(q\circ \phi )J_{\phi }$, and $\mathrm{Area}(S):={\int }_D{J}_{\phi }$.

Here $S$ denotes the patch with its chosen parametrization. Both integrands are bounded and Riemann integrable on the compact Jordan region $D$ by continuity (A continuous real function on a compact Jordan measurable set is Riemann integrable over that set). Boundary values are included in the parameter integral but may be changed on the content-zero boundary without changing its value.

Surface area and scalar surface integrals are unchanged by every regular reparametrization, regardless of orientation sign.

For a regular patch $(D,\phi )$, the parametrization induces on its interior the unit normal $N_{\phi }=\frac{{\phi }_u\times {\phi }_v}{\mathrm{\parallel }{\phi }_u\times {\phi }_v{\mathrm{\parallel }}_2}\mathrm{.}$ The denominator is positive there by regularity and The surface area density is the norm of the cross product of the parameter tangents, and the vector is orthogonal to the tangent plane (The tangent plane of a regular surface patch). Choosing $N_{\phi }$ rather than $-N_{\phi }$ is an orientation.

For a continuous vector field $F$, the flux in the orientation induced by $\phi$ is ${\int }_D(F\circ \phi )\cdot ({\phi }_u\times {\phi }_v)$. This is the scalar Riemann integral of a continuous function on $D$ (Surface area and scalar surface integrals on a regular patch, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$); replacing the orientation by its negative negates the integrand.

An orientation-preserving reparametrization preserves flux and an orientation-reversing reparametrization negates it.

Changing a bounded scalar or flux parameter integrand only on the content-zero boundary of a compact Jordan parameter region preserves integrability and its integral.

In both directions, one of the two bounded functions is integrable if and only if the other is, and then their integrals agree.

A compatible finite patch presentation is a finite list of regular surface patches whose images cover a set $S$, such that for two distinct patches the preimage of their overlap has content zero in each parameter region. For flux, their induced normals must agree at every point of the overlap that is the image of an interior parameter point of both patches. Stating the requirement on the overlap itself is what gives it content: the induced normal of a patch is defined at the images of its interior parameter points, so a requirement imposed only away from the overlap preimages would constrain nothing and would admit opposite normals on patches whose interiors meet along a curve.

For a compatible finite patch presentation, area, scalar surface integrals, and oriented flux are the sums of the corresponding patch values; pairwise overlap preimages have content zero. The presentation is part of the data, so these sums are single-valued without presuming an unproved independence-of-presentation theorem. The content-zero modification result Content-zero parameter-boundary exceptions do not affect surface integrals ensures that seam, pole and endpoint values on a parameter boundary do not affect the individual summands. The content-zero condition on overlap preimages is a separate restriction on the presentation, and what it buys is that no piece of $S$ carrying positive area is counted twice; each summand is an integral over the whole of its own parameter region and is unaffected by the overlaps.

Let $F:U\subseteq {\mathbb{R}}^3\to \mathbb{R}$ be $C^k$, $k\ge 1$, and let $c$ be a regular value. Every point of a regular level surface in ${\mathbb{R}}^3$ lies in the relative interior of a regular surface patch, and the patch tangent plane is the level-set tangent space.

If $F^{-1}(c)$ is empty, the assertion is vacuous.

If a regular level surface $S=F^{-1}(c)\subseteq {\mathbb{R}}^3$ is compact, then finitely many regular surface patches have relative interiors whose union contains $S$. The empty surface is covered by the empty family.

Let $D\subseteq {\mathbb{R}}^2$ be a compact Jordan parameter region and let $g$ be $C^1$ on a neighbourhood of $D$. For the graph $S=\{(x,y,g(x,y)):(x,y)\in D\}$ and every continuous real-valued function $q$ on $S$, $\mathrm{Area}(S)={\int }_D\sqrt{1+\mathrm{\parallel }\mathrm{\nabla }g{\mathrm{\parallel }}_2^2},{\int }_{S}qdS={\int }_{D}q(x,y,g(x,y))\sqrt{1+\mathrm{\parallel }\mathrm{\nabla }g{\mathrm{\parallel }}_2^2}\mathrm{.}$ For a continuous vector field $F$ on $S$, upward flux is ${\int }_{D}F(x,y,g(x,y))\cdot (-g_x,-g_y,1)\mathrm{.}$ For the graph of $g$ over $D$, the downward flux is the negative of ${\int }_{D}F(x,y,g(x,y))\cdot (-g_x,-g_y,1)$.

Let $a<b$, and let $r:[a,b]\to [0,\mathrm{\infty })$ be $C^1$ on a neighbourhood of $[a,b]$, positive on $(a,b)$, and allowed to vanish only at the endpoints. Put $\phi (s,t)=(s,r(s)\cos t,r(s)\sin t),(s,t)\in [a,b]\times [0,2\pi ]\mathrm{.}$ For every continuous real-valued function $q$ on $\phi ([a,b]\times [0,2\pi ])$, the scalar surface integral is ${\int }_{S}qdS={\int }_0^{2\pi }{\int }_a^{b}q(\phi (s,t))r(s)\sqrt{1+r^{\mathrm{\prime }}(s{)}^2}dsdt,$ where $S$ is the patch with its displayed parametrization.

Assume the hypotheses of Scalar surface integrals on a surface of revolution. The surface obtained by rotating $r$ about the axis has area $2\pi {\int }_a^{b}r(s)\sqrt{1+r^{\mathrm{\prime }}(s{)}^2}ds$.