Divergence and curl were defined by derivatives in chapter 2. Here we prove what they measure. Flux adds up the component of a field crossing a surface. Circulation adds up its component along a directed curve. Their local limits and their global integral theorems connect Maxwell's differential equations to experiments with finite regions and loops.
We retain the Euclidean dot and cross products, total differentiability, the chain rule, and equality of continuous mixed partials from chapter 2. All local fields below are at least \(C^1\). A proof for a box or one parameterized patch is given here. A fully general domain theorem needs additional integration machinery; the exact external dependencies are identified where used.
3.1. Adding values over a region
Definition 3.1. Multiple Riemann integrals
For a box \(B=\prod_{j=1}^n[a_j,b_j]\), cut each interval into finitely many subintervals. This forms smaller boxes \(B_\alpha\) of volume \(|B_\alpha|\), the product of their side lengths. Choose a tag \(\xi_\alpha\in B_\alpha\). A tagged sum is \(\sum_\alpha f(\xi_\alpha)|B_\alpha|\). The upper Darboux sum replaces each tagged value by the supremum of the function on that subbox; the lower Darboux sum uses the infimum. These are defined for bounded real-valued functions. Its mesh is the largest diameter of a subbox. The integral \(\int_B f\,dV\) is the number \(I\) for which, for each \(\epsilon>0\), every tagged sum with sufficiently small mesh differs from \(I\) by less than \(\epsilon\). Write \(dA\) in two dimensions and \(dV\) in three. Vector-valued integrals are taken component by component.
Alphabeta: Grid partitions, Darboux sums, Multiple integral.
Theorem 3.2. Continuous functions integrate, and iterated integrals agree
A continuous function on a compact box has this integral. Its iterated one-dimensional integrals in any order equal the multiple integral. Integrals are linear, additive on box subdivisions, and obey \(|\int_B f|\le |B|\sup_B|f|\).
Alphabeta: Continuous integrability, Riemann Fubini, Integral properties.
Proof from uniform continuity
A continuous function on a compact box is uniformly continuous: otherwise pairs \(p_k,q_k\) with \(|p_k-q_k|\to0\) but values separated by a fixed positive amount have a subsequence with \(p_k\to p\) by compactness, hence also \(q_k\to p\), contradicting continuity. It is bounded by the same compactness argument. For sufficiently small mesh, its oscillation in every subbox is less than \(\epsilon/|B|\) if \(|B|>0\). Thus the upper sum minus the lower sum is at most \(\epsilon\). Refinement increases lower sums and decreases upper sums; their supremum and infimum coincide. Tagged sums lie between these bounds, proving integrability. A zero-volume box has zero integral.
For two coordinates, uniform continuity also shows \(x\mapsto\int f(x,y)\,dy\) is continuous, since its value difference is at most the length of the \(y\)-interval times \(\sup_y|f(x,y)-f(x',y)|\). Every inner integral on each strip lies between that strip's subbox lower and upper bounds after subdivision. Integrating these bounds in \(x\) places the iterated integral between the same global lower and upper sums. Their difference tends to zero, proving equality. Repeating this argument handles three or any finite number of coordinates and any order. Linearity, additivity, and the bound hold for the sums and hence for their limits.
Example. On the unit cube, \(\int xyz\,dV=(\int_0^1x\,dx)(\int_0^1y\,dy)(\int_0^1z\,dz)=1/8\). No probabilistic or measure-theoretic interpretation is needed.
The Euclidean compactness lemma proves the compactness and extreme-value facts used in the preceding proof.
A Jordan region is a bounded set whose boundary can be covered by finitely many boxes of arbitrarily small total volume. For a function continuous on its closure, hence bounded and uniformly continuous there, integrate over the region by extending the function as zero outside into an enclosing box. Boxes wholly inside give ordinary oscillation control; only boxes touching the boundary require the bound \(2\sup|f|\) times their total volume. This proves existence and independence of the enclosing box. Graphs of continuous functions on compact boxes have zero volume: divide the base into small boxes, cover each graph segment by a box with height its oscillation plus an arbitrarily small thickness, and use uniform continuity. Smooth balls and the graph regions used below are Jordan regions by this argument applied to their finitely many graph pieces. The exact library definition is the Riemann integral over a Jordan set. This does not assert that every bounded set has a Riemann integral.
Counterexample. The function equal to one on rational numbers and zero on irrational numbers has every upper Darboux sum equal to one and every lower Darboux sum equal to zero on \([0,1]\). Boundedness alone does not ensure a Riemann integral.
3.2. Curves, surfaces, and their orientations
Definition 3.3. Line integral and circulation
A piecewise \(C^1\) path \(\gamma:[a,b]\to U\) is continuous and continuously differentiable on finitely many subintervals, with one-sided endpoint derivatives. Its directed line integral is
\[\int_\gamma F\cdot d\ell=\int_a^b F(\gamma(t))\cdot\gamma'(t)\,dt.\]For a closed path, \(\gamma(a)=\gamma(b)\), write \(\oint_\gamma\) and call it circulation. The scalar line integral \(\int_\gamma f\,ds=\int_a^b f(\gamma(t))|\gamma'(t)|\,dt\) uses unsigned length. Reversing the path changes the sign of the directed integral but not of the scalar length integral. An increasing \(C^1\) reparameterization preserves both, by the one-dimensional substitution rule; a decreasing one reverses directed orientation.
Alphabeta: Line integrals, Gradient theorem.
Example. For \(F=(-y,x,0)\) on \(\gamma(t)=(R\cos t,R\sin t,0)\), \(0\le t\le2\pi\), the integrand is \(R^2\), so circulation is \(2\pi R^2\). The same curve traversed clockwise has circulation \(-2\pi R^2\).
Gradient line integrals. If \(F=\nabla f\), the chain rule and one-dimensional fundamental theorem give \(\int_\gamma F\cdot d\ell=f(\gamma(b))-f(\gamma(a))\), summing over the finitely many smooth pieces. Hence a gradient has zero circulation on every closed path in its domain. This is stronger than merely having zero curl locally.
Definition 3.4. Oriented surface patch and flux
A \(C^1\) parameterized patch is a map \(r:D\subset\mathbb R^2\to\mathbb R^3\). At regular points \(r_u\times r_v\ne0\), it supplies the unit normal \(n=(r_u\times r_v)/|r_u\times r_v|\). The tangent parallelogram has area \(|r_u\times r_v|\,du\,dv\), so define
\[\int_S F\cdot n\,dA:=\int_D F(r(u,v))\cdot(r_u\times r_v)\,du\,dv.\]For an embedded surface, use patches that are one-to-one in their interiors and count each part once. The displayed definition also applies to parameterized surfaces with multiplicity. Choosing the negative normal changes the sign. For a boundary of a solid, outward means pointing from the solid toward its exterior. On a graph \(r(x,y)=(x,y,g(x,y))\), the upward oriented area vector is \((-g_x,-g_y,1)\,dx\,dy\). A rectangular face has its constant unit normal times its ordinary area element.
Alphabeta: Regular surface patches, Normal and flux, Reparameterization.
Worked example for Definition 3.4
Counterexample: A regular parametrization can count the same surface twice
The area factor follows from differentiability: a small parameter rectangle has edge vectors \(r_u\Delta u\) and \(r_v\Delta v\) to first order, and chapter 2 proved their parallelogram area formula. The definition itself is exact. Independence under a general change of surface parameters uses the two-dimensional change-of-variables theorem, identified below; only rectangles, graphs, and explicitly chosen sphere parameters are needed for our local proofs.
Example. The constant field \(F=(0,0,c)\) has flux \(c(b-a)(d-c_0)\) through the upward rectangle \([a,b]\times[c_0,d]\) in the horizontal plane. Its flux through a vertical rectangle is zero. A large tangential field contributes no flux through that surface.
3.3. The local limits, with error estimates
Theorem 3.5. Divergence is outward flux per volume
Let \(F\) be \(C^1\) near \(p\) and let \(B\) be a rectangular box containing \(p\), with positive side lengths, wholly in that neighborhood. Then
\[\int_{\partial B}F\cdot n\,dA=\int_B\nabla\cdot F\,dV.\]Consequently, as the diameter of such boxes tends to zero,
\[\frac{1}{|B|}\int_{\partial B}F\cdot n\,dA\longrightarrow\nabla\cdot F(p).\]The boxes may be arbitrarily thin and may have any fixed orthonormal orientation.
Alphabeta: Local divergence limit.
Proof, including the epsilon–delta remainder
For an axis box \(\prod[a_i,b_i]\), the two faces perpendicular to \(e_1\) contribute
\[\int_{a_2}^{b_2}\int_{a_3}^{b_3}[F_1(b_1,x_2,x_3)-F_1(a_1,x_2,x_3)]\,dx_3\,dx_2 =\int_B\partial_1F_1\,dV,\]by the fundamental theorem and Theorem 3.2. Add the analogous pairs for coordinates two and three. Given \(\epsilon>0\), continuity supplies \(\delta>0\) such that \(|\partial_iF_i(q)-\partial_iF_i(p)|<\epsilon/3\) whenever \(|q-p|<\delta\), for each \(i\). If \(\operatorname{diam}B<\delta\) and \(p\in B\), every point in \(B\) satisfies this bound. Thus
\[\left|\int_{\partial B}F\cdot n\,dA-|B|\nabla\cdot F(p)\right|\le\epsilon |B|.\]This is an explicit bound on the unnormalized remainder, with no assumption on aspect ratios. For a rotated box, transform by a proper orthogonal matrix as in chapter 2. Dot products, lengths, areas, volumes, and divergence are preserved; the same axis proof applies to the transformed field. Reflections also preserve outward flux and divergence when orientation of each face is chosen outward.
Source and sink. For \(F=x-p\), divergence is three and a small box has outward flux \(3|B|\). For its negative, it is \(-3|B|\). A constant nonzero field enters one side and exits another with equal contributions, giving net zero. “Source” means an excess of outward over inward flux at the point; it does not mean that every component points outward. The field \((x,-y,0)\) expands in one direction and contracts in another with net zero.
Theorem 3.6. Curl is circulation per oriented area
Let \(a,b\) be orthogonal unit vectors, \(n=a\times b\), and \(R\) a rectangle in the plane through \(p\), containing \(p\), with positive area \(|R|\), wholly inside the neighborhood on which \(F\) is \(C^1\). Orient its boundary counterclockwise when viewed from the side toward which \(n\) points. Then
\[\oint_{\partial R}F\cdot d\ell=\int_R(\nabla\times F)\cdot n\,dA,\qquad \frac{1}{|R|}\oint_{\partial R}F\cdot d\ell\longrightarrow(\nabla\times F(p))\cdot n.\]Alphabeta: Local curl limit, Antisymmetric derivative.
Proof in an arbitrary orthonormal plane and the remainder bound
Write \(r(s,t)=p+sa+tb\), \(P=F(r)\cdot a\), and \(Q=F(r)\cdot b\). Along the bottom and top edges the combined integral is \(-\int\int\partial_tP\,dt\,ds\); along right and left it is \(\int\int\partial_sQ\,ds\,dt\). This follows by writing all four directed edge integrals and applying the fundamental theorem to their differences. Theorem 3.2 combines them into the area integral of \(\partial_sQ-\partial_tP\). Chain rule gives
\[\partial_sQ-\partial_tP=\sum_{i,j}(b_i a_j-a_i b_j)\partial_jF_i =(\nabla\times F)\cdot(a\times b),\]where the last equality follows by expanding the three cross-product components. This proves the identity for any such plane, independently of any general Stokes theorem.
Given \(\epsilon>0\), choose \(\delta>0\) so that every entry \(\partial_jF_i(q)\) differs from its value at \(p\) by less than \(\epsilon/6\) for \(|q-p|<\delta\). Since \(\sum_i|a_i|,\sum_i|b_i|\le\sqrt3\), the sum of the absolute coefficients \(|b_i a_j-a_i b_j|\) is at most six. Thus, if \(\operatorname{diam}R<\delta\),
\[\left|\oint_{\partial R}F\cdot d\ell-|R|(\nabla\times F(p))\cdot n\right|\le\epsilon |R|.\]Again no bounded aspect-ratio assumption is used. The limit measures the component of curl normal to the plane, not the magnitude of the field.
Rotation and deformation. For \(F=(-y,x,0)\), circulation around any positively oriented horizontal rectangle is twice its area. For \(F=(x,-y,0)\), it is zero despite spatial variation. In a linear field \(F(p+h)=F(p)+Ah\), split \(A=(A+A^{\mathsf T})/2+(A-A^{\mathsf T})/2\). The first matrix is symmetric; the second is antisymmetric and obeys \(((A-A^{\mathsf T})/2)h=\tfrac12(\nabla\times F)\times h\), by component expansion. Curl therefore determines the infinitesimal rigid-rotation part, while the symmetric part can stretch or compress. This does not assert that all nearby field arrows form circular trajectories.
3.4. Global theorems and exactly what has been proved
Theorem 3.7. Stokes on a parameter rectangle
Let \(r:[a,b]\times[c,d]\to U\) be \(C^2\) on a neighborhood of the rectangle, and let \(F\) be \(C^1\) on a neighborhood of its image. Orient the parameter boundary counterclockwise and map it by \(r\). Then
\[\oint_{r(\partial D)}F\cdot d\ell=\int_D(\nabla\times F)(r)\cdot(r_u\times r_v)\,du\,dv.\]If a surface consists of finitely many such patches whose internal edges are shared with opposite directions, their internal line integrals cancel. The remaining boundary therefore satisfies the same formula. Regularity makes this an ordinary surface flux; the parameter identity itself holds even at isolated degenerate points.
Alphabeta: Boundary orientation, Patched surfaces, Stokes theorem.
Worked example for Theorem 3.7
Counterexample: A regular parametrization can count the same surface twice
Proof from the chain rule, without assuming Stokes
Set \(P(u,v)=F(r)\cdot r_u\), \(Q(u,v)=F(r)\cdot r_v\). The same four-edge calculation used above gives boundary integral \(\int_D(\partial_uQ-\partial_vP)\,du\,dv\). Differentiation gives
\[\partial_uQ-\partial_vP= \sum_{i,j}(r_{v,i}r_{u,j}-r_{u,i}r_{v,j})\partial_jF_i(r) +F(r)\cdot(r_{vu}-r_{uv}).\]The last term is zero by the mixed-partial theorem. Expanding the first gives \((\nabla\times F)(r)\cdot(r_u\times r_v)\). Adding finitely many identities proves the patch version, since each shared edge is traversed twice with opposite signs. Subdivision is legitimate because rectangle integrals are additive. No partition of unity has been invoked.
Example. Take \(r(u,v)=(u,v,u^2+v^2)\) on \([0,1]^2\) and \(F=(-y,x,0)\). Curl is \(2e_3\), while \(r_u\times r_v=(-2u,-2v,1)\); the surface integral is two. The directed boundary integral is also two because the field has no vertical component. The surface may bend without changing this circulation.
Theorem 3.8. Divergence theorem
For a bounded solid \(V\) with piecewise \(C^1\) boundary and a \(C^1\) field on a neighborhood of its closure,
\[\int_V\nabla\cdot F\,dV=\int_{\partial V}F\cdot n\,dA,\]with the normal outward. The box case was proved above. Finite unions of boxes follow by cancellation of internal faces. The sphere case is proved next. The general piecewise-smooth statement uses the precise external proof dependency below.
Alphabeta: General divergence theorem.
Full general-domain dependency, with exact proved sources
The definition of a bounded piecewise C¹ domain specifies the boundary class. The complete divergence theorem proof uses boundary graph charts, the fundamental theorem in the normal coordinate, and localization. In particular, its finite ambient partition lemma is a separate proved step, not an implicit assumption in our box argument. General chart independence is supplied by the surface-integral chart-independence lemma.
For domains other than rectangles in parameter space, the complete classical Stokes proof uses the specified elementary Green regions and induced boundary orientations. These external theorems extend the cases proved locally; the words “every smooth surface” are not a conclusion of finite rectangle cancellation alone. Their proofs and prerequisite links are part of the optional general-domain route, not assumed background.
Theorem 3.9. Sphere flux and the ball divergence theorem
For a sphere centered at zero with radius \(R>0\),
\[r(\theta,\phi)=R(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),\quad 0\le\theta\le\pi,\quad0\le\phi\le2\pi,\] \[r_\theta\times r_\phi=R^2\sin\theta\,e_r,\qquad e_r=r/R,\qquad \int_{S_R}F\cdot n\,dA=R^2\int_0^{2\pi}\int_0^\pi F(Re_r)\cdot e_r\sin\theta\,d\theta\,d\phi.\]The sphere has area \(4\pi R^2\). For \(F\in C^1\) on a neighborhood of the closed ball, its flux equals the volume integral of divergence.
Alphabeta: Graph-face flux, Graph-region slicing, Jordan change of variables, Degenerate parameter boundaries.
Proof for balls by slices, including the equator limit
Differentiating the parameterization and taking its cross product gives the stated area vector; integrating \(R^2\sin\theta\) gives \(4\pi R^2\). The parameter seam and poles are sets of zero parameter area and add no flux. To prove the ball theorem, first consider just \(\partial_1F_1\). For \((y,z)\) in the disk \(D_R\), let \(a(y,z)=\sqrt{R^2-y^2-z^2}\). Slicing the ball gives
\[\int_{B_R}\partial_1F_1\,dV =\int_{D_R}[F_1(a(y,z),y,z)-F_1(-a(y,z),y,z)]\,dy\,dz.\]The equality follows from the one-dimensional fundamental theorem on each interval \([-a,a]\) and the same upper/lower-sum squeeze as Theorem 3.2 for the continuous graph boundaries. The two hemispheres are graphs \(x=\pm a(y,z)\). Their outward area-vector first components are respectively \(+1\) and \(-1\) times \(dy\,dz\), since for the upper graph the full vector is \((1,-a_y,-a_z)\), and for the lower it is \((-1,-a_y,-a_z)\). Thus their \(F_1 n_1\) flux equals the displayed disk integral away from the equator.
Use disks \(y^2+z^2\le R^2-\eta^2\) to avoid the graph's divergent derivatives at the equator. The omitted disk annulus has area \(\pi\eta^2\), so the omitted \(F_1n_1\) contribution is at most \(2\pi\eta^2\sup|F_1|\). It tends to zero. The hemisphere graph and angular formula agree on the truncated patches by ordinary two-dimensional substitution; this limited substitution, or full chart independence, is the external theorem identified below. The angular omitted belt has area \(4\pi R\eta\), giving a second direct bound tending to zero for any bounded field. Repeat the slice argument for \(\partial_2F_2\) and \(\partial_3F_3\), and add. Translated balls follow by shifting coordinates.
The only extension beyond box integration used here is slicing a continuous graph region and changing between its two regular surface parameters. The former was justified by the sum squeeze; the latter depends on the proved change-of-variables theorem. This theorem is also needed for general curvilinear volume coordinates below.
Radial example and singularity. If \(F(x)=f(|x|)e_r\), the sphere flux is \(4\pi R^2f(R)\). For \(F=x/|x|^3\), it is \(4\pi\), although divergence is zero at every nonzero point, as chapter 2's product rule shows: \(\nabla\cdot(xr^{-3})=3r^{-3}-3r^{-5}|x|^2=0\). There is no contradiction: this field is undefined at the center, violating the ball theorem's hypotheses. For a shell, inner and outer fluxes cancel with the inner normal directed into the hole. This is the mathematical issue behind a point charge.
3.5. Coordinates that stretch and turn
Definition 3.10. Orthogonal coordinates and scale factors
Let \(x=r(q_1,q_2,q_3)\) be a \(C^2\), one-to-one coordinate map with a differentiable inverse and nonzero determinant. Suppose its three coordinate tangent vectors \(r_{q_i}\) are pairwise orthogonal and positively oriented. Define scale factors \(h_i=|r_{q_i}|>0\), unit vectors \(e_i=r_{q_i}/h_i\), and field components \(F_i=F\cdot e_i\). These are physical components in a changing orthonormal frame, not the coefficients of the unnormalized tangents. Infinitesimal length in coordinate direction \(i\) is \(h_i\,dq_i\); volume factor is \(H=h_1h_2h_3\), and a coordinate face normal to \(e_i\) has area factor \(H/h_i\).
Proposition 3.11. Orthogonal-coordinate formulas
with the other curl components obtained by cyclic permutation, and
\[\Delta f=\frac1H\sum_i\partial_{q_i}\left(\frac H{h_i^2}\partial_{q_i}f\right).\]Proof and its change-of-variables dependency
Chain rule gives \(\partial_{q_i}f=\nabla f\cdot r_{q_i}=h_i\nabla f\cdot e_i\); expansion in the orthonormal frame proves the gradient formula. For divergence, apply the general divergence theorem to a small coordinate cell, using the boundary chart definitions and change-of-variables theorem already identified. Its volume integral is \(\int (\nabla\cdot F)(r)H\,dq\). Its opposing \(i\)-face fluxes are the difference of \(\int (H/h_i)F_i\) at the endpoints of coordinate \(i\). The fundamental theorem combines their sum into \(\int\sum_i\partial_{q_i}((H/h_i)F_i)\,dq\). Divide by the coordinate-cell volume and shrink to the point. Continuity, with precisely the average-versus-center bound used in Theorem 3.5, equates the integrands and proves the formula.
For curl, hold \(q_1\) fixed and use Stokes on a parameter rectangle in \((q_2,q_3)\), which was proved locally. Its tangent area vector is \(h_2h_3e_1\). Its edge integrals use \(F\cdot r_{q_i}=h_iF_i\). The four-edge fundamental-theorem calculation gives the displayed formula after division and shrinking. No general divergence theorem is needed for this curl calculation. Taking divergence of the proved gradient gives the Laplacian formula.
Cylindrical coordinates. Set \(r(\rho,\phi,z)=(\rho\cos\phi,\rho\sin\phi,z)\), with \(\rho>0\) and an angular interval avoiding the seam. Derivatives give \(h_\rho=1,h_\phi=\rho,h_z=1\). Thus
\[\nabla\cdot F=\frac1\rho\partial_\rho(\rho F_\rho)+\frac1\rho\partial_\phi F_\phi+\partial_zF_z,\] \[\Delta f=\frac1\rho\partial_\rho(\rho\partial_\rho f)+\frac1{\rho^2}\partial_\phi^2f+\partial_z^2f.\]For \(F=\rho e_\rho=(x,y,0)\), divergence is two; merely differentiating \(F_\rho\) would incorrectly give one. The expanding circumference supplies the additional term.
Spherical coordinates. Set \(r(r,\theta,\phi)=r(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\), with \(r>0,0<\theta<\pi\) and an angular interval avoiding the seam. Here \(h_r=1,h_\theta=r,h_\phi=r\sin\theta\), in positive order \((r,\theta,\phi)\). Therefore
\[\nabla\cdot F=\frac1{r^2}\partial_r(r^2F_r)+\frac1{r\sin\theta}\partial_\theta(\sin\theta F_\theta)+\frac1{r\sin\theta}\partial_\phi F_\phi,\] \[\Delta f=\frac1{r^2}\partial_r(r^2\partial_rf)+\frac1{r^2\sin\theta}\partial_\theta(\sin\theta\partial_\theta f)+\frac1{r^2\sin^2\theta}\partial_\phi^2f.\]For a radial field, divergence is \(r^{-2}(r^2F_r)'\). This proves the inverse-square example directly. Poles, the origin, and angular seams are coordinate failures, not necessarily singularities of a smooth Cartesian field. At those places, use another chart or Cartesian coordinates.
General nonorthogonal coordinates: divergence, curl, and Laplacian
Let \(r\) be a regular \(C^2\) coordinate map. Choose its coordinate order so that \(\det Dr>0\), giving positive orientation. Define \(g_{ij}=r_{q_i}\cdot r_{q_j}\), the metric matrix, and let \((g^{ij})\) be its inverse. Let \(J=\det Dr=\sqrt{\det(g_{ij})}>0\). Write \(F=\sum_i V^i r_{q_i}\), defining the coefficients \(V^i\) in the coordinate tangent basis. Then
\[\nabla\cdot F=\frac1J\sum_i\partial_{q_i}(JV^i),\qquad \Delta f=\frac1J\sum_{i,j}\partial_{q_i}(Jg^{ij}\partial_{q_j}f).\]Define the lowered components, also called covariant components, by \(F_j=F\cdot r_{q_j}=\sum_k g_{jk}V^k\). They describe the field's pairing with each coordinate tangent and generally differ from the coefficients \(V^j\). Write \(\nabla\times F=\sum_i C^i r_{q_i}\). Then
\[C^1=\frac{\partial_{q_2}F_3-\partial_{q_3}F_2}{J},\qquad C^2=\frac{\partial_{q_3}F_1-\partial_{q_1}F_3}{J},\qquad C^3=\frac{\partial_{q_1}F_2-\partial_{q_2}F_1}{J}.\]Positive chart orientation is required for these curl signs. An orientation-reversing chart would instead use the signed determinant in the denominator.
The volume identity follows from determinant multiplicativity: \(g=(Dr)^{\mathsf T}Dr\), so \(\det g=(\det Dr)^2\). To see the divergence proof, the oriented area vectors of opposing coordinate faces are cross products of the other two tangent vectors; dotting them with \(F\) selects \(JV^i\), with the appropriate face sign. The same coordinate-cell argument and the same stated change-of-variables dependency prove the first formula. Chain rule gives \(\partial_{q_j}f=\sum_i V^i g_{ij}\) for \(F=\nabla f\); solving with the inverse matrix gives \(V^i=\sum_jg^{ij}\partial_{q_j}f\), proving the second. For the curl formula, hold \(q_1\) fixed and apply the locally proved parameter-rectangle Stokes theorem. The edge integrands are \(F_j\,dq_j\), so its four-edge calculation gives the area integral of \(\partial_{q_2}F_3-\partial_{q_3}F_2\). The surface integrand is \((\nabla\times F)\cdot(r_{q_2}\times r_{q_3})=JC^1\), by the scalar triple product: only the \(r_{q_1}\) coefficient contributes. Divide by the parameter rectangle's area and shrink; continuity equates the integrands. Cyclic permutation proves the other two formulas. This curl argument uses our local Stokes proof and requires no volume substitution. In orthogonal coordinates, \(g_{ij}=h_i^2\delta_{ij}\). Writing \(F_i^{\mathrm{phys}}=F\cdot e_i\) for the physical components of Definition 3.10, the lowered components satisfy \(F_i=h_iF_i^{\mathrm{phys}}\). Thus \(V^i=F_i/h_i^2=F_i^{\mathrm{phys}}/h_i\), and the formulas reduce to Proposition 3.11. No general relativity or differential forms are assumed here.
Nonorthogonal example. For the shear map \(r(u,v,w)=(u+v,v,w)\), the metric matrix is \(\begin{pmatrix}1&1&0\\1&2&0\\0&0&1\end{pmatrix}\), its inverse is \(\begin{pmatrix}2&-1&0\\-1&1&0\\0&0&1\end{pmatrix}\), and \(J=1\). Therefore \(\Delta f=2f_{uu}-2f_{uv}+f_{vv}+f_{ww}\). For the physical function \(f(x,y,z)=x^2\), its coordinate expression is \((u+v)^2\); this formula gives \(4-4+2=2\), agreeing with its Cartesian Laplacian. Treating the skew coordinate vectors as orthonormal would give the wrong answer. For the physical swirl \(F=(-y,x,0)\), the lowered components in this chart are \(F_1=-v,F_2=u,F_3=0\). The curl formula gives \(C^1=C^2=0,C^3=1-(-1)=2\), hence \(\nabla\times F=2r_w=(0,0,2)\), agreeing with the Cartesian computation.
3.6. Local information and holes
Proposition 3.12. Zero curl can coexist with nonzero global circulation
On the domain \(U=\{(x,y,z):x^2+y^2>0\}\), the smooth field
\[F=\left(-\frac y{x^2+y^2},\frac x{x^2+y^2},0\right)\]has zero curl, but its circulation around \((\cos t,\sin t,0)\), \(0\le t\le2\pi\), is \(2\pi\). Thus it cannot equal the gradient of a single-valued scalar function on all of \(U\).
The only potentially nonzero curl component is \(\partial_x(x/(x^2+y^2))-\partial_y(-y/(x^2+y^2))\). These derivatives are both \((y^2-x^2)/(x^2+y^2)^2\), so their difference is zero. On the unit circle, \(F=(-\sin t,\cos t,0)=\gamma'(t)\), giving integral \(2\pi\). Gradient circulation vanishes by Definition 3.3's consequence, proving the last assertion. Stokes on the usual spanning disk is inapplicable because the disk intersects the removed axis.
A field line is a differentiable path obeying \(\gamma'(s)=F(\gamma(s))\), or, when only its unparameterized direction matters, with derivative a positive scalar multiple of the field. Zero divergence says that infinitesimal net flux vanishes. It does not say field lines close. The constant field \(e_1\) has zero divergence and straight, unbounded field lines \((s,y_0,z_0)\). The radial inverse-square field has zero divergence away from its removed singularity and radial lines that reach the edge of that domain. Closure requires additional global information.
These distinctions matter when reading Maxwell's equations. Their local statements concern derivatives at a point. Integral statements concern a field smooth on the entire region between a loop or surface and its interior. Missing points, boundaries, and singular sources must be included in the hypotheses before using an integral theorem.
Worked examples
Each example checks the hypotheses and the calculation for the linked statements. The links in the statements return here.
Worked example 1. A box integral from tagged sums
Applies to: Definition 3.1, Theorem 3.2.
On \([0,1]^2\), take the continuous function \(f(x,y)=x+y\). Right-endpoint tags on an \(N\)-by-\(N\) grid give \(N^{-3}\sum_{i,j=1}^N(i+j)=1+1/N\). Changing the tags changes a sum by at most \(2/N\), the maximum oscillation of \(f\) on a subbox, so every fine-grid tagged sum tends to 1. More generally the upper and lower sums on any partition differ by at most \(\sqrt2\) times its mesh, because \(|f(p)-f(q)|\le\sqrt2|p-q|\). Both iterated integrals are 1: \(\int_0^1(x+1/2)dx=1\) and \(\int_0^1(1/2+y)dy=1\). The bound \(|\int f|\le1\cdot\sup f=2\), linearity from the separate integrals of \(x,y\), and additivity by splitting the square at \(x=1/2\) are all directly verified.
Worked example 2. A line integral and the local curl limit
Applies to: Definition 3.3, Theorem 3.6.
For \(F=(-y,x,0)\), the closed unit circle \(\gamma(t)=(\cos t,\sin t,0)\) has \(F(\gamma)\cdot\gamma'=1\), so its circulation is \(2\pi\). Its unsigned length integral with \(f=1\) is also \(2\pi\), since \(|\gamma'|=1\); reversing the path negates only the directed integral. The increasing reparametrization \(t=2s\), \(0\le s\le\pi\), gives integrand 2 and the same result.
On the counterclockwise rectangle \([-a,a]\times[-b,b]\) at \(z=0\), the horizontal edges each contribute \(2ab\) and the vertical edges each contribute \(2ab\). Thus circulation is \(8ab=2|R|\), so circulation per oriented area is exactly \((\operatorname{curl}F)\cdot e_z=2\) for every positive \(a,b\), including arbitrarily thin shrinking rectangles. This checks the local curl theorem’s actual normalization.
Worked example 3. Flux and Stokes on a tilted rectangle
Applies to: Definition 3.4, Theorem 3.7.
Use \(r(u,v)=(u,v,u)\) on \([0,1]^2\), smooth on a neighborhood. Its area vector is \(r_u\times r_v=(-1,0,1)\ne0\), so the unit normal is \((-1,0,1)/\sqrt2\) and the area is \(\sqrt2\). For \(F=(-y/2,x/2,0)\), \(\operatorname{curl}F=e_z\), giving curl flux \(\int_0^1\int_0^1 1\,du,dv=1\). The pullback along the boundary is \(-v\,du/2+u\,dv/2\); its four counterclockwise edge integrals are \(0,1/2,1/2,0\), summing to 1. Thus both sides of Stokes agree with the chosen orientation and the correct area vector. Reversing the patch orientation negates both signs.
Worked example 4. Flux through a box and a sphere
Applies to: Theorem 3.5, Theorem 3.8, Theorem 3.9.
For \(F=(x,y,z)\), \(\operatorname{div}F=3\). A box with side lengths \(a,b,c\) has net flux \(abc\) from each pair of opposite faces, hence \(3abc\). The flux divided by volume is exactly 3, giving the shrinking-box limit about any point. The same field is \(C^1\) on a neighborhood of every closed ball. On the radius-\(R\) sphere its outward component is \(R\), so spherical coordinates give flux \(R^3\int_0^{2\pi}\int_0^\pi\sin\theta\,d\theta d\phi=4\pi R^3\). The volume integral is \(3(4\pi R^3/3)\), the same result. Integrating just the area density \(R^2\sin\theta\) gives \(4\pi R^2\), checking the sphere parametrization as well as both divergence-theorem examples.
Worked example 5. Scale factors in a cylindrical chart
Applies to: Definition 3.10, Proposition 3.11.
Take \(r(\rho,\phi,z)=(\rho\cos\phi,\rho\sin\phi,z)\) with \(\rho>0\) and \(\phi\) in an interval shorter than \(2\pi\). The coordinate tangents are orthogonal, positively oriented, with lengths \(1,\rho,1\); the chart has a smooth inverse on its image. Thus \(H=\rho\). For \(f=\rho^2\), \(\nabla f=2\rho e_\rho\), and \(\Delta f=\rho^{-1}\partial_\rho(\rho\cdot2\rho)=4\), as for the Cartesian function \(x^2+y^2\). The field \(F=\rho e_\rho\) has divergence \(\rho^{-1}\partial_\rho(\rho^2)=2\), while \(G=\rho e_\phi=(-y,x,0)\) has curl component \((\operatorname{curl}G)_z=\rho^{-1}\partial_\rho(\rho^2)=2\), other components zero. The physical component \(G_\phi=\rho\) differs from its coefficient 1 on the unnormalized tangent \(r_\phi\). These calculations check each coordinate formula and the distinction emphasized in the definition.
Worked example 6. A loop encircling the missing axis
Applies to: Proposition 3.12.
For \(F=(-y/(x^2+y^2),x/(x^2+y^2),0)=e_\phi/\rho\), restrict to \(\rho>0\). The cylindrical formula gives \((\operatorname{curl}F)_z=\rho^{-1}\partial_\rho(\rho/\rho)=0\), with the other components zero. On the radius-2 circle \(\gamma(t)=(2\cos t,2\sin t,0)\), \(F(\gamma)=(-\sin t,\cos t,0)/2\) and \(\gamma'=2(-\sin t,\cos t,0)\), so circulation is \(2\pi\), independent of radius. A single-valued potential would give \(\oint d(p\circ\gamma)=0\), contradiction. Stokes cannot be applied to the spanning disk here because the field is undefined on its axis, making the missing domain hypothesis explicit.
Counterexamples and limits of the hypotheses
Each construction below refutes the explicitly stated stronger claim. It does not refute the related theorem with its original hypotheses.
Counterexample 1. A regular parametrization can count the same surface twice
Related statements: Definition 3.4, Theorem 3.7.
Claim refuted. Regularity alone guarantees that a parameter integral is the once-counted flux through the geometric image.
Proof.
Let \(r(\theta,z)=(\cos\theta,\sin\theta,z)\), \((\theta,z)\in[0,4\pi]\times[0,1]\), and \(F(x,y,z)=(x,y,0)\). This map is smooth on a neighborhood, with \(r_\theta\times r_z=(\cos\theta,\sin\theta,0)\), of length 1 everywhere. It is therefore regular and outward-oriented on the unit cylinder. The parameter flux is \(\int_0^1\int_0^{4\pi}1\,d\theta dz=4\pi\). But the same cylindrical side surface is covered once by \([0,2\pi]\times[0,1]\), giving flux \(2\pi\). Almost every surface point has two preimages in the first parameter rectangle. The definition explicitly allows multiplicity; one-to-one interiors or a suitable once-counting patch choice are required to interpret it as ordinary geometric flux. There is no failure of the parameter identity in Stokes’ theorem.
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