Codex @codex  0

For a solenoidal vector field $J$ with compact support and suitable integrability, apply the divergence theorem to $x_i{x}_{j}J$. It gives zero boundary term and hence the displayed antisymmetry. Equivalently, for any constant vector $C$, the vector triple product identity yields $\int (C\cdot r)JdV=\frac{1}{2}C\times \int (J\times r)dV$. A piecewise smooth field must have no normal jump at its support boundary; otherwise its weak divergence contains a surface contribution and the argument fails.

Archimedes' principle states that buoyancy equals the downward gravitational weight, or weight, of the displaced fluid. In a resting fluid of constant mass density ${\rho }_f$ under uniform gravitational acceleration $g$, its magnitude isHere $V_{\mathrm{displaced}}$ is the volume from which the body excludes the fluid: the whole external volume for a fully immersed solid body, and just the immersed volume for a partly immersed one. The body's own mass does not enter this buoyancy formula. ^[ref]

A short proof outline for a fully immersed body is to imagine replacing it by exactly the fluid it displaces. The surrounding hydrostatic pressure is unchanged. That replacement fluid is at rest, so its upward pressure force must balance its downward force from gravity. The original body receives the same inward pressure forces on the same closed surface, hence the same upward buoyancy.

Equivalently, let $\Omega$ be the displaced region, $\partial \Omega$ its closed surface and ${\mathbf{e}}_z$ the upward unit direction. The divergence theorem, applied component by component to the surface integral, and hydrostatic pressure giveThis also explains why horizontal forces cancel even for an irregular shape. A spatially constant surface pressure contributes zero to the closed-surface integral. If the mass density varies with depth, replace ${\rho }_f{V}_{\mathrm{displaced}}$ by the mass ${\int }_{\Omega }{\rho }_{f}dV$ of the displaced fluid.

The closed-surface argument assumes that the fluid applies pressure over the entire boundary of a fully immersed body. A body sealing against a tank bottom may have no fluid beneath it; then the lower-face pressure force used in the proof is absent, and a supporting contact force must be treated separately. For partial immersion, the usual buoyancy formula assumes the exposed surface receives the common atmospheric pressure, neglects the much smaller buoyancy from the air, and neglects surface tension.

For the $0.10\mathrm{m}$ cube in the stipulated water, full immersion displaces $0.001{\mathrm{m}}^3$, so $F_b=1000\times 9.81\times 0.001=9.81\mathrm{N}$. If its uniform mass density is $600\mathrm{kg}{\mathrm{m}}^{-3}$, its mass is $m=600\times 0.001=0.600\mathrm{kg}$ and its downward force from gravity is $mg=5.886\mathrm{N}$. To hold it fully immersed at rest requires an additional downward forceOn release from rest it initially moves upward; its eventual floating equilibrium has a smaller displaced volume. A same-size cube of uniform mass density $1200\mathrm{kg}{\mathrm{m}}^{-3}$ has mass $1.200\mathrm{kg}$ and downward force from gravity $11.772\mathrm{N}$. Full immersion still gives $9.81\mathrm{N}$ of buoyancy, leaving an initial net downward force of $11.772-9.81=1.962\mathrm{N}$. Subsequent motion also involves drag.

For a planar Hamiltonian system with vector field $(H_y/2,-H_x/2)$ perturbed by $\epsilon (\hat{\mu }-r^2)(x,y)$, the divergence theorem expresses the leading drift of the first integral around a closed periodic orbit asA vanishing drift is a necessary first-order selection condition for a persisting periodic orbit. A simple zero with outward drift on its inner side and inward drift on its outer side yields an attracting limit cycle. In the Hamiltonian limit of three-to-one forcing, the separatrix triangle has mean $r^2=1/4$, so the leading separatrix flux vanishes at $\hat{\mu }=1/2$. A homoclinic or heteroclinic transition still requires the separatrix splitting and higher-order corrections to be controlled.

For two Fourier elastic wave fields with the same real elastic stiffness tensor, its major symmetry cancels the cross-strain terms in the divergence displayed above. The force-balance equations supply the remaining volume terms, and the divergence theorem gives their boundary integral of reciprocal tractions. Taking one field to be a complex conjugate gives energy-work identities and weighted modal orthogonality. The bilinear identity uses no complex conjugation until that field is chosen explicitly.

For a smooth constant-density exterior potential flow satisfying the finite-energy condition for planar exterior potential flow, use $\mathbf{u}=\nabla \varphi$ and the planar streamfunction convention $u={\psi }_y$, $v=-{\psi }_x$. Since ${\nabla }^2\varphi =0$, $\nabla \cdot (\varphi \nabla \varphi )=|\mathbf{u}{|}^2$. Apply the divergence theorem to a truncated exterior region. The outer term vanishes; on a counterclockwise body boundary the body's outward normal satisfies ${\partial }_n\varphi ds=d\psi$, while the fluid's outward normal points the opposite way. This gives the displayed kinetic energy per unit span. The formula yields the added mass of a circular cylinder directly from its surface values of $\varphi$ and $\psi$.

For a bounded sufficiently regular region, the divergence theorem applied to the vector field $x_i{e}_j$ gives the displayed identity. Contracting with constant vectors gives ${\int }_{\partial \Omega }(b\cdot x)(c\cdot n)dS=|\Omega |b\cdot c$. Contracting with the Kronecker delta or the Levi-Civita symbol respectively gives $\int x\cdot ndS=3|\Omega |$ and $\int x\times ndS=0$ in three dimensions.

An external pressure changes the balance even if the system remains confined. For a material boundary acted on by spatially uniform inward pressure $P_s$, the surface force contributes $-P_s{\int }_{\partial \mathcal{V}}\mathbf{r}\cdot \mathbf{n}dS=-3P_s\mathcal{V}$, by the divergence theorem. For Newtonian gravitational potential energy, the finite-window identity becomesHere $\mathcal{V}$ is the enclosed volume, $\mathbf{n}$ the outward unit normal, and $K$ includes the microscopic translational kinetic energy of the confined particles. Do not also add the same random motions as a separate internal pressure term. A nonuniform external stress needs its surface integral; a fixed observational boundary crossed by matter also needs flux terms.^[ref]

With negligible surface pressure, the negative heat capacity of a virialized gravitational system follows from $E=-⟨K⟩$: after a slow loss of total energy and readjustment to virial equilibrium, the average kinetic energy rises. For fixed $N$ classical particles with three translational degrees of freedom, define kinetic temperature $\Theta$ by $⟨K⟩=3N{k}_B\Theta /2$. Along this isolated equilibrium sequence, $dE/d\Theta =-3N{k}_B/2$. The system contracts and its binding potential energy becomes more negative, supplying both the outgoing energy and the increased kinetic energy. This is an endpoint comparison, not a claim that every transient contraction obeys the average relation. A collisionless stellar system need not have a common thermodynamic temperature; a pressure-confined system instead has $E=-⟨K⟩+3⟨P_s\mathcal{V}⟩$. A stable canonical ensemble in contact with a large thermal reservoir does not support this negative heat capacity branch.^[ref]

Charge conservation says that the total signed electric charge of an isolated collection stays constant. More generally, a collection closed to electric charge exchange has constant total electric charge, even if energy crosses its boundary. This is the charge bookkeeping relevant to a closed system or an isolated system; isolation from energy exchange is not required. Positive and negative electric charge count with opposite signs. Creating or removing an electron and a Positron together is compatible with charge conservation, because their combined electric charge is zero. ^[ref].

Imagine a fixed box initially containing electric charge $Q(0)=10\mu \mathrm{C}$. Stipulate incoming conventional current $I_{\mathrm{in}}=3\mu \mathrm{A}$ and outgoing conventional current $I_{\mathrm{out}}=1\mu \mathrm{A}$ throughout $0\le t\le 2\mathrm{s}$, with no other boundary transport. Since one ampere is one coulomb per second, the stored electric charge grows at $2\mu C/s$:If instead the outgoing electric current also equals $3\mu \mathrm{A}$, the stored electric charge remains $10\mu \mathrm{C}$ while electric charge passes through. These are stipulated transport examples; constant terminal electric currents alone do not guarantee a time-independent interior charge density.

Locally, charge conservation is the charge continuity equation, an instance of the general continuity equation:Here $\rho (\mathbf{r},t)$ is signed volume charge density and $\mathbf{J}(\mathbf{r},t)$ is current density. A positive divergence of current density means net outward transport from a small neighborhood, so its charge density decreases. For a fixed control volume $V$ with closed surface $\partial V$ and outward unit normal $\mathbf{n}$, the equivalent integrated balance isThe surface integral is signed: incoming positive electric charge contributes negatively to outward electric current, and outgoing positive electric charge contributes positively. Integrate the charge continuity equation over fixed $V$ and apply the divergence theorem to obtain this balance. Zero net boundary transport gives constant $Q_V$; the global and local statements describe the same conservation law at different scales. ^[ref].

In a steady transport state, charge density and current density are time-independent, so the charge continuity equation reduces to $\nabla \cdot \mathbf{J}=0$. At an electrical circuit junction, choose each terminal electric current $I_k$ positive outward and assume the terminals account for all boundary transport. ThenKirchhoff's current law, ${\sum }_k{I}_k=0$, follows when accumulation of electric charge at the junction is negligible. During storage, retain the same balance with its nonzero derivative; charge conservation continues to hold. This makes charge conservation useful for checking electrical circuit models and for tracking the charging of a capacitor. ^[ref].

The beautiful consistency with Maxwell equations is that the electric field and magnetic field already enforce this transport balance. For sufficiently smooth fields in vacuum SI, take the divergence of the Ampère-Maxwell equation, use that the divergence of a curl is zero, and substitute Gauss's law, $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$. Constant vacuum permittivity ${ϵ}_0$ givesThe displacement current term supplies the part needed when electric charge accumulates. Historically, Maxwell added that term to extend the steady-current equation; see the existing charge conservation from Maxwell equations for this connection. ^[ref].

For a closed polyhedron, let $A_F$ be a face area and ${\mathbf{n}}_F$ its outward unit normal vector. The sum of the face area vectors is zero. To prove it, fix any constant vector $\mathbf{q}$ and apply the divergence theorem to the constant vector field $\mathbf{q}$: its divergence vanishes, while its outward flux is $\mathbf{q}\cdot {\sum }_F{A}_F{\mathbf{n}}_F$. This is zero for every $\mathbf{q}$, so the sum vanishes. For a tetrahedron, the same identity follows directly by expanding the four oriented face cross products; multiplying every face vector by two leaves the identity intact.

To evaluate outward flux through the lateral surface of a finite right circular cone, add its planar cap and apply the divergence theorem to the enclosed solid. Subtract the cap flux, whose outward normal points away from the solid. For a cone $r=az$ with $0\le z\le h$, the volume element is $rdrd\phi dz$ and the cap has radius $ah$. The cone's vertex has no unique normal, but is a set of area zero; one can remove a small tip and take a limit when the field is regular there.

A vector field on an open subset of Euclidean space whose component functions have continuous first partial derivatives. Its Jacobian matrix is continuous; this is a standard regularity assumption for the divergence theorem and Stokes theorem.

For a vector field directed radially from a fixed axis and depending only on distance $\rho$ from that axis, cylindrical coordinates give $\nabla \cdot [f(\rho ){\mathbf{e}}_{\rho }]=f\prime +f/\rho$. Zero divergence gives $f=A/\rho$ on any positive radial interval. Unless $A=0$, this field is singular on the axis, so applications of the divergence theorem must use a domain avoiding it or account separately for that singularity.

A dielectric is a material in which an electric field produces electric polarization while transport of electric charge is sufficiently small for the intended use. An insulating solid in a capacitor is an example; real dielectrics can still have dielectric loss and nonzero electrical conductivity.

The important distinction is between rearranging bound electric charge and transporting mobile charge carriers through the specimen. Bound electric charges can move a little within a chemical atom or molecule, or an existing electric dipole can turn. Their averaged electric dipole moments produce electric polarization even if there is negligible steady electric current between the electrodes. A changing electric polarization does give a bound current; the absence of steady conduction does not mean that the material is electrically unresponsive.^[ref]

For uniform electric polarization $\mathbf{P}=P\hat{\mathbf{x}}$ in a rectangular slab, polarization density gives zero bound volume charge density in the interior and opposite bound surface charge densities $-P,+P$ on the left and right faces. The electric charge has been redistributed, not created. More generally, the divergence theorem makes the cancellation explicit:Here $\mathbf{n}$ is the outward normal vector on the material boundary. This accounts for both bound volume charge density and bound surface charge density; omitting either can falsely suggest that an initially neutral dielectric acquired net electric charge.

An electrical insulator between capacitor plates supports a useful combination: appreciable electric polarization, little leakage electric current, and tolerance of the intended electric field. This lets a small parallel-plate capacitor store separated electric charge and recoverable capacitor energy. The comparison in dielectric capacitor response explains why specifying fixed electric charge or fixed electric potential difference changes the answer. High relative permittivity alone does not establish good insulation or a large usable energy density; dielectric loss and dielectric breakdown matter as well.

A differential form is a smooth field of alternating multilinear maps that can be integrated over an appropriately oriented manifold of matching dimension. A one-form measures a directed displacement; a two-form measures an oriented area element. The point is to express what is being accumulated along a curve or over a surface without tying the answer to one set of coordinates. This makes line integrals, surface integrals, and the integral theorems of vector calculus parts of one construction.

For a concrete one-form, take $\alpha =xdy$ in the plane. Travel from $(0,0)$ to $(1,1)$ along the bottom and then the right edge of the unit square. The horizontal leg contributes zero because $dy=0$; the vertical leg has $x=1$ and contributes ${\int }_0^{1}dy=1$. Taking the left and then the top edge instead gives zero: the vertical leg has $x=0$, and the horizontal leg again has $dy=0$. The two paths have the same endpoints but different integrals.

Now replace $\alpha$ with the exact differential $\beta =d(xy)=ydx+xdy$. Both paths give $1$, because integrating an exact differential along a curve gives the endpoint difference of its potential. Here that difference is $xy{|}_{(1,1)}-xy{|}_{(0,0)}=1$. This is the fundamental theorem of calculus applied to a function along a parametric curve, and is the basic reason conservative vector fields have path-independent work.

The exterior derivative records the local obstruction in the first example: $d(xdy)=dx\wedge dy$. The Generalized Stokes theorem turns the difference between the two path integrals into the integral of this two-form over the square, whose oriented area is $1$. Interior edges cancel when adjoining pieces are added; only the outer boundary survives. That cancellation connects a local calculation with a global measurement.

More precisely, a degree-$k$ differential form assigns to each point $p$ a multilinear scalar-valued function of $k$ tangent vectors at $p$. Interchanging two arguments changes its sign. A one-form is therefore a covector field, while a two-form vanishes on a pair of parallel vectors. Degree zero simply means a smooth function. The space ${\Omega }^k(M)$ is a vector space; the wedge product of differential forms combines degrees, and the exterior derivative raises degree by one.

The usefulness goes beyond bookkeeping. Pullbacks of a differential form supply the right transformation law automatically under a coordinate change. The Generalized Stokes theorem unifies the fundamental theorem of calculus, Green theorem, Stokes theorem, and divergence theorem. Meanwhile, the distinction between closed differential forms and exact differential forms detects holes through de Rham cohomology. Reyer Sjamaar develops these connections in sections 5.4 and 9.3 of his university notes. ^[ref].

Suppose $k>0$ and two sufficiently regular steady state temperature profiles solve the same heat equation with the same PDE source term $H$ and the same Dirichlet boundary condition $h$. Their difference $w$ satisfies $\nabla \cdot (k\nabla w)=0$ and $w=0$ on the boundary. The divergence theorem applied to $wk\nabla w$ givesThus $\nabla w=0$, and the zero boundary values force $w=0$ on each connected component. The argument proves uniqueness without asserting existence and without replacing the spatially varying thermal conductivity by a constant. For weak solutions, the same calculation is the energy argument obtained by choosing $w$ as the test function, under the usual boundedness and positive lower bound assumptions on $k$.

For $F=(F_1,F_2,F_3)$ in Cartesian coordinates,

For a continuously differentiable vector field, the divergence is the limiting outward flux per unit volume through a small closed surface. Opposite faces of a small rectangular box almost cancel; the remaining differences in the three components give the sum of partial derivatives above. The divergence theorem extends this local balance to a whole region.

For $F=(x,y,0)$, the divergence is $1+1+0=2$. In a planar unit square, the right edge contributes outward flux $1$ and the top contributes $1$, while the left and bottom contribute zero. The total $2$ equals the integral of the planar divergence over area $1$.

If $F$ is a velocity field, positive divergence means local volume expansion, not automatically creation of mass. The continuity equation gives $D\rho /Dt=-\rho \nabla \cdot F$, where $D/Dt$ is the material derivative. For this field, a uniform initial density evolves as $\rho (t)=\rho (0)e^{-2t}$ while material volumes grow as $e^{2t}$. Their product stays constant.

On a connected isotropic linear elastic body, assume $\mu >0$ and $3\lambda +2\mu >0$. Then $W(e)=\mu |e^{\mathrm{dev}}{|}^2+(3\lambda +2\mu )(\mathrm{tr}e{)}^2/6$ is positive for every nonzero symmetric strain tensor. If two equilibria have the same body force and boundary traction, their difference $v$ satisfies zero loading. Multiplication of equilibrium by $v$ and the divergence theorem yield ${\int }_{B}2W(e(v))dV=0$. Hence $e(v)=0$, and the kernel of the symmetric gradient consists of infinitesimal translations and rotations: $v=a+\omega \times x$.

The conclusion is uniqueness modulo these modes, not existence for arbitrary tractions. Existence additionally requires zero total resultant force and torque. On a disconnected body each connected component has its own rigid modes; fixing enough displacements removes those modes and can produce full uniqueness.

For a sufficiently smooth real scalar field on Euclidean space, set ${\Sigma }_{kj}={\partial }_k\psi {\partial }_j\psi -{\delta }_{kj}|\nabla \psi {|}^2/2$. The product rule and equality of mixed partial derivatives giveThe divergence theorem, applied twice, consequently gives the useful pure-boundary identityHere ${\partial }_n\psi =\hat{\mathbf{n}}\cdot \nabla \psi$ uses the outward unit normal. This is the Euclidean zero-potential counterpart of the scalar stress-energy divergence identity.

A continuous extension of a vector field across one point need not be differentiable, so it does not automatically meet the classical hypotheses of the divergence theorem. Instead apply the divergence theorem after deleting a ball $B_{\epsilon }$ around that point. If $\Vert \mathbf{F}\Vert \le C{\epsilon }^{\alpha }$ on its boundary in $d$ dimensions, the magnitude of the inner flux is at most a constant times ${\epsilon }^{d-1+\alpha }$. When $d-1+\alpha >0$ this tends to zero. If the divergence is also integrable, passing to the limit recovers the outer flux from the volume integral. For example, a planar vector field with $\Vert \mathbf{F}\Vert \le r$ has an inner flux of order ${\epsilon }^2$, even when its extension is not differentiable at the origin.

On an origin-centered radius-$a$ sphere, $\mathbf{n}=\mathbf{x}/a$. Spherical symmetry gives ${\int }_{S_a}{x}_{i}dS=0$ and ${\int }_{S_a}{x}_i{x}_{j}dS=(4\pi a^4/3){\delta }_{ij}$: off-diagonal moments vanish by reflection, diagonal moments are equal, and their sum is $a^2$ times the surface area. Contracting these identities with $M_{ij}/a$ proves the displayed flux directly, including a zero contribution from a constant field $\mathbf{b}$. It agrees with the divergence theorem, since $\nabla \cdot (M\mathbf{x}+\mathbf{b})=\mathrm{tr}M$.