Gauss's Law in Differential Form

The integral form of Gauss's law is a magnificent instrument with a strange blind spot: it talks about bags. Draw a closed surface, and the law reports the total charge inside — but "inside a bag" is a statement about a whole region, and physics since Faraday has been driving toward statements about points. What does the field do right here? What does the charge right here force it to do? This article shrinks the bag until it answers: divide the flux out of a tiny box by the box's volume, send the volume to zero, and the ratio becomes the divergence — the flux-per-volume machine built back in the vector-calculus tier. Gauss's law then collapses into three symbols and a source:

\nabla\cdot\mathbf{E} = \rho/\varepsilon_0

This is the same physics as the integral form — the two are exactly interconvertible through the divergence theorem — but it is a profoundly different kind of statement. It holds point by point, ties the field's local geometry to the charge density at that very spot, and asks nothing about surfaces, symmetry, or the rest of the universe. It is the form in which Gauss's law joins Maxwell's equations, the form field theory quantizes, and the form every numerical solver enforces cell by cell. It is also where two of your E0 investments pay their dividends at once: the divergence theorem powers the conversion, and the three-dimensional delta function absorbs the point charge's infinity without flinching. The chapter closes here with electrostatics' first local law. One more local statement — the vanishing curl, next article — and the Helmholtz theorem will certify that you know everything a static field can do.

Key takeaways

The differential form of Gauss's law: $\nabla\cdot\mathbf{E}=\rho/\varepsilon_0$ — the divergence of the field at a point equals the charge density at that same point, over $\varepsilon_0$ .
It follows from the integral form by the arbitrary-volume argument: if $\int_V(\nabla\cdot\mathbf{E}-\rho/\varepsilon_0)\,\mathrm{d}\tau=0$ for every volume, the integrand vanishes identically.
The two forms are one law: divergence theorem left, charge-density integral right — convert freely, choose by problem.
Direct verification from Coulomb's law runs through the E0 delta identity $\nabla\cdot(\hat{\boldsymbol{\eta}}/\eta^2)=4\pi\delta^3(\boldsymbol{\eta})$ : the divergence integral sifts the density at the field point.
In empty space $\nabla\cdot\mathbf{E}=0$ : field lines are conserved locally — even the visibly "spreading" $1/r^2$ field has zero divergence away from its source.
Point charges appear as delta functions, $\rho = q\,\delta^3(\mathbf{r}-\mathbf{r}_0)$ ; surface charges make $\mathbf{E}$ jump and are handled by boundary conditions, not by differentiating across the sheet.
Divergence alone does not determine the field: statics needs the curl equation too — the Helmholtz theorem's two questionnaires.
Key idea: shrink the Gaussian bag to a point and Gauss's law becomes local — the field's out-spreading per unit volume, at each point, is the charge density sitting there.

The core insight

Gauss's law in differential form, $\nabla\cdot\mathbf{E}=\rho/\varepsilon_0$ , states that the divergence of the electrostatic field at each point equals the local charge density over $\varepsilon_0$ — the local, point-by-point expression of flux–charge accounting, exactly equivalent to the integral form via the divergence theorem.

Where this lives today

The differential form is the version of Gauss's law that machines enforce. Every finite-difference and finite-element electrostatics solver imposes $\nabla\cdot\mathbf{E}=\rho/\varepsilon_0$ (usually via the potential, as Poisson's equation — the very next tier) on each of millions of mesh cells; the Yee algorithm underlying FDTD electromagnetic simulation is celebrated precisely because its staggered grid conserves the discrete divergence exactly, so charge never leaks numerically. Plasma "divergence cleaning" and constrained-transport schemes in astrophysical MHD codes exist to police the magnetic twin $\nabla\cdot\mathbf{B}=0$ — a whole algorithmic literature devoted to keeping a divergence honest. In fundamental physics, the local law's status keeps rising: in quantum electrodynamics it survives as the Gauss constraint, an operator equation selecting the physical states of the theory, and lattice gauge theorists enforce it configuration by configuration when they compute the proton's mass from first principles. And gravitational-lensing maps of dark matter are, at bottom, divergence X-rays: measure the field (light deflection), differentiate, and read off the invisible density — worked example 2 at galaxy-cluster scale.

The physical picture

The integral law audits bags; the differential law audits points. Recall what divergence is — the E0 balloon test: enclose a point in a tiny box, measure the net flux escaping, divide by the box's volume, shrink. The limit is $\nabla\cdot\mathbf{E}$ , the field's local faucet-density. Now run Gauss's law on the shrinking box: the flux out is always $Q_{\text{enc}}/\varepsilon_0$ , and $Q_{\text{enc}}/V \to \rho$ as the box closes on the point. Flux-per-volume on the left, charge-per-volume-over- $\varepsilon_0$ on the right — the differential law is nothing but Gauss's law read through a microscope. Where there is positive charge, the field is locally a faucet; where negative, a drain; where vacuum, sealed pipework, however violently the field may vary.

Figure

Shrink the bag to a point — flux per volume becomes divergence, and Gauss's law goes local

Hold two calibrating pictures. First, the point charge: its field visibly fans outward everywhere, yet $\nabla\cdot\mathbf{E}=0$ at every point except the charge itself — the E0 balloon test on $\hat{\mathbf{r}}/r^2$ found the weakening magnitude exactly cancelling the geometric spreading. All of the divergence is concentrated in one infinitely dense spike at the origin, and the E0 delta-function article gave that spike its legal name: $\nabla\cdot(\hat{\mathbf{r}}/r^2) = 4\pi\delta^3(\mathbf{r})$ . The differential Gauss law is that identity, dressed for electrostatics: charge density is the delta spike, scaled by $q/\varepsilon_0$ . Second, the uniformly charged sphere of the previous article. Inside, the field ramps linearly — and a linear radial field has constant divergence, matching the constant $\rho$ filling the ball. Outside, the $1/r^2$ tail has zero divergence, matching the vacuum. The kink at the surface is precisely where $\rho$ switches off. The differential law reads that profile like an X-ray: wherever the field's local out-spreading is nonzero, charge is present, in exact proportion — worked example 1 runs the numbers.

The law, localized

Start from the integral form, which the previous article established for arbitrary surfaces, and write its right side as a volume integral of the density:

\oint_{\partial V}\mathbf{E}\cdot\mathrm{d}\mathbf{a} = \frac{1}{\varepsilon_0}\int_V \rho\,\mathrm{d}\tau \tag{1}

where: $\partial V$ is the boundary of an arbitrary volume $V$ . The left side is flux through a closed surface — exactly the shape the divergence theorem converts. Applying it (the E0 theorem: flux through the skin equals the volume integral of the divergence) turns (1) into a statement comparing two volume integrals over the same arbitrary region, and the arbitrariness is then lethal to any disagreement between the integrands. The result:

\nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0} \tag{2}

where: everything is evaluated at a single point — the divergence of the total field here, the charge density here. No surface, no symmetry, no rest-of-universe. Equation (2) is Maxwell's first equation in the form he wrote it, and the first derivation below walks the conversion with the care the "arbitrary volume" step deserves. For a point charge at $\mathbf{r}_0$ , the density is the delta spike $\rho = q\,\delta^3(\mathbf{r}-\mathbf{r}_0)$ , and (2) reduces to the E0 identity — the second derivation closes that loop from the Coulomb side. The two forms are strictly interconvertible, and fluency means switching without ceremony:

you have	reach for	because
symmetric source, want $E$ at finite distance	integral form + Gaussian surface	the solver mode of the last article
field formula, want the charge that made it	differential form: compute $\nabla\cdot\mathbf{E}$	an X-ray for sources — worked example 2
point-by-point law for theory building	differential form	locality; the form PDEs, relativity and QFT use
conservation statement over a region	integral form	global bookkeeping, no derivatives needed
surface charge layer	neither naively — boundary conditions	$\rho$ is a delta sheet; $\mathbf{E}$ jumps by $\sigma/\varepsilon_0$

Derivation AFrom bags to points — the arbitrary-volume argument

We convert the integral law into the differential law with full logical care. The two working parts: the divergence theorem (E0), which rewrites the flux as a volume integral, and the observation — subtler than it looks — that an equation holding for every volume forces the integrands to agree point by point. The second part is where hands get waved in lesser treatments; we wave nothing. The payoff is not just equation (2) but the template: every "integral law ⇒ local law" conversion in physics (mass conservation, energy conservation, all of Maxwell) runs on exactly this argument.

Step 1. Rewrite the flux side. For any volume $V$ with boundary $\partial V$ , the divergence theorem gives

\oint_{\partial V}\mathbf{E}\cdot\mathrm{d}\mathbf{a} = \int_V (\nabla\cdot\mathbf{E})\,\mathrm{d}\tau

Why is this legal here? The theorem requires $\mathbf{E}$ continuously differentiable in $V$ . For smooth charge distributions it is; for point and surface charges, see Step 4 — the delta-function formalism is exactly the license extension.

Step 2. Set the two volume integrals against each other. Substituting into (1) and gathering:

\int_V \left(\nabla\cdot\mathbf{E} - \frac{\rho}{\varepsilon_0}\right)\mathrm{d}\tau = 0 \qquad\text{for every volume } V

Step 3. Kill the integrand. Suppose the bracket were positive at some point $P$ . By continuity it stays positive in a small ball around $P$ ; choose $V$ to be that ball, and the integral over it would be strictly positive — contradiction. Same argument, mirrored, for negative. Hence the bracket vanishes everywhere:

\nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0}

Why can't the integrand oscillate its way to zero? Over some particular volume it could — positive patches cancelling negative ones. The force of the argument is the word "every": the volumes themselves hunt down any nonzero patch. One equation per volume, infinitely many volumes, is enough to pin a function.

Step 4. Patrol the edge cases. At a point charge, $\mathbf{E}$ is not differentiable and Step 1's hypothesis fails classically — but the delta-function calculus of E0 repairs it: with $\nabla\cdot(\hat{\boldsymbol{\eta}}/\eta^2)=4\pi\delta^3(\boldsymbol{\eta})$ admitted as a distributional identity, the theorem and the conclusion survive with $\rho$ containing delta spikes. At a surface charge, $\rho$ is a delta sheet and $\mathbf{E}$ jumps; the differential law there is read as the boundary condition $E^{\perp}_{\text{above}}-E^{\perp}_{\text{below}} = \sigma/\varepsilon_0$ — the pillbox-limit statement, formalized in the next tier.

Go deeper · Derivation B · 2 minThe hard direction — computing the divergence of the Coulomb field directly

Derivation BThe hard direction — computing the divergence of the Coulomb field directly

The first derivation assumed the integral law and localized it. This one earns equation (2) from the ground up: start from the superposition integral for the field of an arbitrary charge distribution — Coulomb's law in continuum form — and compute its divergence honestly. Classically the computation looks doomed ( $\nabla\cdot$ of an integral whose integrand is singular), but the E0 delta identity converts the doom into a one-line sift. This is the calculation that makes the E0 investment in $\delta^3$ pay off in full, and it is the standard route by which field theory verifies that a proposed field law has the right sources. Track carefully which coordinates each operator touches — the $\mathbf{r}$ vs $\mathbf{r}'$ hygiene of the continuous-distributions article is load-bearing here.

Step 1. Write the field of an arbitrary static distribution (the continuous-distributions master formula), with $\boldsymbol{\eta} = \mathbf{r}-\mathbf{r}'$ :

\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\int \rho(\mathbf{r}')\,\frac{\hat{\boldsymbol{\eta}}}{\eta^2}\,\mathrm{d}\tau'

Step 2. Take the divergence at the field point, and slide $\nabla$ under the integral:

\nabla\cdot\mathbf{E} = \frac{1}{4\pi\varepsilon_0}\int \rho(\mathbf{r}')\;\nabla\cdot\!\left(\frac{\hat{\boldsymbol{\eta}}}{\eta^2}\right)\mathrm{d}\tau'

Why may $\nabla$ pass $\rho(\mathbf{r}')$ ? Because $\nabla$ differentiates with respect to $\mathbf{r}$ (the field point) while the integration and the density live on $\mathbf{r}'$ (the source point) — different variables entirely. This is exactly the step that dies if the prime hygiene has been sloppy.

Step 3. Deploy the delta identity. The divergence of the inverse-square unit field with respect to the field point is the E0 result:

\nabla\cdot\!\left(\frac{\hat{\boldsymbol{\eta}}}{\eta^2}\right) = 4\pi\,\delta^3(\boldsymbol{\eta}) = 4\pi\,\delta^3(\mathbf{r}-\mathbf{r}')

Why a delta and not zero? Away from $\boldsymbol{\eta}=0$ the balloon test gives exactly zero (spreading cancels weakening); the entire divergence hides in the singular point, and the E0 pincer/regularization proofs measured its weight: $4\pi$ . This one identity is the whole difficulty of the calculation, and it was paid for in advance.

Step 4. Sift. The delta collapses the integral at $\mathbf{r}'=\mathbf{r}$ :

\nabla\cdot\mathbf{E} = \frac{1}{4\pi\varepsilon_0}\int\rho(\mathbf{r}')\,4\pi\,\delta^3(\mathbf{r}-\mathbf{r}')\,\mathrm{d}\tau' = \frac{\rho(\mathbf{r})}{\varepsilon_0}

The field's divergence at $\mathbf{r}$ knows only the charge density at $\mathbf{r}$ : every other source element contributed a divergence-free $1/\eta^2$ fan. Locality was hiding inside Coulomb's law all along; the delta function is the scalpel that exposes it. Dimensional check: $[\rho]/\varepsilon_0 = \mathrm{(C/m^3)/(C^2N^{-1}m^{-2})} = \mathrm{(V/m)/m}$ ✓ — the correct units for the divergence of a field.

Figure

The X-ray view of the charged sphere — divergence reads the source straight off the field profile

Limiting cases

case	result	meaning
vacuum ( $\rho=0$ )	$\nabla\cdot\mathbf{E}=0$	lines locally conserved — sealed pipework, however curved
point charge	$\rho = q\,\delta^3(\mathbf{r}-\mathbf{r}_0)$	all divergence in one spike of weight $q/\varepsilon_0$
uniform ball interior	$\nabla\cdot\mathbf{E} = \rho/\varepsilon_0$ , constant	the linear ramp's constant out-spreading
surface charge	$\mathbf{E}$ jumps by $\sigma/\varepsilon_0$ across the sheet	differential law becomes a boundary condition
integrate (2) over any $V$	integral form recovered	divergence theorem run backwards — no information lost
add $\nabla\times\mathbf{E}=\mathbf{0}$ (next article)	field fully determined (with decay)	Helmholtz: divergence + curl + boundary = everything

Common trapThe field is spreading out here, so its divergence must be positive — right?

Wrong in the most instructive way. "Spreading arrows" is a picture of the field's direction geometry; divergence weighs direction and magnitude, and the two can conspire. The point charge is the canonical trap: its field fans outward everywhere, yet $\nabla\cdot\mathbf{E}=0$ at every vacuum point — the $1/r^2$ weakening exactly refunds the geometric spreading, as the E0 balloon test showed. All the divergence sits in the delta spike at the charge. The converse trap also bites: a field of parallel arrows with growing magnitude, $\mathbf{E} = kx\,\hat{\mathbf{x}}$ , spreads nowhere visually yet has divergence $k$ — a uniform charge density must be filling that region. Trust the operator, not the sketch: compute $\nabla\cdot\mathbf{E}$ (in the right coordinates — the spherical formula $\frac{1}{r^2}\partial_r(r^2E_r)$ for radial fields), and let the arrows keep their poetry.

Worked examples

1 · X-raying the charged sphere (confidence builder)

The previous article found, for a uniform ball of charge ( $Q = 80$ nC, $R = 10$ cm): $\mathbf{E} = \dfrac{kQr}{R^3}\hat{\mathbf{r}}$ inside and $\mathbf{E} = \dfrac{kQ}{r^2}\hat{\mathbf{r}}$ outside. Verify that the differential law reads back the correct source, with numbers.

Step 1. The tool: for a radial field, $\nabla\cdot\mathbf{E} = \dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 E_r\right)$ — the spherical divergence formula from the curvilinear-coordinates article.

Step 2. Inside: $r^2E_r = kQr^3/R^3$ , so $\nabla\cdot\mathbf{E} = \dfrac{1}{r^2}\cdot\dfrac{3kQr^2}{R^3} = \dfrac{3kQ}{R^3}$ — constant, as a uniform density demands. Numerically: $3(8.99\times10^9)(80\times10^{-9})/(10^{-3}) = 2.16\times10^6\ \mathrm{V/m^2}$ .

Step 3. Check against $\rho/\varepsilon_0$ : the ball's density is $\rho = 3Q/4\pi R^3 = 1.91\times10^{-5}\ \mathrm{C/m^3}$ , so $\rho/\varepsilon_0 = (1.91\times10^{-5})/(8.854\times10^{-12}) = 2.16\times10^6\ \mathrm{V/m^2}$ . ✓ (The two agree identically: $3kQ/R^3 = 3Q/4\pi\varepsilon_0R^3 = \rho/\varepsilon_0$ since $k = 1/4\pi\varepsilon_0$ .)

Step 4. Outside: $r^2E_r = kQ$ , a constant, so $\nabla\cdot\mathbf{E} = 0$ — vacuum, correctly reported, even though the field there visibly fans outward. The X-ray sees charge only where charge is; the kink in $E(r)$ at the surface is the divergence switching off.

Answer: inside, $\nabla\cdot\mathbf{E} = 2.16\times10^6\ \mathrm{V/m^2} = \rho/\varepsilon_0$ exactly; outside, zero — the differential law reads the sphere's anatomy straight off the field profile.

2 · From field to source (exam level)

Measurements in a spherical region find a radial field $E_r = a r^2$ with $a = 500\ \mathrm{V/m^3}$ (and no angular components). Find the charge density producing it, and the total charge within $r = 10$ cm — by two independent routes.

Step 1. Apply the X-ray: $\nabla\cdot\mathbf{E} = \dfrac{1}{r^2}\dfrac{\partial}{\partial r}(r^2\cdot ar^2) = \dfrac{1}{r^2}\cdot4ar^3 = 4ar$ . So

\rho(r) = 4\,\varepsilon_0\,a\,r

— a density growing linearly from zero at the centre. At $r = 10$ cm: $\rho = 4(8.854\times10^{-12})(500)(0.10) = 1.77\times10^{-9}\ \mathrm{C/m^3}$ .

Step 2. Route one for the total charge — integrate the density over the ball:

Q = \int_0^{0.1}\!\!4\varepsilon_0 a r\cdot4\pi r^2\,\mathrm{d}r = 16\pi\varepsilon_0 a\cdot\frac{r^4}{4}\Big|_0^{0.1} = 4\pi\varepsilon_0 a\,(0.1)^4 = 5.56\times10^{-12}\ \mathrm{C}

Step 3. Route two — the integral law on the bounding sphere: $\Phi = E_r(0.1)\cdot4\pi(0.1)^2 = (500)(0.01)(4\pi)(0.01) = 0.628$ V·m, and $Q = \varepsilon_0\Phi = (8.854\times10^{-12})(0.628) = 5.56\times10^{-12}$ C. ✓

Step 4. Read the agreement: the two routes are the divergence theorem run in both directions — differentiate the field and integrate the density, or skip both and read the flux. Their forced agreement, for any radius, is the content of the theorem; their numerical agreement here ( $5.6$ pC) is your arithmetic passing its audit.

Answer: $\rho(r) = 4\varepsilon_0 a r = (1.77\times10^{-8}\ \mathrm{C/m^4})\,r$ ; total charge inside 10 cm is $5.6$ pC, confirmed identically by density integration and by flux.

Origins · 1 minJames Clerk Maxwell and how this was discovered

James Clerk Maxwell

1831–1879 · Cambridge and the Cavendish Laboratory

Maxwell's deepest allegiance was to locality — the conviction, inherited from Faraday, that physics happens point by point in the field, not by ledger entries across finite gaps. The differential form of Gauss's law is that conviction made syntax: where the integral form audits a whole bag at once, $\nabla\cdot\mathbf{E}=\rho/\varepsilon_0$ says what space is doing here, now, with no reference to anywhere else. In the 1873 Treatise, and in the 1864 dynamical theory before it, Maxwell assembled electromagnetic physics entirely from such local statements — a set his followers (Heaviside above all, compressing twenty quaternion equations into the four vector ones) polished into the Maxwell equations of every modern textbook, with this article's law standing first. The wager on locality paid beyond anything Maxwell lived to see: relativity requires local laws (instantaneous bag-audits don't transform), and quantum field theory promoted his first equation into an operator constraint that holds in every quantum state. The microscope that shrank the bag never stopped zooming.

History

1813

— Gauss proves the divergence relation for inverse-square attraction — the mathematical bridge between the two forms

1828

— Green's essay develops the local potential theory of electricity, differential statements ahead of their experimental era

1864

— Maxwell's "Dynamical Theory of the Electromagnetic Field" states electromagnetism as local field equations

1873

— The Treatise organizes the subject around point-by-point laws; the divergence equation stands first

1885

— Heaviside recasts Maxwell's system into the four vector equations, giving the differential Gauss law its modern face

1929

— Weyl and the birth of gauge theory: the local Gauss law becomes a constraint that survives quantization

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §2.2.1–2.2.2 · Jackson, Classical Electrodynamics, 3rd ed., §1.3–1.4 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §2.9–2.11 · Zangwill, Modern Electrodynamics, §3.3–3.4 · Feynman Lectures on Physics, Vol. II, Ch. 4–5 Key numerical anchors for this article: the sphere X-ray gives $3kQ/R^3 = \rho/\varepsilon_0 = 2.16\times10^6\ \mathrm{V/m^2}$ for $Q=80$ nC, $R=10$ cm; the $E_r = ar^2$ example yields $\rho = 4\varepsilon_0ar$ , $\rho(0.1\,\mathrm{m}) = 1.77\times10^{-9}\ \mathrm{C/m^3}$ , and $Q = 4\pi\varepsilon_0a(0.1)^4 = 5.56\times10^{-12}$ C confirmed by both routes; the operative tool for radial fields is the spherical divergence formula $\frac{1}{r^2}\partial_r(r^2E_r)$ ; the boundary-condition jump across a surface charge is $\sigma/\varepsilon_0$ (pillbox limit, formalized in the next tier). Constant used throughout: $\varepsilon_0 = 8.854\times10^{-12}$ F/m.

Builds on: Gauss's Law in Integral Form · The Divergence (Gauss) Theorem · The Dirac Delta in Three Dimensions · Continuous Charge Distributions Leads to: The Curl of the Electrostatic Field · Poisson's Equation · Applications of Gauss's Law and Symmetry · Electrostatic Potential

If the two forms are exactly equivalent, why does physics bother keeping both?

Because equivalence of content is not equivalence of usability, in either direction. The integral form is the practical solver (symmetric sources, finite regions, no derivatives of possibly-rough fields) and the natural language of conservation ("what left the box"). The differential form is the theory-builder: it is local — the property relativity demands, since "flux through a bag now" is not even well-defined across reference frames — it composes into PDEs with the other Maxwell equations, and it quantizes. The professional pattern: derive and reason in differential form, compute specific symmetric answers in integral form, and cross the divergence-theorem bridge without noticing you did. Every conservation law in physics (charge, energy, momentum, probability) maintains the same two dialects for the same reasons.

How can the divergence be zero in vacuum when field lines are so obviously diverging?

Vocabulary is the culprit: "diverging" in everyday speech means directions fanning apart; the operator weighs net flux per volume, where fanning directions can be exactly refunded by weakening magnitude. Run the honest balloon test on a $1/r^2$ field at a vacuum point: the outer face of a small box has weaker field but is farther out along a widening cone; the books balance to zero — the E0 divergence article computed this and the flux article's projection argument is the same fact globally. What zero vacuum divergence asserts is field-line conservation between sources, which the "spreading" picture never contradicted: the lines spread, but none are born. If the arrows-versus-operator tension ever resurfaces, recompute; the operator has never lost.

What does the differential law even mean at a point charge, where E blows up?

Classically, nothing — the field is not differentiable there, and both sides of (2) are meaningless as ordinary functions. The delta-function formalism is the repair, and it is exact rather than hand-wavy: treat $\nabla\cdot(\hat{\mathbf{r}}/r^2) = 4\pi\delta^3(\mathbf{r})$ as a statement about integrals (both sides give $4\pi$ against any test volume containing the origin — the E0 proofs), and the law holds with $\rho = q\,\delta^3(\mathbf{r}-\mathbf{r}_0)$ . The mathematics is Schwartz's distribution theory, built decades after Maxwell precisely to make such equations honest. Physically, the message is tidy: a "point charge" is the idealization of density concentrated below any resolution — and the delta is the bookkeeping symbol for exactly that idealization.

How does the differential form handle a charged sheet, where the field jumps?

By confessing that a derivative across the jump does not exist and taking the pillbox limit instead. An idealized surface charge is a delta sheet in $\rho$ ; correspondingly the perpendicular field component jumps by exactly $\sigma/\varepsilon_0$ across it (integrate the law over a vanishing pillbox straddling the sheet — the volume term keeps only the sheet's charge). The resulting statement, $E^\perp_{\text{above}} - E^\perp_{\text{below}} = \sigma/\varepsilon_0$ , is called a boundary condition, and it is the differential law's official representative wherever fields are discontinuous. The next tiers use it constantly — conductor surfaces, dielectric interfaces — so file the principle now: smooth regions get the PDE, interfaces get its pillbox shadow.

Given ρ everywhere, does ∇·E = ρ/ε₀ determine E?

No — and the counterexample is one line: add to any solution a constant field $\mathbf{E}_0$ (or any curl of anything); the divergence is unchanged. A vector field has three components; one scalar equation cannot pin them. The missing information is exactly what the Helmholtz theorem itemized in tier E0: you need the curl as well (statics supplies $\nabla\times\mathbf{E}=\mathbf{0}$ , the next article's business) plus decay or boundary conditions. Then — and only then — the field is unique, and equals the Coulomb superposition integral. This is why "the two local laws of electrostatics" travel as a pair, and why the potential formulation (which builds the zero curl in automatically) is about to take over the subject.

Which form should I reach for on an exam?

Decision tree, honestly ranked. Field from symmetric charge → integral form, Gaussian surface, two lines. Charge from a given field formula → differential form: one divergence computation X-rays the source (worked example 2), massively faster than inverting any integral. Conceptual questions about what fields can do ("could this field exist in vacuum?") → differential form: compute $\nabla\cdot\mathbf{E}$ (and, soon, $\nabla\times\mathbf{E}$ ); nonzero divergence in a stated vacuum is an instant contradiction. Flux through a specified surface → integral form, possibly via solid angles. The single most examinable subtlety spans both: zero divergence in a region does not mean zero field there — it means no sources there, and the field may be passing through on business from elsewhere.

Is there a differential form for the magnetic field too?

Yes, and the pair is the deepest short poem in physics: $\nabla\cdot\mathbf{B} = 0$ — always, everywhere, with no known exceptions. The zero on the right is the statement that magnetic charge does not exist: no magnetic faucets, no drains, every magnetic field line closes on itself or runs forever. Cut a bar magnet and you get two dipoles, never a north and a south in isolation. Whether that zero is fundamental or merely unbroken is a live experimental question — Dirac showed in 1931 that even one magnetic monopole anywhere would explain electric charge quantization, and searches (MoEDAL at the LHC among them) continue. Until one turns up, Gauss's electric law has a source and his magnetic law has a silence, and the asymmetry shapes all of electromagnetism.