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The center-to-center extent is $18\mathrm{m}$ on each side and $18\sqrt{2}=25.46\mathrm{m}$ diagonally, while the shortest nonzero baseline is $6\mathrm{m}$. At $\lambda =0.02\mathrm{m}$, characteristic scales areThese describe the finest synthesized scale, a largest recoverable scale estimate, and the individual-dish field of view, respectively. Along an array side the longest baseline is only $18\mathrm{m}$, giving $3.82^{\prime }$ rather than the diagonal estimate. Projection, weighting and sidelobes determine an actual synthesized beam.

There is no sharp universal largest-scale cutoff: the commonly used $0.6\lambda /b_{\min }$ convention would give $6.88^{\prime }$. For an ideal uniformly illuminated circular dish, the primary-beam full width at half maximum is approximately $1.02\lambda /D=17.5^{\prime }$, while the first-null angular radius is $1.22\lambda /D=21.0^{\prime }$. These definitions should not be mixed. Uniform lattice spacing also produces grating sidelobes; the primary beam restricts the useful field but does not supply the missing zero-baseline flux.

The wavelength corresponds to about $15\mathrm{GHz}$, in the low-frequency decrement part of the thermal Sunyaev-Zeldovich effect. Hot cluster electrons inverse-Compton scatter CMB photons. In the Rayleigh-Jeans limit,The signal probes integrated electron pressure. Short baselines give sensitivity to extended arcminute-scale cluster structure, while the primary field covers the cluster and surrounding sky. The array's several-arcminute resolution suits many such structures; it is not equally well matched to every cluster size.

Simultaneous cross-correlations reject uncorrelated receiver noise and provide spatial information; compact radio sources can be constrained with the relatively longer baselines and separated from extended structure. Broad frequency coverage helps sensitivity provided channel smearing is controlled. Correlated atmospheric emission, calibration error, primordial CMB fluctuations and radio-source contamination remain relevant. The lack of zero and very short spacings also means that total integrated cluster flux requires a model or complementary data. An interferometer is therefore useful for the spatially resolved decrement, rather than automatically measuring every cluster's full pressure integral.

An observed catalogue is the result of a population, an instrument and a selection rule. Its astronomical selection function can depend on flux, color, size, surface brightness, sky location, cadence and follow-up success. The relevant cut is often on noisy measured quantities. A model that selects on noiseless true flux instead can predict a different sample even when it uses the same nominal threshold.

Malmquist bias occurs because brighter members of a population are visible through a larger volume in a flux-limited survey. For a uniform Euclidean population, $r_{\max }\propto \sqrt{L}$ and $V_{\max }\propto L^{3/2}$. The selected luminosity distribution is therefore weighted toward large $L$. For an intrinsic Gaussian absolute-magnitude distribution with mean $M_0$ and variance ${\sigma }_M^2$, multiplying by $10^{-0.6M}$ and completing the square gives the Gaussian absolute-magnitude Malmquist shiftThis assumes the stated simple population, a complete flux cut and no additional boundaries. Exactly identical noiseless standard candles have a distance truncation, not a luminosity spread to reweight. Scatter, extinction, cosmological volume and source evolution need their own model.

Lutz-Kelker bias concerns parallax errors, spatial population weighting and selected calibration samples. Uniform spatial density gives $p(r)\propto r^2$ locally, hence $p(\varpi )\propto {\varpi }^{-4}$ on the physically valid range. More distant objects can scatter into a measured-parallax cut. In addition, inversion $r=1/\hat{\varpi }$ is nonlinear and gives misleading estimates at low signal-to-noise; a negative measured parallax is possible even for a positive true distance. A posterior instead combines the parallax likelihood with a justified population prior. An unbounded uniform-volume prior is improper here; a finite population model or a proper distance prior for parallax inference is necessary.^[ref]

Bayes' theorem makes the assumptions explicit. If $\eta$ denotes population parameters, $x$ latent properties and $S$ the catalogue inclusion event, a selected-sample likelihood has the formThis selection-normalized astronomical likelihood accounts for the model-dependent probability of entering the catalogue. For individual inference, selection information already determined by the fully conditioned measured data must not be multiplied in a second time. The sampling design, inferential target and actual conditioned information determine the appropriate expression.

Other effects include Eddington bias, where steep number counts and measurement scatter move more faint objects upward than bright objects downward, and surface-brightness selection that misses diffuse galaxies despite substantial total flux. Dust extinction affects both detectability and inferred luminosity. Variable-source detection depends on cadence and duration; spectroscopic follow-up can favor easy-to-measure redshifts or particular colors. Blends, calibration drift and redshift errors can mimic population changes.

Examples include magnitude-limited supernova samples used in distance-redshift fits, where luminosity scatter and follow-up must be modeled, and radial-velocity planet surveys, which favor large-amplitude, short-period systems and leave inclination degeneracies. Forward modeling through the survey pipeline, injection-and-recovery estimates of completeness, calibrated noise and properly modeled censored or nondetected observations address these effects. A universal correction or an unexamined inverse-volume weight does not replace an explicit generative and selection model.

A charge-coupled device converts absorbed photons into stored charge in semiconductor potential wells. An exposure accumulates a pixelated image; clocked electrodes transfer rows toward a serial register and output amplifier. Quantum efficiency, pixel sampling and optical throughput connect collected photons to the expected signal. In ordinary optical imaging it primarily counts photoelectrons rather than independently measuring every photon's energy.

The CCD noise budget combines source and sky shot noise, dark-current fluctuations and read noise. For an aperture containing $n$ pixels, known background expectation $N_b$, total dark expectation $N_d$, source expectation $N_s$ and per-pixel read noise $r$,Estimating the background from a finite separate region adds its own uncertainty. Cooling reduces dark current; longer integration reduces the relative importance of one read but can encounter saturation, cosmic rays or changing conditions. Full well capacity, amplifier gain, linearity, blooming and charge-transfer efficiency limit useful detector dynamic range.

Bias frames establish the electronic offset; matched dark frames characterize time- and temperature-dependent dark signal. Flat fields calibrate pixel-to-pixel response and illumination, with appropriate attention to wavelength. Bad-pixel masks, repeated exposures and dithering help reject defects and cosmic rays. A calibrated PSF and astrometric solution are needed for reliable photometry and positions. High quantum efficiency and stable linear integration are advantages, but an uncalibrated count array is not yet an astronomical flux map.

Delay-compensated geometry. Choose an east-west baseline vector $\mathbf{b}=b\hat{\mathbf{x}}$ and let the upward unit vector be $\hat{\mathbf{z}}$. A ray direction at zenith angle $\beta$ in their plane has $\mathbf{s}(\beta )=(\sin \beta ,0,\cos \beta )$. Its signed geometric path difference is $\mathbf{b}\cdot \mathbf{s}=b\sin \beta$. Removing the phase-centre delay $b\sin \theta /c$ leaves the residual pathWith the receiver ordering chosen to give positive phase for this path, a complex correlator measuresThe reverse ordering complex-conjugates the expression without changing its information. A single real multiplier supplies its cosine component; the stated exponential represents an in-phase/quadrature correlator. Equal cable lengths to the central correlator remove a fixed instrumental difference but do not remove the changing geometric delay. Pointing the antennas and electronically compensating the path are separate operations.

Resolved uniform source. Contributions from mutually incoherent sky elements add as brightness-weighted complex visibilities. Put $I(\alpha )=I_0$ over $-\Delta \alpha /2<\alpha <\Delta \alpha /2$ and zero elsewhere, and define $q=kb\cos \theta$. Integrating givesThereforeThe zero-argument value is defined by continuity as one. Normalization by $V(0)=I_0\Delta \alpha$ isolates the visibility shape from total flux. The first null occurs when the source spans one fringe period, $\Delta \alpha =\lambda /(|b\cos \theta |)$. A negative real visibility beyond a null is a phase reversal, not negative surface brightness.

Path-compensation precision. A finite observing band sets a coherence envelope. For a rectangular band of width $\Delta \nu$, an uncorrected residual delay $\delta \tau =\delta \ell /c$ multiplies the correlation byCoarse delay steps must leave $\Delta \nu |\delta \tau |\ll 1$ if decorrelation is to be small. Thus a coarse path-step scale must be well below the coherence length $c/\Delta \nu$, with the precise bound determined by the tolerated amplitude loss and maximum quantization residual. A digital fine-delay or phase-rotation system can separately correct the central-frequency phase.

If there is no such carrier-phase correction, the steps must also be a small fraction of $\lambda$ to avoid large untracked phase jumps or time-averaging loss. For a step $d_{\ell }$ rounded to the nearest setting, $|\delta \ell |\le d_{\ell }/2$ and the maximum carrier-phase error is $\pi d_{\ell }/\lambda$. Bandwidth controls coarse group-delay decorrelation; wavelength controls the precision of untracked carrier phase. No numerical delay step can be inferred from the central frequency alone without a bandwidth, loss tolerance and compensation architecture. Earth rotation additionally sets how rapidly the delay setting must be updated.

Largest sampled angular scale. For a near-zenith view take the shortest projected baseline as $b_{\min }\simeq 0.5\mathrm{m}$. At $\nu =30\mathrm{GHz}$, $\lambda =c/\nu =9.9931\times 10^{-3}\mathrm{m}$. The inverse shortest spatial frequency givesThis is an order-of-magnitude largest fringe scale. A convention defining maximum recoverable source size as $0.6\lambda /b_{\min }$ would instead give about $41$ arcminutes; source profile, weighting and projected baseline coverage determine the operational value. The antenna diameter and primary beam can further restrict the usable field but were explicitly neglected in the source's initial idealization. Ten antennas supply $10\times 9/2=45$ pairwise baselines; their number improves sampling and sensitivity, not the inverse-shortest-baseline scale by a factor of ten. The longest baseline is not supplied, so the finest synthesized resolution is not numerically fixed.

A suitable observing programme. A compact centimetre-wave array is well suited to CMB temperature anisotropy on degree to sub-degree scales sampled by its baselines, and, with sufficiently long baselines and suitable foreground control, thermal Sunyaev-Zeldovich structure in galaxy clusters. At $30\mathrm{GHz}$ the thermal Sunyaev-Zeldovich signal is a decrement relative to the CMB. The compact baselines retain sensitivity to extended brightness features; the spatial frequencies measured by each baseline have a direct statistical interpretation in CMB analysis.

Compared with lower radio frequencies, synchrotron foregrounds are generally reduced, while thermal dust rises toward higher frequencies; observing near this centimetre-wave atmospheric window is therefore useful, although foregrounds are not absent. Correlation and phase switching reject mean uncorrelated receiver offsets and some instrumental drifts. Receiver thermal noise still contributes variance, and atmosphere, ground pickup or cross-talk can be correlated and require control. Additional long baselines can identify compact radio-source contaminants for subtraction, but the question does not specify enough array geometry to guarantee that capability. There is no zero baseline in ordinary pairwise cross-correlation, so the array does not directly measure the sky monopole or arbitrary larger-scale smooth emission.

Atmospheric turbulence produces time-dependent refractive-index fluctuations and phase errors across the telescope pupil. A large aperture's long-exposure point spread function is broadened approximately to $0.98\lambda /r_0$, where $r_0$ is the Fried parameter, rather than the diffraction scale $\lambda /D$. For fixed turbulence, $r_0\propto {\lambda }^{6/5}$, so the seeing angle improves weakly as ${\lambda }^{-1/5}$. The atmospheric coherence time scales with $r_0$ divided by an effective wind speed. This temporal scale concerns changing turbulence, not the optical bandwidth's coherence time.

Reduce the disturbance. A good high-altitude site, favourable weather, observing near the zenith, thermal control and dome ventilation reduce atmospheric and local seeing. They improve the uncorrected image but do not remove all high-order wavefront distortion. Active optics slowly maintains the main mirror's figure and alignment; it is distinct from the rapid adaptive correction of atmospheric phase errors.

Correct the wavefront. Adaptive optics uses a nearby natural guide star or artificial laser guide star, a wavefront sensor, a real-time controller and a deformable mirror. A Shack-Hartmann sensor estimates local wavefront slopes; mirror actuators apply the opposite optical-path distortion fast enough to track the atmosphere. A separate fast steering element or the mirror can correct image motion. In an ideal well-corrected direction this approaches diffraction-limited resolution and increases the concentration of light in the central image.

Finite guide-star flux, actuator sampling, control latency and anisoplanatism leave residual errors. A laser beacon samples a finite-height cone rather than the full distant-source cylinder, and a conventional laser reference does not determine absolute tip-tilt, so a natural reference remains useful. Multiple guide stars and multiple conjugate mirrors can widen the corrected field, with extra sensing and reconstruction complexity. These are fundamental geometry and control limits, not claims of a universal corrected sky fraction.^[ref]

Freeze and reconstruct. Short exposures can freeze the instantaneous speckle pattern. Speckle interferometry uses calibrated Fourier statistics to recover diffraction-scale structure; power-spectrum information alone loses phase, while bispectrum or other phase-sensitive measurements support image reconstruction. Lucky imaging selects sharp short frames, registers them and combines them, trading photon efficiency for improved image quality. Tip-tilt alignment alone removes image motion rather than all high-order aberration. These methods require sufficiently bright sources or references and can incur substantial read-noise penalties. Deconvolving a long-exposure image is useful only where a known point spread function and adequate signal preserve information; it cannot recover arbitrarily lost spatial frequencies. Moving to space avoids atmospheric seeing, although it changes the observing platform and its limitations rather than correcting a ground-based telescope.

Weak transitions in dilute gas. A forbidden emission line is a transition absent in the electric-dipole approximation because of its selection rules, but permitted through weaker channels such as magnetic-dipole or electric-quadrupole radiation, sometimes aided by state mixing. It is not an impossible transition. Its upper atomic level is relatively long lived, so collisions can deactivate it before a photon is emitted in dense gas. Low-density astrophysical plasmas allow radiative decay to compete successfully.

In a schematic two-level statistical equilibrium,The critical density is $n_{\mathrm{crit}}=A_{ul}/q_{ul}$ for this simple level system; multiple radiative and collisional channels require summed rates. At $n_e\ll n_{\mathrm{crit}}$ each collisional excitation is likely to produce radiation, so ${ϵ}_{ul}\propto n_l{n}_e{q}_{lu}h\nu$. At high density collisional de-excitation suppresses the line relative to this low-density expectation. The rate need not decrease absolutely with increasing density; the suppression is relative to the naive excitation scaling.

A temperature-sensitive ratio. For a Maxwell-Boltzmann electron distribution,where ${\rm Y}$ is an effective collision strength. Lines from upper levels with different excitation energies therefore have a strongly temperature-dependent ratio. Using one ion reduces sensitivity to abundance and ionization fraction compared with comparing unrelated species.

A standard example is doubly ionized oxygen: the $4959$ and $5007$ angstrom lines descend from ${}^1{D}_2$ toward the ${}^{3}P$ ground term, while the $4363$ angstrom line connects ${}^1{S}_0$ to ${}^1{D}_2$. The higher excitation threshold makes the auroral $4363$ line relatively stronger in hotter gas. In the low-density Maxwellian approximation, with slowly varying collision strengths,The numerical form is a useful approximation, not a universal exact atomic law; a multilevel atomic calculation with current collision strengths is needed for precise work.^[ref]

In an active galactic nucleus these lines commonly diagnose the relatively dilute narrow-line gas; dense broad-line gas can collisionally quench them. Measure and deblend the fluxes, subtract continuum and stellar features, correct differential extinction, and check density independently. The $5007/4959$ ratio mainly tests radiative branching and calibration because both lines share one upper level; it is not the required temperature thermometer. Temperature gradients, multiple density components, shocks, ionization structure and departures from the Maxwellian electron distribution can bias a single inferred temperature. Without a measured ratio and density regime, no particular AGN temperature is numerically determined here.

Measure counts in electrons so that photon and detector noise can be combined directly. For source electrons $S$, $n$ aperture pixels, total time $t$, sky rate $b$ per pixel, dark-current rate $d$ per pixel, $M$ reads with rms read noise $r$ per pixel, and an independently estimated mean background per pixel of variance ${\sigma }_{\hat{B}}^2$, an ideal aperture-sum budget isThe last term represents the propagated calibration uncertainty if it can be treated as a random variance; systematic residuals can instead bias the measurement and need an explicit error model. For a background estimated from $n_b$ independent, equally exposed sky pixels, one approximation is ${\sigma }_{\hat{B}}^2=[(b+d)t+M{r}^2]/n_b$. Correlated resampling, detector cross-talk or a spatially structured sky invalidates the independent-pixel simplification.

Photon counting and sky. Source photon shot noise has variance $S$ and cannot be calibrated away. Longer integration or greater throughput improves its fractional error as $S^{-1/2}$. Sky photons, including moonlight, airglow and scattered light, add their own Poisson fluctuations. A dark site, appropriate observing time and filter, good baffling and a smaller or optimally weighted source aperture reduce sky contributions. Better seeing or adaptive optics can concentrate a point source into fewer noise-bearing pixels. Sky subtraction removes the mean background but not the random sky photons already recorded, and the sky estimate adds uncertainty of its own.

Dark and read noise. Thermal dark current contributes a mean and shot noise. Cooling reduces its rate; well-matched dark frames remove a reproducible mean and map hot pixels, but do not undo the science exposure's random dark-current fluctuations. Each read adds electronic noise. At fixed total integration, many short exposures add more read variance than fewer long exposures, so use exposures long enough to become sky limited when saturation, cosmic rays and temporal changes permit. Multiple non-destructive reads can help on detectors that support them. Subtracting a bias frame corrects an electronic offset, not the stochastic read noise.

Calibration and non-Poisson effects. Pixel sensitivity differences, illumination gradients, fringes, nonlinearity, imperfect charge transfer and electronic pattern noise leave calibration errors. High-count flat fields, stable operating conditions, linearity calibration, dithering and appropriate detector models help, while the finite signal-to-noise of calibration frames propagates into the result. Cosmic-ray hits, hot pixels, satellite trails and scattered-light artefacts are not ordinary stationary Gaussian noise: identify or mask them, often using several exposures. Overaggressive clipping can remove real transient or compact-source signal.

Atmospheric scintillation adds correlated intensity fluctuations, especially in short bright-star measurements; seeing changes and guiding errors move flux between pixels or apertures. Differential photometry using appropriate reference stars and stable apertures reduces common variations but does not guarantee removal of chromatic or position-dependent effects. For a constant flux-limited source in independent stationary noise the final signal-to-noise usually grows as $\sqrt{t}$ once background limited; a systematic floor or correlated sky can stop that improvement. Distinguish improvement of calibration, which removes reproducible bias, from elimination of intrinsically random photon arrivals, which is impossible.

The quantities $n_i(\tau ,t)$ are cell-cycle age densities: $n_i(\tau ,t)d\tau$ counts biological cells whose time since entering phase $i$ lies in $[\tau ,\tau +d\tau ]$. Here $\tau$ is time spent in the current phase, rather than time since birth. In this two-stage approximation to the cell cycle, a biological cell passes from G1 phase to G2 phase, then completes cell division and produces two daughters entering G1 phase.

The time derivative measures change at fixed phase age. The age derivative transports biological cells towards larger $\tau$ at unit speed, because $d\tau /dt=1$. Thus each equation is an age-structured population equation. The nonnegative hazard function ${\Gamma }_i(\tau )$ means that, conditional on still being in phase $i$ at age $\tau$, the probability of leaving during the next $dt$ is ${\Gamma }_i(\tau )dt+o(dt)$. Its product with $n_i$ is the loss from that phase-age class; it is not an additional death term.

The boundary condition at $\tau =0$ specifies the incoming flux. Every G1 phase exit contributes one entrant to G2 phase, whereas every G2 phase exit contributes two entrants to G1 phase. The initial conditions specify the phase-age composition at $t=0$. Consequently integration over phase age gives the useful population balanceEach completed cell division increases the total number of biological cells by one.

The unit integrals of the profiles make $N_i(t)={\int }_0^{\infty }{n}_i(\tau ,t)d\tau$ the phase populations, and $f_i$ their normalized cell-cycle age densities. Assume the age tails vanish and the following integrals are finite. For a fixed profile define its mean exit hazard function byThese constants have units of inverse time. Integrating the age-structured population equations and using the boundary conditions givesThe incoming fluxes are $n_1(0,t)=2{\Lambda }_2{N}_2$ and $n_2(0,t)={\Lambda }_1{N}_1$, soThe separable-profile closure for an age-structured population has a qualification: arbitrary age-dependent hazard functions do not give this constant-coefficient ordinary differential equation for arbitrary initial conditions. Without fixed $f_i$, the effective exit rates are $\int {\Gamma }_i{n}_{i}d\tau /N_i$ and can vary with time. Moreover, an exact positive separation of variables with both profiles fixed forces $N_2/N_1$ to be constant through the incoming fluxes, hence selects a common exponential mode. The displayed ordinary differential equation is the necessary integrated system for such profiles; it is also exact for arbitrary populations when both hazard functions are age independent.

Write $A$ for the population matrix and assume ${\Lambda }_1,{\Lambda }_2>0$. Its characteristic polynomial isThe two eigenvalues have opposite signs. The dominant eigenvalue isand the other eigenvalue is $\nu =-({\Lambda }_1+{\Lambda }_2)-\mu <0$. A positive eigenvector for $\mu$ satisfiesFor any nonzero nonnegative population vector in this constant-coefficient system, the coefficient of the positive eigenvector is positive: a positive left eigenvector annihilates the other eigenvector and has positive pairing with that population vector. Thus $N_i(t)=C_i{e}^{\mu t}+O(e^{\nu t})$, with $C_i>0$, and $N_i^{\prime }/N_i\to \mu$. The long-time leading term therefore obeys $N_i^{\prime }=\mu N_i$.

Substituting that exponential mode into the age-structured population equation givesDefine the phase-residence survival function and its discounted integral byThen the normalized cell-cycle age densities areSince integration of $f_i^{\prime }=-(\mu +{\Gamma }_i)f_i$ gives $f_i(0)=\mu +{\Lambda }_i$, this is equivalentlyThe constants must be consistent with both incoming fluxes. PutThe boundary conditions for amplitudes $n_i=e^{\mu t}{\varphi }_i$ give ${\varphi }_1(0)=2H_2{\varphi }_2(0)$ and ${\varphi }_2(0)=H_1{\varphi }_1(0)$. Hence the Two-stage Euler-Lotka equation isIt determines $\mu$ directly from the hazard functions; the formula in terms of ${\Lambda }_i$ is not permission to choose those constants independently of the profiles. If each phase is eventually left with probability one, $H_i(0)=1$ and $H_i(\mu )$ decreases to zero, giving a unique positive root under the usual finite, non-instantaneous residence-time assumptions. More generally positive growth requires $2H_1(0)H_2(0)>1$. Nonnegative hazard functions alone do not ensure it: if one phase has zero exit hazard function, growth cannot have the asserted positive exponent. Convergence of the complete age distribution to these profiles needs the usual mixing assumptions on phase-residence times; the two-dimensional eigenvalue calculation establishes the long-time claim for the fixed-profile system.

The bacteria disperse by diffusion with coefficient $D_B$. Their additional chemotaxis flux isIt points up the nutrient concentration gradient, and its negative spatial derivative enters the balance equation. The positive $N_0$ prevents singular sensitivity at zero nutrient concentration; increasing the background nutrient concentration reduces the drift response to a fixed gradient. The logistic growth equation term $rB(1-B/K)$ represents net reproduction with low-population growth rate $r$ and carrying capacity $K$.

The nutrient undergoes diffusion with coefficient $D_N$. Its production term $a{B}^2/(B_0^2+B^2)$ is an increasing, bounded response to the bacteria, reaching half its maximum $a$ at $B=B_0$. It has the algebraic form of a Hill equation with exponent two; the equation alone does not specify a molecular mechanism. The term $dBN$ removes nutrient at a rate proportional to both bacteria abundance and nutrient concentration.

The full fluxes areThe homogeneous Neumann boundary conditions make both fluxes vanish at the endpoints, so neither bacteria nor nutrient crosses the boundaries. The initial conditions specify their starting spatial distributions. This model includes nutrient production by bacteria, rather than a separate external supply term.

Take the membrane particle flux density to be positive from cytoplasm into the cell nucleus. With membrane permeability $\alpha$,For fixed compartment volumes, conservation of mass and the absence of exterior flux giveSubtract the two concentration derivatives:This linear ordinary differential equation has solutionIts relaxation time is $\boxed{T=V_c{V}_n/[\alpha A_n(V_c+V_n)]}$. Here $\alpha$ has dimensions length/time, so $\alpha A_n/V$ has dimensions inverse time. A positive initial concentration difference decays monotonically toward zero; a negative one approaches zero from below.

For a continuously differentiable vector field on a volume $V$ with piecewise smooth boundary $S$, the divergence theorem stateswhere $\hat{\mathbf{n}}$ is the outward unit normal. For the polynomial field here,Therefore the outward flux through the unit cube isAs a direct boundary check, the $x=1$ face contributes $2$, the $y=1$ face contributes $-1$, and the $z=1$ face contributes $1/2$. All three opposite faces contribute zero, reproducing the result and fixing the orientation sign.

For $r=\sqrt{x^2+y^2+z^2}>0$, the radial vector field has components $F_i=x_{i}f(r)$. The product rule givesThus for $f(r)=r^{n-1}$,When $n=-2$, $\mathbf{F}=\hat{\mathbf{r}}/r^2$ and the ordinary divergence is zero at every nonzero radius. However, the field is singular at the origin. Its outward flux through a radius-$R$ sphere is $(1/R^2)(4\pi R^2)=4\pi$, so one cannot apply the smooth-field divergence theorem across the origin and conclude that this flux vanishes.

In the language of distributions, the complete distributional divergence identity is $\boxed{\nabla \cdot (\hat{\mathbf{r}}/r^2)=4\pi {\delta }^{(3)}(\mathbf{r})}$. Here ${\delta }^{(3)}$ is the three-dimensional Dirac delta function. Without that generalized interpretation, the correct statement is simply zero divergence away from the origin and undefined ordinary divergence at the singular point. Removing a small ball around the origin restores the ordinary theorem: the inner boundary has inward radial orientation and cancels the outer flux.

The divergence theorem states that, for a sufficiently smooth vector field $\mathbf{F}$ on a volume $\Omega$ with piecewise smooth closed boundary $\partial \Omega$,Here $\hat{\mathbf{n}}$ is the outward unit normal. The left-hand side is the total outward flux; the right-hand side adds the local divergence throughout the volume. The orientation and the requirement to include the complete closed boundary are essential.

Read the given vector field as the three-dimensional field $\mathbf{F}=(x,y,0)$. Its divergence is the sum of the three matching-coordinate partial derivatives:The absence of a $z$ component does not make the three-dimensional flux vanish: the field still flows outward through surface normals with horizontal components.

The enclosed volume lies above $z=0$ and below $z=4-x^2-y^2$. In cylindrical coordinates, its limits are $0\le r\le 2$, $0\le \theta \le 2\pi$, $0\le z\le 4-r^2$, with volume element $rdzdrd\theta$. The divergence theorem givesThus the volume is $8\pi$, and the outward flux is $\boxed{16\pi }$.

A direct check separates the boundary into its cap and base. On the base, $\mathbf{F}\cdot (-\hat{\mathbf{z}})=0$. On the graph $z=4-x^2-y^2$, the upward vector area element is $(2x,2y,1)dxdy$, so the cap flux is ${\iint }_{x^2+y^2\le 4}2(x^2+y^2)dxdy=16\pi$, in agreement with the volume calculation.

The divergence theorem includes every part of the closed boundary, including the two vertical cut faces. Its volume integrand simplifies because the terms linear in $x$ cancel:In cylindrical coordinates, the remaining integration region is $0\le r\le 1$, $0\le z\le 1-r^2$, $\pi /2\le \theta \le 2\pi$. With the Jacobian determinant $r$, the outward flux isThe angular range is three quarters of a full revolution. The volume calculation already accounts for the base and cut faces, so no additional surface terms should be appended.

For a continuously differentiable vector field $\mathbf{F}$ near a piecewise smooth oriented surface $S$, Stokes theorem statesThe circulation around the whole boundary equals the flux of the curl through the surface. The unit normal vector and boundary direction must be compatible with the right-hand rule: curling the right-hand fingers in the boundary direction points the thumb along the normal. Reversing both orientations reverses both integrals.