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A dark, opaque circular planet of radius $R_p$ blocks specific intensity over its overlap with the projected star. With $k=R_p/R_{\star }$ and no transit dilution, the instantaneous exoplanet transit depth is exactlyWhen $k\ll 1$, the entire planetary disk is inside the stellar disk, and specific intensity varies little across that disk, small-planet transit weighting reduces toHere $z$ is the projected centre separation divided by $R_{\star }$. The numerator samples the local specific intensity; the denominator normalizes by the whole star's mean specific intensity. The small-size argument and the distinction between full overlap and ingress are explicit in the original analytic treatment. ^[ref]

For a short approximately straight crossing at constant transverse velocity $v_{\perp }$, write $x=v_{\perp }t/R_{\star }$ with $t=0$ at mid-transit, and let $b$ be the transit impact parameter. Then $z^2=x^2+b^2$ and ${\mu }_p=\sqrt{1-b^2-x^2}$. With stipulated $u=0.6$ and $k=0.05$, a central crossing has mid-transit exoplanet transit depth approximately $0.003125$, whereas $b=0.8$ gives $0.002375$. A uniform disk would give $k^2=0.0025$ for either fully overlapping crossing. The brighter central chord produces a deeper dip and the off-centre chord a shorter crossing at the same $v_{\perp }$.

At fixed $z<1$ away from the edge, a two-dimensional Taylor's theorem expansion over the symmetric planetary disk cancels the first derivatives; the first correction to the blocked radiant flux is of order $k^4$ if the second derivatives remain bounded. This error estimate is not uniform at the edge, where the radial derivative of the linear limb-darkening law diverges. Ingress, egress, a grazing exoplanet transit, and a crossing of a starspot require the overlap integral or a suitable more complete model. Inferring $k=\sqrt{\delta }$ without the local-to-mean specific intensity ratio can bias the estimated planet radius.

For $r=-\lambda >0$ and vanishing solid-to-liquid diffusivity ratio, $H(r)=\sqrt{\pi }r{e}^{r^2}\mathrm{erfc}(-r)$ gives the interfacial liquid concentration. It increases from zero to $C_s$ as $r$ increases. Its derivative is $C_s{H}^{\prime }/(1+H{)}^2$, which decreases from $\sqrt{\pi }{C}_s$ to zero. For a direct proof, write $A=H/r=\sqrt{\pi }{e}^{r^2}\mathrm{erfc}(-r)$. Then $A^{\prime }=2rA+2$, so $A\ge \sqrt{\pi }+2r$. Differentiation gives $2H^{\prime 2}-(1+H)H^{\prime \prime }=2[(1+r^2+2r^4)A^2+r(4r^2-1)A+2r^2-2]>0$: the bracket increases with $A$, and substitution of its lower bound gives only positive polynomial coefficients, including constant $\pi -2$. Thus the concentration is strictly concave as a function of melting speed parameter $r$. Combining this with a linearized thermal balance produces one fold on the melting branch and, with the freezing branch, an interval of three possible interface speeds.

Smooth convergence on a compact interval requires uniform convergence of the functions and of their derivatives of every order separately, as measured by the derivative seminorms. The need for several scales is visible inFor $n\ge 2$, $p_0(f_n)=1/n$ because $t=\pi /(2n)$ lies in $[0,1]$. The derivative has supremum norm one for every $n$, already at $t=0$. Thus $f_n$ converges uniformly to zero, but fails the test for smooth convergence to zero. Andreas Buttenschoen, Applied Functional Analysis, spaces of continuous and differentiable functions discusses this example and the family of smoothness measurements.

More generally, for any fixed $K\ge 0$, $h_n(t)=\sin (nt)/n^{K+1}$ has $p_K(h_n)\le 1/n$ but $p_{K+1}(h_n)=1$ for $n\ge 2$. If a norm $N$ induced the smooth topology, its continuity would give $N(f)\le C{p}_K(f)$ for some finite $K$, while continuity of $p_{K+1}$ for $N$ would give $p_{K+1}(f)\le C^{\prime }N(f)$. Substituting $h_n$ contradicts these two bounds. Smooth convergence therefore cannot be reduced to one norm. Its beauty is that every finite order can be controlled without fixing a final order in advance.

For the first-order Cauchy-Riemann operator, the equation ${\varphi }_t-i{\varphi }_x=0$ admits the entire functions ${\varphi }_n(x,t)=e^{-\sqrt{n}}\cos (n(x+it))$. Every fixed-order derivative of the initial data converges uniformly to zero, since $n^r{e}^{-\sqrt{n}}\to 0$. Nevertheless ${\sup }_x|{\varphi }_n(x,t)|=e^{-\sqrt{n}}\cosh (n|t|)\to \infty$ for each $t\ne 0$. Thus even convergence of all initial derivatives does not give continuous dependence on initial data away from the initial line. Analytic existence and smooth-data stability are different properties.

A penalized function estimate balancing a residual loss against an integrated derivative penalty. For squared-error loss and the second derivative roughness penalty, the estimate is a cubic smoothing spline. Higher derivative orders give other spline degrees.

Suppose an integrable orthonormal scaling function has a low-pass filter of a multiresolution analysis that is $C^{p-1}$ near $\pi$, and its Fourier transform is $C^{p-1}$ near zero. If the associated wavelet has $p$ integrable vanishing moments, then $m^{(k)}(\pi )=0$ for $k<p$. Indeed, moment differentiation of the Fourier transform makes ${\hat{\psi }}^{(k)}(0)=0$, while $|\hat{\phi }(0)|=1$. In $\hat{\psi }(2t)=e^{-it}\overline{m(t+\pi )}\hat{\phi }(t)$, division by the nonzero smooth factor proves the conclusion. With only continuity of that factor one still obtains a Peano zero, but not automatically higher ordinary derivatives.

Smoothness describes the degree of differentiability of a function. The class $C^k$ has continuous derivatives through order $k$; a smooth function is of class $C^{\infty }$.

For a Sobolev space function, almost every line parallel to a coordinate axis gives an absolutely continuous restriction. An absolutely continuous function has derivative zero almost everywhere on each of its level sets. Slicing and Fubini's theorem give the assertion in higher dimension. If $w\in W^{2,p}$, apply it to $w$ and then to each component of its gradient to get $D^{2}w=0$ almost everywhere on $\{w=0\}$. This supplies the contact differentiation used in noncontact of ROF level boundaries.

A motivating measurement used the McMath solar optical telescope at Kitt Peak. The Sun's image drifted across a $10\mathrm{mm}$-high, $0.2\mathrm{mm}$-wide optical slit; a double-pass spectrograph selected wavelength intervals from $0.0015\mathrm{nm}$ wide at $303\mathrm{nm}$ to $0.007\mathrm{nm}$ at $1099\mathrm{nm}$. An EMI 9750 QB photodetector or silicon photodiode supplied a digitized signal: $1\mathrm{kHz}$ samples were summed into 4096 positions per crossing, including dark and off-limb sky readings. ^[ref]

Controls included daily realignment of optical telescope mirrors, wavelength settings between spectral lines insensitive to the rotational Doppler shift, selection of detectors without detectable hysteresis, and measured electronic nonlinearity corrections below $0.1\%$ of maximum signal. Limb-position calibration combined centres of gravity of the edge derivative profiles with almanac semidiameters; the preliminary edge shifts were consistent with astronomical seeing of approximately $5$ arcseconds full width at half maximum. ^[ref]

The decreasing centre-to-limb readout gives a specific intensity profile, constraining stratification of the photosphere under a radiative transfer model. Astronomical seeing, instrumental scattering of light, and solar granulation limit this inference; brightness alone does not uniquely determine temperature versus geometric height. These observations need not follow the stipulated linear limb-darkening law.

A BV space function is special when its derivative has no Cantor part of a bounded-variation derivative: $Du=\nabla udx+[u]{\nu }_u{\mathcal{H}}^{n-1}\lfloor J_u$. The remaining singular part is supported on the jump set of a bounded-variation function. This space accommodates sharp interfaces and is the natural setting for the Mumford–Shah functional.

The spherical derivative compares the mass of shrinking centered open balls with their Lebesgue measure, when the limit exists. This is a density of one measure relative to Euclidean volume, rather than the relative density of a set within one fixed measure. A singular finite Borel measure has spherical derivative zero Lebesgue almost everywhere.

For a smooth real function on the unit two-sphere, integration of the covariant-derivative commutation formula gives the displayed identity. Explicitly, ${\nabla }^a{\nabla }_a{\nabla }_{b}w={\nabla }_b{\Delta }_{S}w+{\mathrm{Ric}}_b{}^c{\nabla }_{c}w$, and the unit sphere has $\mathrm{Ric}=g$. Integrating by parts once and then again proves the formula. It controls all angular second derivatives by the Laplace-Beltrami operator without singular estimates at coordinate poles.

For $P_0>0$, $\alpha \ne 0$, $T\ge T_0$ and physical $T>0$, the atmospheric pressure-temperature profile $P=P_0\exp (\alpha \sqrt{T-T_0})$ is equivalent toThe branch restriction is essential: squaring does not create a second physical branch. Its temperature gradient is $dT/d\log P=2\sqrt{T-T_0}/\alpha$. Thus, in hydrostatic equilibrium, $\alpha <0$ gives an atmospheric thermal inversion, while $\alpha >0$ gives outward cooling. At $T=T_0$ the inverse branch has zero derivative; $\alpha =0$ gives constant pressure and cannot describe a nontrivial hydrostatic vertical interval.

A point is stationary for a differentiable function or functional when its first derivative or first variation vanishes in every admissible direction.

Suppose $h$ is differentiable and $F^{\prime }>0$ on its range. The multivariable chain rule shows that $F\circ h$ has exactly the same stationary points as $h$. Since $F$ is strictly increasing, the type of every local maximum, local minimum, and saddle point is preserved. If both functions are twice differentiable, then at a stationary pointso the Hessian matrix signature is unchanged. The hypothesis $F^{\prime }>0$ matters for the converse about stationary points: a strictly increasing function such as $F(s)=s^3$ can have a zero derivative and create new stationary points in a composition.

The statistical tangent space is the closed linear span in L2 space of a statistical tangent set. Taking this closure makes orthogonal projection available and ensures that a continuous derivative specified on attainable score functions extends to the whole space.