A straight line through a point $a$ in nonzero direction $t$ is the set of vectors $r=a+st$ for real parameter $s$. If $|t|=1$, changes in the parameter equal signed distances along the line. Two such lines intersect exactly when their two parameter formulas give a common vector for some choice of their parameters.

A secant line is a straight line passing through two distinct points of a curve. For a graph $y=f(x)$ at $x=a$ and $x=a+h$, its slope is the difference quotient $(f(a+h)-f(a))/h$. As the second point approaches the first, a limiting secant line can give the tangent line.

In Cartesian coordinates, the slope of a nonvertical straight line is its vertical change divided by its horizontal change. The graph $y=mx+c$ has slope $m$; a vertical straight line has no finite slope. For a tangent line to a differentiable function, the slope is the derivative.

Suppose a family $A\subseteq C(K,\mathbb{R})$ on a compact space is closed under pointwise maximum and minimum. If a target $f$ can be approximated arbitrarily closely at each pair of points by a member of $A$, it can be approximated in the supremum norm. For a fixed first point, take finitely many pairwise approximants whose neighborhoods cover $K$ and take their maximum: it lies above $f-\epsilon$ everywhere and below $f+\epsilon$ near that first point. Take finitely many such first-point neighborhoods covering $K$ and take the minimum of their maxima. The result is within $\epsilon$ of $f$ everywhere. Starting with tolerance $\epsilon /2$ ensures a strict uniform approximation error smaller than $\epsilon$.

The overdamped filament bending equation diagonalizes in the complete free-free bending spectrum. Positive modes decay at rates $A{k}_n^4/\zeta$, while translation and tilt remain fixed. The long-time shape is the affine least-squares projection of the initial height. Free-end boundary values conserve the total height integral and its first spatial moment. For square-integrable initial data the expansion gives the smoothing semigroup solution.

The integral ${\int }_{\mathbb{R}}{e}^{-t^2}(z-t{)}^{-1}dt$ in the lower half-plane extends to the displayed entire function. In the upper half-plane the continued function differs from the directly evaluated real-contour integral by $2\pi i{e}^{-z^2}$. Its algebraic expansion is $\sqrt{\pi }{z}^{-1}{\sum }_{m\ge 0}(2m-1)!!/(2z^2{)}^m$. The real axes are switching lines for the exponentially small term under the Stokes-multiplier convention; the diagonal rays $\arg z=\pi /4+j\pi /2$ are equal-magnitude boundaries for the exponential and algebraic mechanisms. Stating both the residue jump and the sector domains prevents confusion between switching and dominance conventions.

The saddle and endpoint asymptotic expansions of this entire function have algebraic coefficients $C{\omega }^j/z$ in sectors $(2j-1)\pi /3<\arg z<(2j+1)\pi /3$, where $C=\Gamma (1/3)/3$ and $\omega =e^{2\pi i/3}$. Deforming the contour onto the decay ray $\arg (zt)=2j\pi /3$ gives its complete integral $C{\omega }^j$. The endpoint contribution is $-e^{-z^3}(3z^3{)}^{-1}{\sum }_{r\ge 0}(-1{)}^r(2/3{)}_r/z^{3r}$. The algebraic coefficient switches at $\arg z=\pi /3,\pi ,5\pi /3$; the equal-exponential-magnitude anti-Stokes lines are the intervening rays $\pi /6+j\pi /3$.

Two sectorial fundamental solutions of the same linear system differ on their overlap by a constant invertible right factor, since $(Y_j^{-1}{Y}_{j+1}{)}^{\prime }=0$. With a common formal normalization these factors encode the exponentially small ambiguities invisible to the asymptotic expansion. They are called Stokes matrices. Equivalently their inverses act on column vectors of solution coefficients; the convention must be specified when ordering products.

At an unramified irregular singular point with distinct leading eigenvalues, a sectorial fundamental solution is a fundamental matrix whose entries are holomorphic functions asymptotic to a prescribed formal matrix on a sector of the logarithm cover. For competing cubic exponentials, the usual construction uses six overlapping sectors with basic angle $\pi /3$. A formal power series need not converge, and two sectorial realizations may differ by exponentially small terms across a sector boundary.

For Stokes flow without body forces, $\nabla \cdot \mathbf{\sigma }=0$. Symmetry of the Newtonian fluid stress tensor also gives $\nabla \cdot (\mathbf{x}\times \mathbf{\sigma })=0$, interpreted componentwise as the angular-momentum flux. The divergence theorem makes the resultant force and torque the same on any two homologous enclosing surfaces lying in a nonsingular fluid region. In a boundary perturbation of a nearly spherical particle, evaluating each perturbation field on a fixed enclosing surface avoids spurious geometric force terms: the geometric terms from changing the surface normal and area cancel those from evaluating the base stress at the displaced surface.

Extend the velocity rigidly through a moving sphere in an incompressible tube flow. The extended velocity is continuous and divergence-free, so its total flux is independent of height. Quiescent far-field conditions make this flux zero. The sphere contributes $V_z{A}_s$ through a cross-section; the rotational contribution integrates to zero over its circular slice. Thus the fluid flux compensates the solid displacement. A descending sphere generates upward fluid flux in every section cutting its interior. This statement assumes a localized flow with no imposed throughflow and negligible tube-end effects.

For a stationary cylinder of radius $a$ in the far-field shear $(\Gamma y,0)$, the stream function is $\psi =(\Gamma /4)[r^2-2a^2\log (r/a)-a^2-(r^2-2a^2+a^4/r^2)\cos 2\phi ]$. It solves the biharmonic equation and the no-slip boundary condition. The fluid exerts torque $-2\pi \mu \Gamma a^2$ per unit length, because the angular part of the surface traction averages to zero and the mean tangential traction is $-\mu \Gamma$.

The Oseen approximation retains advection of a small disturbance by a uniform incident velocity $\mathbf{U}$, but neglects advection of the disturbance by itself. Its steady equations areA Stokes disturbance of a sphere decays as $Ua/r$, so the ratio of background advection to viscous diffusion grows as $Ur/\nu$. Thus the approximation resolves the outer region $r\sim \nu /U$, where the Stokes flow expansion is nonuniform, while self-convection remains smaller by $a/r$.

Two identical radius-$a$ spheres moving together through a viscous fluid at separation $b\gg a$ entrain each other. The leading far velocity of one sphere at the other is $3aU/(4b)$ for separation perpendicular to the velocity, and $3aU/(2b)$ for parallel separation. Replacing $U$ by its velocity relative to this induced flow in Stokes law gives the displayed drag reductions relative to $D_0=6\pi \mu aU$, with errors of order $(a/b{)}^2$. Reflection symmetry and linearity imply equal drag on the two spheres in either geometry.

With body-outward normal $n$, put $F=-\int \sigma ndS$, ${\Sigma }_{ij}=\int x_i(\sigma n{)}_{j}dS$, $U_{ij}=\int u_i{n}_{j}dS$, and $M=\Sigma -2\mu U$. Taylor expansion of the exterior boundary integral representation of Stokes flow gives a Stokeslet from $F$, a rotlet from $G_j=-{ϵ}_{jkl}{\Sigma }_{kl}$, a source from $Q=\mathrm{Tr}U$, and a stresslet from $S=(M+M^T)/2-I\mathrm{Tr}M/3$. Thus $F$ and $G$ are the force and torque exerted on the fluid, while $Q$ is the instantaneous displaced-volume rate. A volume-preserving deforming body has $Q=0$.

For a two-dimensional decaying Stokes flow in $y>0$, prescribe boundary Fourier mode $(U,V)e^{ikx}$. The boundary shear stress and normal stress are the displayed diagonal map, with the common exponential understood. For $k>0$ and $V=0$, the Unscaled Papkovich–Neuber representation gives $u=U(1-ky)e^{ikx-ky}$, $v=-ikUy{e}^{ikx-ky}$ and $p=-2i\mu kU{e}^{ikx-ky}$. Differentiation proves the tangential traction and vanishing normal stress. The normal-velocity calculation supplies the second diagonal entry. The map converts a viscous exterior into a nonlocal boundary resistance proportional to $|k|$.

A layer of half-thickness $h_0$, dynamic viscosity $\mu$ and density $\rho$, between semi-infinite fluids of viscosity $\lambda \mu$ and densities $\rho \mp \Delta \rho$, relaxes a symmetric thickness Fourier mode. In the plug-like extensional viscosity regime, mass conservation and $4\mu (h{u}_x{)}_x=h\Delta \rho g{h}_x-{\sigma }_{xz}^{+}$ together with the Fourier traction map for a viscous half-space give the displayed rate, where $\kappa =|k|h_0$. Inner longitudinal extension dominates for $\lambda \ll \kappa$; outer tangential resistance dominates for $\kappa \ll \lambda$ while the plug approximation remains valid.

Calcium chloride is $\mathrm{CaC}{\mathrm{l}}_2$. One doubly charged calcium ion balances two singly charged chloride ions, a direct example of the stoichiometry required by electroneutrality.

It forms several hydrates and interacts strongly with water. Its hydration state affects dissolution energetics and the mass-to-amount conversion; dissolved-ion counts must also be distinguished from counts of solid formula units.

For positive stocks satisfying $d{S}_t^i/S_t^i={\sum }_j{\sigma }_{ij}d{W}_t^j+{\mu }_{i}dt$, let $q=n^{-1}1$ and $V=\sigma {\sigma }^T$. The Itô formula givesThus its volatility is ${\sigma }^{T}q$. This geometric average is generally not the value of the constant equal-weight self-financing portfolio: its instantaneous drift is $q^T\mu -\mathrm{tr}(V)/(2n)+q^{T}Vq/2$, whereas the fully invested equal-weight portfolio has drift $q^T\mu$.

A proportional continuous dividend payment at rate $\delta$ per unit time relative to the ex-dividend stock price. The total gain of one share is $d{S}_t+\delta S_{t}dt$. Under a risk-neutral measure with short rate $r$, a geometric stock diffusion therefore has ex-dividend drift $(r-\delta )S_t$. A fixed cash dividend at a specified date is a different convention and generally requires an ex-dividend price jump.