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Consider two dilute populations of spheres with the same buoyant density and radii $b$ and $2b$. Adopt the fixed linear density gradient and constant local centrifugal acceleration of linear-gradient isopycnic focusing, with constant dynamic viscosity and negligible diffusion. Stokes's law makes $V/f$ proportional to $b^2$, so doubling radius multiplies the migration rate parameter by four, although the mass increases eightfold.

Use $u=z/L$, $\tau =t/T$ and $q=({\rho }_b-{\rho }_0)/(GL)=2$. The liquid occupies $0\le u\le 1$, so the matching density position lies beyond its outer boundary. Both populations begin at $u(0)=0.1$; choose ${\lambda }_A=0.15$ and ${\lambda }_B=0.6$. Their band-center trajectories areAt $\tau =0.5$, they lie at approximately $0.2373$ and $0.5924$; at $\tau =1$, at $0.3647$ and $0.9573$. Equal buoyant density has not prevented a useful finite-time separation. Solving $u_B(\tau )=1$ using the natural logarithm gives first boundary arrival at $\tau =\ln (1.9)/0.6\simeq 1.0698$. The corresponding arrival for A is $\ln (1.9)/0.15\simeq 4.2790$. Trajectories describe migration only until contact with the boundary; subsequent collection into a centrifugation pellet is not represented by extending the curve beyond $u=1$.

This is the useful contrast with isopycnic centrifugation: stopping time helps determine the recovered positions in rate-zonal centrifugation, whereas a stationary matching density position sets the limiting band center in isopycnic centrifugation. Real resolution also depends on initial zone width, diffusion and variations in particle properties; distinct calculated centers alone do not prove that fractions will be pure.

Take the original sphere on the tube axis, with diagonal positive hydrodynamic resistance matrix $R_0$ and axial force $W>0$. Add a freely moving neutral sphere and extend its rigid velocity through its interior. The resulting field is admissible in the original domain, so its dissipation obeys $D>q^T{R}_{0}q\ge R_{zz}{V}_z^2$ for any finite nonoverlapping placement of the extra sphere. Since only the first sphere does external work, $D=W{V}_z$, giving the displayed bound without assuming zero transverse velocity or rotation. Strictness follows because equality would require the original nontrivial analytic flow to have zero strain throughout the added ball.

For a nonrotating force-free sphere moving with speed $U$ in a stationary tube, use its moving frame: the tube wall has velocity $-U$. In the parabolic lubrication gap $h=h_0+x^2/(2a)$, $h_0=ϵa(1+\lambda \cos \theta )$, moving-boundary lubrication flux gives $q=-h^3{p}_x/(12\mu )-Uh/2$. The wall-shear and pressure-drop balance in a tube rules out an $O({ϵ}^{-3/2})$ net pressure jump. Applying this common pressure recovery to each angular strip gives $q=-2U{h}_0/3$. The next balance yields the displayed positive localized pressure drop. Meanwhile the integrated leading shear stress on the sphere vanishes in each strip; this is consistent with negligible leading torque, and does not mean the local shear or pressure force vanishes.

For a smooth passing encounter of a force-driven sphere with a fixed sphere, isotropy and reflection symmetry imply $\mathbf{U}=[m_{\perp }(R)I+(m_{\parallel }(R)-m_{\perp }(R))\mathbf{nn}]\mathbf{F}$. Along a force axis $X$, $U_X$ is even and the transverse velocity is odd in $X$. Uniqueness then makes the trajectory fore-aft symmetric and restores the incoming impact parameter downstream. This consequence of kinematic reversibility of Stokes flow remains valid with lubrication resistance, provided the particles remain smooth, noncontacting and subject only to the stated forces. Surface roughness and contact can invalidate these assumptions.

Fourier's law predicts the local conductive heat flux from the local temperature gradient. In a continuum approximation with isotropic thermal conductivity $k>0$, its vector form and one-dimensional form areThe heat flux $q$ has units $\mathrm{W}{\mathrm{m}}^{-2}$, the temperature gradient has units $\mathrm{K}{\mathrm{m}}^{-1}$ and the thermal conductivity has units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$. The rate of heat transfer $\dot{Q}$ has units $\mathrm{W}$. These units distinguish a local material response from the total power crossing a surface.

The minus sign makes heat flow toward lower temperature: for $k>0$, $\mathbf{q}\cdot \nabla T=-k|\nabla T{|}^2\le 0$. In the worked bar in thermal conduction, $dT/dx=-200\mathrm{K}{\mathrm{m}}^{-1}$ therefore gives $q=+400\mathrm{W}{\mathrm{m}}^{-2}$. Reversing the end temperatures reverses the heat flux. A zero temperature gradient gives zero conductive heat flux in this model.

Fourier's law is a constitutive relation expressing measured material behavior; it is not a consequence of conservation of energy alone. It applies locally and does not require a steady state. The straight-line bar solution additionally assumes constant thermal conductivity, constant cross-sectional area, no sources or side losses, and fixed end temperatures. Anisotropic materials require the directional response described by the thermal conductivity tensor. The scalar $k$ used in the examples expresses the stated isotropy assumption. ^[ref]

Historically, Joseph Fourier's Théorie analytique de la chaleur was published in 1822. In Chapter II, article 112, he writes the heat crossing a spherical surface as the product of thermal conductivity, elapsed time, surface area and the negative radial temperature gradient; article 113 balances the incoming and outgoing heat to obtain the heat equation for a sphere. The enduring idea is this connection between a local flow rule and a global prediction of changing temperature. ^{[ref] [ref]}

If $X$ is an $(n-1)$-connected based CW complex, $n\ge 2$, suspension induces an isomorphism ${\pi }_k(X)\to {\pi }_{k+1}(\Sigma X)$ for $k\le 2n-2$ and a surjection for $k=2n-1$. For a sphere, $\Sigma S^n\cong S^{n+1}$, giving the stable range ${\pi }_k(S^n)\cong {\pi }_{k+1}(S^{n+1})$.

Rolling a sheet into a circular cylinder bends it without changing its internal lengths and angles. Fitting that same sheet smoothly onto a sphere requires distortion of its first fundamental form. Gaussian curvature distinguishes these situations: it measures the intrinsic curvature of a smooth surface, even though its simplest definition uses how the surface bends in Euclidean space.

The Gaussian curvature at a point is the determinant of the shape operator, or the product of the two principal curvatures:Here $I=Ed{u}^2+2Fdudv+Gd{v}^2$ is the first fundamental form and $II=ed{u}^2+2fdudv+gd{v}^2$ is the second fundamental form, with $S=-dN$. Regularity makes $EG-F^2>0$. Reversing the unit normal negates $e,f,g$ and both principal curvatures, leaving Gaussian curvature unchanged. ^[ref]

The sign describes bending at the point: positive Gaussian curvature means the principal curvatures have the same nonzero sign, as at an elliptic point; negative Gaussian curvature means opposite signs, as on a saddle; zero Gaussian curvature means at least one principal curvature vanishes. Zero Gaussian curvature at a single point does not imply that a neighborhood is a plane. ^[ref]

Four worked models make the signs concrete:

The patches use a common length unit, with $R=1$ for the sphere and circular cylinder. The marked red and blue curves have principal directions at the black point; the arrow is the chosen unit normal. For physical lengths, principal curvature and mean curvature have units of inverse length, while Gaussian curvature has units of inverse squared length. Rescaling a surface by $\lambda >0$ divides its principal curvatures by $\lambda$ and its Gaussian curvature by ${\lambda }^2$. Writing the saddle as $h=(x^2-y^2)/(2L)$ restores an explicit length scale: $K(0)=-1/L^2$.

The Theorema Egregium says that Gaussian curvature depends only on the first fundamental form and its first two coordinate derivatives. A local isometry therefore preserves Gaussian curvature, even if the shape operator and the principal curvatures change. ^[ref]

Here is the outline, rather than a full proof. The Gauss formula splits second derivatives of a parametric surface into tangential Christoffel symbols and the second fundamental form. The Christoffel symbols are determined by the first fundamental form and its first derivatives. Commuting the next partial derivatives gives the Gauss equation, which identifies $\det S$ with the intrinsic sectional curvature computed from the Riemann curvature tensor. Thus the apparently extrinsic product $k_1{k}_2$ is determined by the Riemannian metric. ^[ref]

One can see the distinction directly by rolling a plane strip with coordinates $(u,v)$ into the circular cylinder $X(u,v)=(R\cos (u/R),R\sin (u/R),v)$. Both have $I=d{u}^2+d{v}^2$, so the map is a local isometry, and is one-to-one on an open strip of width less than $2\pi R$. Extending $u$ over all real numbers wraps around periodically; it does not give a global isometry between the entire plane and the entire circular cylinder. A nonempty open patch of a sphere cannot be flattened by a local isometry into a plane, because its Gaussian curvature $1/R^2$ cannot become zero. The obstruction concerns preservation of the first fundamental form, rather than a particular drawing or choice of seam.

A Gaussian surface is a closed surface used with the integral form of Gauss's law to relate electric flux to enclosed electric charge. It need not coincide with the surface of any charged body. Choosing it to match a symmetry can make the normal component of the electric field constant on a portion of it, allowing that component to be taken outside a surface integral. A concentric sphere works for an isolated spherically symmetric charge distribution; an arbitrarily placed sphere generally does not.

Zero enclosed electric charge implies zero total electric flux. Inferring zero electric field additionally needs information such as the spherical symmetry used in the electric field of a uniformly charged spherical shell. The imagined boundary does not itself block or redirect the electric field.

Gauss's law relates the electric field on a closed surface to the net electric charge inside it. In SI units, with outward unit normal $\mathbf{n}$,Here $\rho$ describes a volume charge density, and ${\epsilon }_0$ is the vacuum permittivity; for ideal point or surface charges, use the enclosed electric charge directly rather than an ordinary volume-density integral. The left side is electric flux: the surface integral of the outward normal component of the electric field. Field directed outward contributes positively; field directed inward contributes negatively. The equality uses net signed electric charge, so positive and negative enclosed charges can cancel. The Gaussian surface is an imagined accounting boundary, not necessarily a material object. See OpenStax, explaining Gauss's law.

Where the electric field and charge density are smooth, the local statement is $\nabla \cdot \mathbf{E}=\rho /{\epsilon }_0$. The divergence theorem connects this divergence to the surface integral. This is the satisfying connection between local sources and global accounting: a surface can register how much net electric charge it encloses without revealing where that charge is located within it. A point charge requires care with the singular field at its position; the integral formulation remains useful with a boundary that avoids the charge.

The law fixes total electric flux, rather than every value of the electric field. In an uncharged region, a uniform electric field $\mathbf{E}=(E_0,0,0)$ crosses a cube of side $\ell$. Its left and right faces contribute $-E_0{\ell }^2$ and $+E_0{\ell }^2$; the other four faces contribute zero. Thus the net electric flux is zero while the electric field is nonzero everywhere. Charges outside a Gaussian surface can change the field on it without changing its net electric flux. Conversely, fields cannot generally be reconstructed from enclosed electric charge alone. Electrostatics also requires the electric field to be the negative gradient of an electric potential, with appropriate boundary information. This distinction is emphasized in Feynman lectures II.5, application of Gauss' law.

Spherical symmetry supplies the extra information needed for a particularly simple calculation. An isolated spherically symmetric charge distribution in vacuum produces a radial electric field, constant in magnitude on each concentric sphere. Its electric flux is therefore $4\pi r^2{E}_r(r)$, givingOutside the distribution, doubling $r$ quadruples the surface area and divides the radial electric field by four, while the electric flux stays $Q/{\epsilon }_0$. The inverse-square dependence is an area-dilution result. The electric field of a uniformly charged ball and electric field of a uniformly charged spherical shell show that the same total charge can give different interior fields. These symmetry arguments and their limits are developed in OpenStax, applying Gauss's law.

A geodesic is a curve whose tangent vector is parallel along itself under the Levi-Civita connection. With an affine parameter it has zero covariant acceleration:This is the Riemannian manifold version of moving straight ahead with constant speed. Its coordinate form is the geodesic equation. In Euclidean space it becomes $\ddot{\gamma }=0$, giving straight lines. On an embedded surface with its induced metric, the ordinary acceleration of such a curve is normal to the surface: all acceleration is used to follow the surface, rather than to turn within it.

On a unit round sphere, the great circle $\gamma (t)=(\cos t,\sin t,0)$ has $\ddot{\gamma }(t)=-\gamma (t)$, which is normal to the sphere. Two points at $t=0$ and $t=\pi /3$ can be joined along that circle by the short arc of length $\pi /3$ or the long arc of length $5\pi /3$. Each admits a constant-speed geodesic parametrization, yet only the short arc realizes their great-circle distance.

Every geodesic minimizes arc length on sufficiently short subarcs. A long segment can cease to be globally minimizing, as the sphere example shows; the first loss from a starting point is described by the cut locus. Thus “locally straight” and “shortest route between these endpoints” are distinct assertions. On a circular cylinder, different winding gives the same distinction, made explicit by Riemannian distance on a circular cylinder.

This provides a useful division of work. The geodesic equation predicts a path from an initial point and tangent vector; comparing arc lengths decides which candidate solves a global shortest-route problem. Variations of energy of a curve give the equation as an Euler-Lagrange equation, offering a proof outline and a connection with classical mechanics. ^[ref].

A geometric locus is the set of points satisfying a specified geometric or algebraic condition in an ambient space. For example, points at a fixed positive distance $r$ from $p$ form a circle in the plane and a sphere in three dimensions. In the complex plane, $|z-z_0|=r$ has Cartesian equation $(x-\mathrm{Re}{z}_0{)}^2+(y-\mathrm{Im}{z}_0{)}^2=r^2$, proving the circle description. A locus can also be a union, such as $x(y-1)=0$, which consists of two lines; domain restrictions in the original equation must be retained when simplifying it.