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For a sheet moving with $u(x,0)=\alpha x$ through otherwise stationary fluid, the layer thickness is $\delta =\sqrt{\nu /\alpha }$. Withthe boundary-layer equations reduce toThe exact solution is $f=1-e^{-\eta }$.

The twisted cubic is a nondegenerate projective variety, so every hypersurface containing it has degree at least two. At least two hypersurfaces are needed to cut out a curve in ${\mathbb{P}}^3$, making their degree product at least four rather than the curve's degree three.

A projective variety is nondegenerate when it is contained in no hyperplane of its ambient projective space.

Zeeman's theorem says that the complement of an $m$-twist spin fibers over the circle with fiber the punctured $m$-fold cyclic branched cover of the original knot. In particular, the $1$- and $-1$-twist spins are unknotted.

This identifies the complement structure of a twist-spun knot.

A closed-cover intersection statement for a simplex is equivalent to the nonexistence of a boundary-valued map that preserves every face.

Normalized distances to three closed sets give continuous barycentric coordinates; a covering forces at least one coordinate to vanish, placing the image on the simplex boundary.

An equilateral triangle has three equal side lengths.

An acute triangle has all three interior angles strictly smaller than a right angle.

An anti-trapped surface is the time reverse of a trapped surface: both future-directed orthogonal null expansions are positive.

A scalar conservation law in one space dimension has the form $u_t+{\partial }_{x}f(u)=0$. While a classical solution exists, it obeys the quasilinear transport equation $u_t+f^{\prime }(u)u_x=0$.

A linear transport equation has a prescribed velocity field and is linear in the transported unknown.

For the transport field $b(t,x)$, a characteristic curve solvesThe chain rule converts differentiation of $u(t,X(t))$ into the transport operator applied to $u$.

The Lorenz condition ${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0$ reduces the vacuum field equation to $\square {\bar{h}}_{\mu \nu }=0$. A plane-wave amplitude therefore satisfies $k^{\mu }{H}_{\mu \nu }=0$ and $k^{\mu }{k}_{\mu }=0$.

For the conserved scalar stress tensor above,

A toric variety $X_{\Sigma }$ is obtained by gluing the affine toric varieties associated with the cones of a fan $\Sigma$. It contains an algebraic torus as a dense open orbit.

A fan in toric geometry is a finite collection of rational strongly convex cones in toric geometry in a real vector space with a lattice, closed under taking faces of a polyhedral cone, such that the intersection of two cones is a face of each. A fan determines a toric variety.

A fan $\Sigma$ in $N_{\mathbb{R}}$ is a finite collection of strictly convex rational polyhedral cones closed under taking faces, such that the intersection of two cones is a face of each.

An algebraic torus of dimension $n$ is an algebraic group isomorphic over the base field to $({\mathbb{G}}_m{)}^n$. Its character and cocharacter lattices are dual free abelian groups $M$ and $N$.

A locally convex space is a topological vector space with a neighbourhood basis at zero consisting of convex sets. Equivalently, its topology is a seminorm topology generated by some family $\mathcal{P}$ of seminorms. We allow non-Hausdorff spaces here; a separating family of seminorms gives precisely the Hausdorff case. Every normed vector space is locally convex, but smooth convergence generally requires a whole family of measurements.

The geometric and measurement descriptions fit together: seminorm sublevel sets are convex, while the Minkowski functional of an open, convex zero neighbourhood invariant under all scalar multiplications with $|\lambda |\le 1$ is a continuous seminorm. Such neighbourhoods can be chosen as a neighbourhood basis. ^[ref].

Local convexity makes Hahn-Banach theorem and separation of a point and an open convex set available. It turns geometric distinctions into continuous linear functionals, connecting convex sets to the continuous dual space. For a seminorm topology generated by $\mathcal{P}$, a linear functional $\ell$ is continuous exactly when there are $p_1,\dots ,p_m\in \mathcal{P}$ and $C>0$ withThus one continuous linear functional needs only finitely many of the available measurements. For example, point evaluation of a derivative on the space of smooth functions on a compact interval is controlled by a suitable derivative seminorm.

Every finite-dimensional real vector space has a unique Hausdorff topology making it a topological vector space. Choosing a basis identifies it with ordinary Euclidean space; every linear map between such spaces is continuous.

The vertex of a double cone is not a surface point because its punctured neighborhood disconnects into two pieces.