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A hypersurface in an $n$-dimensional smooth manifold is a smooth submanifold of dimension $n-1$. Locally, a regular level set $\{\varphi =0\}$ with $\nabla \varphi \ne 0$ is a hypersurface whose normal vector is $\nabla \varphi$.

An embedded submanifold $M\subseteq N$ is a subset whose inclusion is an embedding. Around every point there are coordinates on $N$ in which $M$ is a coordinate plane.

An embedded submanifold is a submanifold for which the inclusion gives the subspace topology, unlike a merely immersed submanifold.

A smooth parametrized curve is regular when its velocity never vanishes.

Regularity is an additional differential condition on a smooth curve, not a synonym for an arbitrary curve.

Two nonzero functions with disjoint supports have identically zero pointwise product. Separated continuous tent functions or smooth bumps provide examples.

Normal frequencies satisfy $\det (\mathsf{K}-{\Omega }^2\mathsf{M})=0$, and each null vector gives the fixed amplitude ratio of a mode.

The Hessians of kinetic and potential energy at equilibrium give the symmetric mass matrix $\mathsf{M}$ and stiffness matrix $\mathsf{K}$.

For two masses hung in series from springs of natural length $\ell$, the lower equilibrium spring tension is $m_{2}g$ and the upper tension is $(m_1+m_2)g$. Hence their equilibrium lengths are

Angular frequency measures phase advance per unit time. For $x(t)=A\cos (\omega t+\phi )$ it is $\omega =2\pi /T$, where $T$ is the period.

At an equilibrium, a positive-definite Hessian matrix of the potential gives stable small oscillations. If a continuous symmetry supplies zero modes, positive definiteness on the directions transverse to the symmetry gives stability of the relative configuration.

Natural inflation uses a periodic potentialIts potential slow-roll parameter obeyson the branch where the tangent is positive.

The number of e-folds between field values ${\varphi }_i$ and ${\varphi }_f$ iswhen the field rolls monotonically down the potential.

Hilltop inflation begins near a local maximum of the inflaton potential. Fora positive field rolls away from the maximum at $\varphi =0$.

The potential slow-roll parameter isUnder the slow-roll equations, ${\dot{\varphi }}^2/(2V)\simeq {ϵ}_V/3$.

For $V(\varphi )=\lambda {\varphi }^4/4$ with $\lambda >0$, use natural units and the unreduced Planck mass $M_P=G^{-1/2}$. The slow-roll equations give $H\simeq \sqrt{2\pi \lambda /3}{\varphi }^2/M_P$ and $\dot{\varphi }\simeq -M_P\sqrt{\lambda /(6\pi )}\varphi$. Integrating gives the displayed exponential field decay, valid on either sign branch, andThe potential slow-roll parameters require ${\varphi }^2\gg M_P^2$. The expression is valid only during slow-roll inflation; its formal late-time saturation does not describe the subsequent reheating or radiation era.

For $V(\varphi )=m^2{\varphi }^2$ in units where $3H^2\simeq V$ and for positive $\varphi$,and

When $\beta N>\nu$ and $f>0$, the positive equilibrium isWriting $a=\beta I^{\ast }>0$, the reduced $(S,I)$ Jacobian has characteristic polynomialBoth roots have negative real part. Its discriminant is negative for sufficiently small $f$ and positive for sufficiently large $f$, giving respectively a stable focus and a stable node.

When $N>N_c=(\mu +\nu )/\beta$, the positive equilibrium isOn the invariant plane $X+Y+Z=N$, the $(X,Y)$ Jacobian at this equilibrium has negative trace $-(\mu +\beta Y^{\ast })$ and positive determinant $\beta Y^{\ast }(\mu +\nu )$, so both eigenvalues have negative real part and the equilibrium is locally asymptotically stable.

The singular values of a matrix measure stretching of Euclidean norm. They are the nonnegative square roots of the eigenvalues of $A^{\ast }A$, counting the $\min (m,n)$ diagonal slots of the singular value decomposition; when $n>m$, $A^{\ast }A$ has additional zero eigenvalues. Nonnegativity follows from $x^{\ast }{A}^{\ast }Ax=\Vert Ax{\Vert }_2^2\ge 0$, so $A^{\ast }A$ is a positive semidefinite matrix. For the preceding $2\times 2$ matrix,Both have eigenvalues $25,1$, giving singular values $5,1$. In particular,Here the operator norm uses Euclidean norm: no unit vector stretches more than fivefold. Zero singular values identify lost directions; the number of positive singular values equals the matrix rank. This summarizes both amplification and information loss with nonnegative numbers.^[ref]

Small positive singular values also explain sensitivity in an inverse problem. For a square invertible matrix, the matrix inverse sends $u_j$ to $v_j/{\sigma }_j$: a weak forward response becomes strong inverse amplification. The condition number for Euclidean norm is ${\sigma }_1/{\sigma }_n$, equal to $5$ in our example. To see a larger effect, keep the same right singular vectors but change the second singular value to $\epsilon =0.01$:The output perturbation has Euclidean norm $0.001$, but the recovered input changes by $0.1$. Its hundredfold amplification follows from the direction of the perturbation, not just its size. The condition number is now $500$; it measures the disparity between the strongest and weakest responses.^[ref]

A partial function is lambda-definable when one lambda term maps Church numerals in its domain to the numeral of the output and produces no numeral on inputs outside its domain.

At type $\sigma \to \sigma \to \sigma$, the Church Booleans are $\top =\lambda x.\lambda y.x$ and $\perp =\lambda x.\lambda y.y$.