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For an open set $U\subseteq {\mathbb{R}}^n$ and $1\le p\le \infty$, the first-order Sobolev space iswhere each $D_{i}u$ is a weak derivative. Its standard norm is $\Vert u{\Vert }_{W^{1,p}}=\Vert u{\Vert }_p+{\sum }_i\Vert D_{i}u{\Vert }_p$.

A smooth manifold is orientable when it admits an atlas whose transition maps have positive Jacobian determinant wherever they are defined.

This atlas criterion expresses orientability for a differentiable manifold.

Every open cover of a smooth paracompact manifold admits a smooth partition of unity subordinate to it: nonnegative smooth functions with locally finite supports in the cover whose sum is one.

A map between smooth manifolds is smooth when its expression in every pair of coordinate charts is a smooth function.

A map between smooth manifolds is smooth when its coordinate expressions in compatible charts have derivatives of every order. The condition is invariant under smooth chart changes and expresses smoothness in charts.

An abstract smooth surface is a two-dimensional smooth manifold. Thus it is a Hausdorff second-countable space with an atlas of plane-valued coordinate charts whose transition maps are smooth.

A smooth hypersurface in an $n$-dimensional smooth manifold is an embedded submanifold of codimension one. Locally it can be written as $\{\varphi =0\}$ withat every point of that zero set. The regular level set theorem gives this local description, and its tangent space is $\ker d{\varphi }_p$.

The covector $d\varphi$ is defined without a metric. If the ambient manifold has a Riemannian metric, its metric dual gradient $\nabla \varphi$ is a normal vector, since $g(\nabla \varphi ,v)=d\varphi (v)=0$ for every tangent $v$. A pseudo-Riemannian metric gives the analogous normal, which can be null. A global scalar defining function need not exist, as the normal-bundle obstruction to being a regular level set shows.

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.

For a monotonic slow-roll trajectory, the positive number of e-folds accumulated between initial and final field values isThis follows by inserting $\dot{\varphi }\simeq -V^{\prime }/(3H)$ into $dt=d\varphi /\dot{\varphi }$ and using the slow-roll equations. The sign of $V^{\prime }$ fixes the direction of the field's motion. On a branch with $V^{\prime }>0$ and ${\varphi }_i>{\varphi }_f$, the potential slow-roll parameter gives the equivalent formFor $V^{\prime }<0$, the field instead increases and the limits of this positive square-root form must be reversed. The signed $V/V^{\prime }$ formula treats both cases directly. It requires a rolling branch without an intervening zero of $V^{\prime }$ where the change of variables fails, and it estimates the evolution only while the slow-roll approximation applies.

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