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For a thin disk in the harmonic vertical gravitational potential ${\Omega }_K^2{z}^2/2$, hydrostatic equilibrium and $p=c_s^2\rho$ with height-independent isothermal sound speed give the displayed Gaussian density. For perturbations with adiabatic index $\gamma$, the Brunt–Väisälä frequency is $N^2=(1-1/\gamma ){\Omega }_K^2{z}^2/H^2$. An isothermal background can therefore support buoyancy for adiabatic perturbations; isothermal perturbations instead have $\gamma =1$ and zero buoyancy frequency.

In a reflection-symmetric astrophysical disk, the vertical mass coordinate measures mass per unit area above the midplane. It satisfies $dm=\rho dz$, so ${\partial }_m={\rho }^{-1}{\partial }_z$ inside the material. If the total surface density is $\Sigma$, the upper face is $m=\Sigma /2$; the normalized coordinate $\zeta =2m/\Sigma$ ranges from zero to one. Integrals weighted by mass density become ordinary integrals in $m$, which is especially useful for radiative diffusion and the density-weighted mean kinematic viscosity.

For $n\ge 1$, a family of $2^{n-1}$ vertices in the Boolean hypercube has these minimum and maximum external vertex boundary sizes. The Harper theorem gives the minimum: in odd dimension take the lower half of the levels, and in even dimension add the middle-level sets containing coordinate one to all lower levels. The family of even Hamming weight attains the maximum, since every remaining vertex has an adjacent even-weight vertex.

For a specified set $S$ of vertices, a rooted separation is a pair $(A,B)$ with $A\cup B=V(G)$, $S\subseteq A$, and no edges between $A\setminus B$ and $B\setminus A$. Its order is $|A\cap B|$. It avoids a branch set $C$ when $A\cap C=\varnothing$. A graph with vertex connectivity at least $|S|$ has no rooted separation of order below $|S|$ avoiding a nonempty branch set: both sides of such a separation would be nonempty after deleting its separator.

A proper vector subspace of a vector space is one strictly smaller than the ambient space. In a finite-dimensional vector space, this is equivalent to having smaller dimension: a basis of the subspace can be extended to a basis of the ambient space, and strict inclusion requires at least one additional vector.

A connection is a linear map $\nabla :\Gamma (E)\to {\Omega }^1(B;E)$ satisfying $\nabla (fs)=df\otimes s+f\nabla s$. In a local frame it has the form $d+A$ for a matrix-valued one-form $A$.

A Clifford module bundle is a vector bundle over a Riemannian manifold with a fibrewise Clifford multiplication. A compatible connection on a vector bundle satisfies $[{\nabla }_X,c(\alpha )]=c({\nabla }_X\alpha )$ for the Levi-Civita connection on covectors. On a graded Clifford module bundle, Clifford multiplication exchanges the two grading summands. The associated Dirac operator is $D={\sum }_{i}c(e^i){\nabla }_{e_i}$. Its fibres are modules over the fibres of the Clifford algebra bundle.

A vector bundle over $S^n$ can be constructed by gluing two trivial bundles over its hemispheres with a transition map $\gamma :S^{n-1}\to GL(k,\mathbb{R})$. Extend this map constantly in an equatorial collar to obtain local vector bundle trivializations. A transition map into the special orthogonal group gives compatible fiber orientations and a fiber metric. Homotopies of transition maps give isomorphic bundles.

For a rank-$k$ smooth vector bundle, take the exterior power ${\Lambda }^r{E}_p$ in each fibre. A local frame gives a trivialization with fibre ${\Lambda }^r{\mathbb{R}}^k$, and a transition matrix $A$ acts by ${\Lambda }^{r}A$. Functoriality preserves the cocycle identities, producing a smooth vector bundle of rank $\left(\genfrac{}{}{0ex}{}{k}{r}\right)$, or the zero bundle when $r>k$. Its top exterior power is the determinant line bundle.

The vector bundle whose fibre at $p$ consists of linear maps $E_p\to E_p$. If a local frame changes by $e^{\prime }=eg$, the coefficient matrix of an endomorphism changes by $T^{\prime }=g^{-1}Tg$, giving smooth fibre-linear transitions. For a line bundle this bundle is canonically the trivial scalar bundle, because every fibre endomorphism is scalar multiplication. A connection on a vector bundle induces the endomorphism bundle connection.

A framing is a continuous ordered basis in every fiber, equivalently a trivialization of the vector bundle. A stable framing trivializes the bundle after adding trivial real summands, up to further stabilization and homotopy. For a smooth map it is the stable normal data that is framed, not necessarily the tangent bundle of either manifold.

Every vector bundle on the projective line over a field splits as a direct sum of line bundles. Choose the largest integer $a$ with $H^0(E(-a))\ne 0$. A corresponding map $\mathcal{O}(a)\to E$ is saturated, since saturation otherwise increases its degree. Inductively split its quotient as ${⨁}_i\mathcal{O}(b_i)$. Twisting the exact sequence by $-a-1$ and using $H^1(\mathcal{O}(-1))=0$ gives $b_i\le a$. Hence all extension groups $H^1(\mathcal{O}(a-b_i))$ vanish, and the sequence splits. The multiset of degrees is determined by the dimensions of all twisted global-section spaces.

A line subbundle is an injective map from a line bundle into a vector bundle whose quotient is locally free. Equivalently the injection splits locally. On a smooth algebraic curve, saturation of a rank-one coherent subsheaf produces a saturated line subbundle on a smooth curve.

Choose a fiber metric on a real rank-$r$ vector bundle $E\to B$. Its Thom space is the pointed quotient spaceobtained by collapsing the entire sphere bundle to one basepoint. For a compact Hausdorff space $B$, it is also the one-point compactification of the total space of $E$. A single fiber becomes $D^r/S^{r-1}\cong S^r$.

For the trivial vector bundle $B\times {\mathbb{R}}^r$, the quotient is $B_{+}\wedge S^r$, where $B_{+}$ adjoins a disjoint basepoint and $\wedge$ denotes the smash product. In rank zero the convention is $D(E)/\varnothing =B_{+}$; this keeps the basepoint even though the sphere bundle is empty.

For an oriented rank-$r$ vector bundle over a CW complex, its Thom class restricts to the positive generator on every fiber sphere. Multiplication by that class gives the Thom isomorphismThus the Thom space packages the degree shift of cohomology into one pointed space.^[ref]

A symplectic vector bundle has a smooth fiberwise symplectic form, making each fiber a symplectic vector space. Its structure group reduces to the symplectic group. A symplectic normal bundle is an important example.

The fiberwise direct sum of two vector bundles over the same base. It is formed from the fiber product of their total spaces, with the direct-sum linear structure in each fiber. Transition matrices are block diagonal.

// Target: geometry-and-topology.bigb

A vector bundle is globally generated when its global sections span every fibre. The evaluation map in the display expresses this condition. On a projective curve its space of global sections is finite-dimensional, so a finite collection of sections suffices.

A G-structure on a real rank-$m$ vector bundle is a principal $G$ reduction of its frame bundle. Equivalently it has local frames whose transition matrices take values in $G$, with the reduction consisting of their right $G$-orbits. An orthogonal structure on a real vector bundle is equivalent to a fiber metric; a special orthogonal reduction is equivalent to that metric together with an orientation of a vector bundle.

For a smooth manifold $M$, the tensor bundle $T_s^{r}M=(TM{)}^{\otimes r}\otimes (T^{\ast }M{)}^{\otimes s}$ is a vector bundle whose smooth sections of a vector bundle are tensor fields of type $(r,s)$. Its transition functions of a vector bundle are tensor products of the tangent bundle transition matrices and their inverse transposes. Symmetric tensors and differential forms arise from the symmetric and alternating subbundles.

// Target: fiber-bundle.bigb

On an $n$-dimensional smooth manifold, the density bundle $|{\Lambda }^n{T}^{\ast }M|$ is the real line bundle whose coordinate transformation uses the absolute value of the Jacobian determinant. A density can therefore be integrated without an orientation of a smooth manifold. A Riemannian metric gives the positive density $\sqrt{\det (g_{ij})}|d{x}^1\cdots d{x}^n|$, including on a nonorientable manifold.

// Target: fiber-bundle.bigb