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A static fluid in a uniform downward gravitational field obeys $dp/dz=-\rho g$.

For constant gravitational acceleration and temperature,

For a hyperbolic triangle, $\sinh a/\sin \alpha =\sinh b/\sin \beta =\sinh c/\sin \gamma$.

A hyperbolic triangle is bounded by three hyperbolic geodesics and may have vertices on the ideal boundary.

A Lambert quadrilateral is a hyperbolic quadrilateral with three right angles. If its remaining angle is $\theta$ and the two sides incident with that angle have lengths $a,b$, then

The hyperboloid model of the curvature-minus-one hyperbolic plane isIts metric is the restriction of the Minkowski bilinear form to tangent planes, and its geodesics are intersections with two-dimensional linear subspaces through the origin.

The map $z\mapsto i(1+z)/(1-z)$ is an isometry from the Poincare disc to the upper half-plane.

The upper half-plane carries the hyperbolic metric $(d{x}^2+d{y}^2)/y^2$.

The hyperbolic plane is the simply connected two-dimensional geometry of constant curvature minus one.

If every Jacobian eigenvalue has negative real part, the equilibrium is locally asymptotically stable.

The $n=2$ eigenspace has basis

For the $z$ component of position,requires $\Delta m=0$ and $\Delta \ell =\pm 1$. The first condition follows from $[L_z,z]=0$; the second follows from the vector-operator angular-momentum relation together with odd parity.

Forwith the boundary conditions $y(0)=y^{\prime }(0)=0$, $y^{\prime \prime }(L)=0$, and $-A{y}^{\prime \prime \prime }(L)=F$, the higher-order Euler-Lagrange equation isIts solution splits as

If a time-dependent perturbation remains diagonal in a fixed eigenbasis, Hamiltonians at different times commute. Each component therefore acquires the exact phase $\exp [-i\int E_{\pm }(t)dt/\hslash ]$, with no time ordering required.

Two pure states can be distinguished with certainty in one measurement exactly when their inner product is zero.

For ${\rho }_j=|{\alpha }_j⟩⟨{\alpha }_j|$, projecting onto the positive and negative eigenspaces of ${\rho }_0-{\rho }_1$ attains

The maximal averaging operator has weak type $(1,1)$, controlling the measure of points where local averages are large.

Liouville integrability means having $n$ almost-everywhere independent first integrals in mutual Poisson involution.

For a Hamiltonian phase space of dimension $2n$, Liouville integrability means $n$ functionally independent conserved quantities with pairwise vanishing Poisson brackets. This is the Hamiltonian specialization of integrable systems.

A canonical transformation preserves the symplectic form and hence all Poisson brackets. In one degree of freedom, an invertible map $(q,p)\mapsto (Q,P)$ is canonical exactly when $\{Q,P{\}}_{q,p}=1$.

For a functional $H[u]$, its variational derivative is defined byafter integrating derivatives of $v$ by parts and discarding the boundary terms.