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Orthogonal coordinates have pairwise orthogonal coordinate tangent directions. Their metric is diagonal, with scale factors of orthogonal coordinates $h_i=|\partial \mathbf{x}/\partial q_i|$. Spherical coordinates and cylindrical coordinates are examples.

In the open first quadrant this is one-to-one, with $x=\sqrt{(u+v)/2}$, $y=\sqrt{(u-v)/2}$ and Jacobian determinant $-8xy$. Its image satisfies $u>v$ and $u+v>0$. In particular $1\le v\le u\le U$ is a triangular image for a circular-hyperbolic region. A factor $xy$ in the original integrand cancels the inverse-Jacobian factor $1/(8xy)$; moreover $x^4-y^4=uv$ and $4x^2{y}^2=u^2-v^2$. Boundary degeneracy on $y=0$ is confined to a measure-zero set and must not be confused with an interior failure of invertibility.

On the positive quadrant, the inverse coordinate change is $x=\sqrt{uv}$, $y=\sqrt{v/u}$. Its Jacobian determinant is $1/(2u)>0$. Regions bounded by positive constant-ratio lines and constant-product hyperbolas become rectangles in these coordinates. An integrand $x/y=u$ then cancels the Jacobian factor, making weighted area integrals particularly simple. The positivity restriction selects a unique square-root branch and fixes orientation.

On $x>0$, $0<y<1$, $0<z<x$, this change of variables is a bijection onto $0<\gamma <\beta <\alpha$. Its inverse is $x=\sqrt{\alpha \beta }$, $y=\sqrt{\beta /\alpha }$, $z=\gamma \sqrt{\alpha /\beta }$, and its forward Jacobian determinant is $2x$. Thus $xdxdydz=\frac{1}{2}d\alpha d\beta d\gamma$. Passing to increments $(\alpha -\beta ,\beta -\gamma ,\gamma )$ then maps the ordered cone to the positive octant, turning an exponential with linear exponent into a product of three one-dimensional integrals.

With thermal diffusivity $\kappa$ and magnetic diffusivity $\eta$, the standard Chandrasekhar number uses $\nu \eta$ in its denominator. A thermal-diffusion alternative uses $\nu \kappa$ and is $\zeta =\eta /\kappa$ times the standard parameter. In thermal-time magnetoconvection equations, the linear magnetic-force coefficient is $\sigma Q_{\kappa }=\sigma \zeta Q_{\mathrm{Ch}}$. Specifying the convention prevents a missing diffusivity factor in a dispersion relation.

An ultrarelativistic degenerate electron gas has $P=K{\rho }^{4/3}$, so it forms an index-three stellar polytrope. Insert $P_c=c_{3}G{M}^2/R^4$ and ${\rho }_c=3a_{3}M/(4\pi R^3)$ into that equation of state: the radius cancels, leaving the displayed mass. For fully ionized helium, $n_e=\rho /(2m_p)$ and $K=(3/(2\pi ){)}^{1/3}hc/(16m_p^{4/3})$. This is the ideal limiting Chandrasekhar mass, rather than a claim that every ordinary white dwarf has that mass.

For differentiable maps $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ and $g:{\mathbb{R}}^m\to {\mathbb{R}}^r$, the multivariable chain rule multiplies their Jacobian matrices:The outer $r\times m$ matrix multiplies the inner $m\times n$ matrix, producing an $r\times n$ matrix. This order records that the inner linear map acts first. Componentwise, the matrix multiplication sums the contributions of all intermediate coordinates. Existence of separate partial derivatives alone is not the required differentiability hypothesis.

A linear approximation replaces a differentiable function near a reference input by its value plus the first-order change supplied by its derivative. In one dimension, this is the tangent line $f(a)+f^{\prime }(a)(x-a)$. The chain rule combines such local approximations through function composition; it does not assert their equality to the original functions over a finite interval.

For uniformly spaced abscissae, quarter and three-quarter averages applied to constant or affine samples leave the same polynomial. For $P(x)=a{x}^2+bx+c$ on spacing $h$, both new subgrids lie on $P(x)+3a{h}^2/16$. For degree $n\ge 3$, their candidate polynomial formulas differ with leading term $(h^3/32)P^{\prime \prime \prime }$, which has a nonzero highest-degree coefficient. Thus preservation of polynomial degree for arbitrary sufficiently long data sequences holds exactly for degrees zero, one and two, despite quadratic values changing.

This centered quadratic cardinal B-spline weights each original control ordinate at its uniformly spaced abscissa. Its translates sum to one. It and its first derivative match at half-integer knots, while the second derivative can jump. The limit graph is ${\sum }_i{y}_i\beta ((x-x_i)/h)$; one ordinate influences an interval of total length $3h$. Refinement of a quadratic adds $a{h}^2/4$ in the limit, yielding this basis by extending each three-point neighborhood quadratically.

For a finitely presented base group $G$ and subgroup $H$, the extension $⟨G,t\mid [t,h]=1(h\in H)⟩$ is finitely presented exactly when $H$ is finitely generated. Sufficiency follows by imposing commutation with finitely many subgroup generators. For necessity, any finite set normally generating the defining kernel uses only finitely many centralizing relators. If their subgroup $H_0$ is proper in $H$, Britton's lemma makes $t^{-1}ht{h}^{-1}$ nonidentity in the partial extension for $h\in H\setminus H_0$, contradicting the claimed kernel equality.

// Target: algebra.bigb

For a length-$r$ central series $1=G_0\le \cdots \le G_r=G$, induction from the top gives ${\Gamma }_{r-i+1}(G)\le G_i$, and induction from the bottom gives $G_i\le Z_i(G)$. Consequently the lower central series terminates at ${\Gamma }_{c+1}=1$ exactly when the upper central series reaches $Z_c=G$.

// Target: algebra.bigb

Forward time with centered advection has amplification magnitude greater than one at nonzero modes. Under a fixed nonzero Courant number, its powers blow up as the time step vanishes. If instead $\Delta t/h^2$ stays bounded, then $\log |G|\le c^2\Delta t^2/(2h^2)$, so $|G{|}^n\le \exp (c^{2}T\Delta t/(2h^2))$ is uniformly bounded on fixed time intervals. This severe parabolic refinement supplies stability without contraction; it does not validate the usual hyperbolic-step choice.

For $f\in C^3[x-h,x+h]$, the Peano kernel for the error of the central secant slope has one sign. Integrating its absolute value gives the displayed constant $h^2/6$, and a cubic polynomial attains it. At $x=1$, $h=1$, the unnormalized order-two kernel is $-{\theta }^2/2$ on $[0,1]$ and $-(2-\theta {)}^2/2$ on $[1,2]$; the integral representation includes the factor $1/2!$.

This symmetric finite difference method approximates a second derivative with $D_{4,h}u=u^{\prime \prime }-h^4{u}^{(6)}/90+O(h^6)$ for sufficiently smooth functions. Matching the constant, second-derivative and fourth-derivative coefficients of a symmetric five-node Taylor expansion determines the displayed weights uniquely; its sixth-derivative coefficient is nonzero. The Fourier symbol is $-4h^{-2}{\sin }^2(\theta /2)[1+{\sin }^2(\theta /2)/3]\le 0$. Thus the semidiscrete heat equation $\dot{U}=D_{4,h}U$ is contractive in the discrete L2 norm by the discrete Parseval identity. This says nothing by itself about stability of a subsequently chosen time update.

Collapse two distinct parallel circles of a torus to two separate points. Cutting between the circles gives two annuli, each of which becomes a two-sphere with the two collapsed points as its poles. The resulting CW complex has two vertices, two edges connecting them, and two two-cells with zero cellular boundary. Its homology is as displayed, with all higher groups zero; it is homotopy equivalent to $S^1\vee S^2\vee S^2$.

The product cells of finite CW complexes give the tensor product of their cellular chain complexes. The boundary formula is the displayed signed product rule. The sign comes from moving the outward normal of the second factor past the $p$ oriented tangent directions of the first factor. This is the cellular counterpart of the Eilenberg–Zilber theorem.

A cellular chain whose cellular boundary is zero.

A finite linear combination, with coefficients in $R$, of the oriented $n$-cells of a CW complex.

For a CW complex, the qth cellular chain group is the relative homology group of successive skeleta, a free abelian group with one generator for each q-cell. The attaching maps determine the cellular boundary, and the resulting cellular chain complex computes singular homology.