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If a real algebraic number $\alpha$ has degree $d\ge 3$, then for every $\epsilon >0$ there are only finitely many reduced fractions $p/q$ such thatThe Thue-Siegel auxiliary polynomial proof chooses two such approximations, constructs an auxiliary polynomial nonzero at their pair, and compares its denominator lower bound with the upper bound supplied by its high-order zero at $(\alpha ,\alpha )$.

Let $F(X,Y)=P(X)+YQ(X)$ with $P,Q\in \mathbb{Z}[X]$ linearly independent, ${\mathrm{deg}}_{X}F\le n$, and $H(F)\le C^n$. For every $\delta >0$, if a reduced rational $p/q$ has sufficiently large denominator in terms of $C$ and $\delta$, then $F(X,y)$ has multiplicity at most $\delta n+1$ at $p/q$, for every fixed real $y$.

Indeed, the nonzero Wronskianhas degree below $2n$ and height at most $C_0^n$. A zero of multiplicity $r$ of $F(X,y)$ forces a zero of multiplicity at least $r-1$ of $W$. By Gauss lemma for polynomials, $(qX-p{)}^{r-1}$ then divides $W$ in $\mathbb{Z}[X]$, so $q^{r-1}$ divides the leading coefficient of $W$ and is at most $C_0^n$.

An isopycnal displacement is the vertical displacement $\xi$ of a constant-density surface. In linear stratified quasi-geostrophic flow,

A quasi-geostrophic vertical mode separates horizontal and vertical dependence of the streamfunction. With rigid horizontal boundaries and constant stratification, the vertical eigenfunctions are $\cos (n\pi z/H)$.

Canonical energy variance equals ${\partial }_{\beta }^2\log Z=k_B{T}^2{C}_V$.

Fourier's law predicts the local conductive heat flux from the local temperature gradient. In a continuum approximation with isotropic thermal conductivity $k>0$, its vector form and one-dimensional form areThe heat flux $q$ has units $\mathrm{W}{\mathrm{m}}^{-2}$, the temperature gradient has units $\mathrm{K}{\mathrm{m}}^{-1}$ and the thermal conductivity has units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$. The rate of heat transfer $\dot{Q}$ has units $\mathrm{W}$. These units distinguish a local material response from the total power crossing a surface.

The minus sign makes heat flow toward lower temperature: for $k>0$, $\mathbf{q}\cdot \nabla T=-k|\nabla T{|}^2\le 0$. In the worked bar in thermal conduction, $dT/dx=-200\mathrm{K}{\mathrm{m}}^{-1}$ therefore gives $q=+400\mathrm{W}{\mathrm{m}}^{-2}$. Reversing the end temperatures reverses the heat flux. A zero temperature gradient gives zero conductive heat flux in this model.

Fourier's law is a constitutive relation expressing measured material behavior; it is not a consequence of conservation of energy alone. It applies locally and does not require a steady state. The straight-line bar solution additionally assumes constant thermal conductivity, constant cross-sectional area, no sources or side losses, and fixed end temperatures. Anisotropic materials require the directional response described by the thermal conductivity tensor. The scalar $k$ used in the examples expresses the stated isotropy assumption. ^[ref]

Historically, Joseph Fourier's Théorie analytique de la chaleur was published in 1822. In Chapter II, article 112, he writes the heat crossing a spherical surface as the product of thermal conductivity, elapsed time, surface area and the negative radial temperature gradient; article 113 balances the incoming and outgoing heat to obtain the heat equation for a sphere. The enduring idea is this connection between a local flow rule and a global prediction of changing temperature. ^{[ref] [ref]}

The temperature gradient is the gradient $\nabla T$ of the temperature field. It points in the direction of fastest temperature increase.

Tensor product need not commute with an infinite direct product. For example,The element $(1\mathrm{mod}n{)}_n$ has infinite order in the product, so it survives rational localization.

For modules over a commutative ring, there is a natural isomorphismgiven by $f\mapsto (m\mapsto (n\mapsto f(m\otimes n)))$.

Every balanced map $b:M\times N\to P$ factors uniquely through an $R$-module homomorphism $\tilde{b}:M{\otimes }_{R}N\to P$ satisfying $\tilde{b}(m\otimes n)=b(m,n)$.

A map $b:M\times N\to P$ is balanced when it is additive in each variable and satisfies $b(rm,n)=b(m,rn)$.

The symmetric square is the quotient of $V\otimes V$ by $v\otimes w-w\otimes v$, and records symmetric tensors of rank two.

The exterior square is the quotient of $V\otimes V$ by tensors $v\otimes v$, and records antisymmetric tensors of rank two.

Einstein notation sums automatically over an index repeated once up and once down, or twice in Euclidean coordinates.

An isotropic tensor is invariant under every proper orthogonal change of basis. A rank-four isotropic tensor has the formRank-five isotropic pseudotensors are formed by multiplying one Kronecker delta by one Levi-Civita symbol and permuting the five indices.

A tensor changes sign whenever two indices are interchanged. In three dimensions, a rank-two antisymmetric tensor is $A_{ij}=a_k{ϵ}_{kij}$, a rank-three one is $A_{ijk}=a{ϵ}_{ijk}$, and every totally antisymmetric tensor of rank greater than three vanishes.

A second-rank Cartesian tensor is antisymmetric when its components form a skew-symmetric matrix, $T_{ij}=-T_{ji}$. Orthogonal coordinate changes preserve this property because $T^{\prime }=RT{R}^T$ implies $(T^{\prime }{)}^T=R{T}^T{R}^T=-T^{\prime }$.

A covariant tensor is multilinear in vector arguments and transforms with one covariant index for each argument.

A real distribution $u$ is positive when $u[\varphi ]\ge 0$ for every nonnegative test function $\varphi$. Every positive distribution has order zero and is represented locally by a positive measure.

For a function with derivatives of every order at $a$, its Taylor series is the power seriesIts partial sums are Taylor polynomials. At $a=0$ it is called a Maclaurin series. Forming the coefficients, proving that its partial sums form a convergent sequence, and proving that their limit equals $f(x)$ are separate tasks. A smooth function provides all the coefficients, but a real analytic function is additionally represented locally by the resulting power series.

For the exponential function, every derivative of $e^x$ at zero equals one, givingHere equality holds for every real $x$: the Taylor remainder is bounded by $e^{|x|}|x{|}^{N+1}/(N+1)!$, which tends to zero for each fixed $x$. At $x=1$, the degree-five Taylor polynomial is $163/60\approx 2.71666667$, and the absolute difference from $e$ is less than $e/720$. The factorial denominator eventually overwhelms every fixed power, allowing local derivative information to reconstruct this function throughout the real line.

The geometric series shows why the region of validity matters. For $f(x)=1/(1-x)$, the Taylor series at zero is ${\sum }_{k\ge 0}{x}^k$, with radius of convergence one. For $x\ne 1$, its finite Taylor polynomial satisfies the exact identityAt $x=1/2$ and $N=5$, the remainder is $1/32$. At $x=0.9$, the same degree-five polynomial has value $4.68559$ and remainder $5.31441$, although the Taylor series still converges to $10$ there. Convergence of the infinite expression does not guarantee accuracy from a particular small number of terms. For $|x|>1$, even the terms fail to tend to zero, so this power series cannot represent the function. The standard expansions and the conditions for convergent series are discussed in OpenStax, working with Taylor series.

A stronger caution comes from the flat functionIt is a smooth function with $h^{(n)}(0)=0$ for every $n$, so every Taylor polynomial at zero is zero and its Taylor series converges everywhere to zero. Nevertheless $h(1/2)=e^{-4}>0$. Off zero, repeated differentiation produces a polynomial in $1/x$ times $e^{-1/x^2}$; the exponential decay makes each required limit at zero vanish. This explains how all the local coefficients can miss a positive function arbitrarily close to the centre. Real analyticity supplies the additional condition that rules out this behavior.