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For a smooth real function satisfying the displayed equation with positive integer $r$, differentiating makes $f^{(j+r)}=f^{(j)}$. On any compact interval, all derivatives are therefore bounded by the maximum of the bounds of $f,f^{\prime },\dots ,f^{(r-1)}$. The Taylor formula with integral remainder consequently has error at most $M{R}^{n+1}/(n+1)!$, which tends to zero uniformly for $|x|\le R$. Hence the function equals its globally convergent Taylor series. For $r=3$ and initial derivatives $(1,0,0)$, it is ${\sum }_{k\ge 0}{x}^{3k}/(3k)!$.

A Taylor polynomial replaces a function near a chosen point by a polynomial with matching derivatives. If $f$ has derivatives through order $n$ at $a$, defineThe factorial denominator makes $T_{n,a}^{(j)}(a)=f^{(j)}(a)$ for $0\le j\le n$. This determines a unique polynomial of degree of a polynomial at most $n$. Order zero matches the value, order one also matches the tangent line, and order two also incorporates the second derivative. Matching more local information is useful because polynomials are easy to evaluate, differentiate, and integrate. The distinction between the finite Taylor polynomial and the infinite Taylor series is developed in OpenStax, Taylor and Maclaurin series.

For a computed example, let $f(x)=\sqrt{1+x}$ and expand at zero. Its first three derivatives giveAt $x=0.1$, the linear approximation is $1.05$ and the quadratic approximation is $1.04875$, compared with $\sqrt{1.1}\approx 1.048808848$. The quadratic correction captures the downward bending of the square root and reduces the absolute difference from about $0.00119115$ to $0.0000588482$.

The Taylor remainder turns this convenient approximation into a controlled calculation. On $[0,0.1]$, $f^{\prime \prime \prime }(x)=3/[8(1+x{)}^{5/2}]$ is positive and at most $3/8$. The Taylor theorem with Lagrange remainder therefore givesConsequently $1.04875<\sqrt{1.1}\le 1.0488125$ follows without using the displayed decimal evaluation as evidence. The Taylor polynomial and the Taylor remainder play different roles: the former supplies an answer that is easy to compute, while the latter specifies what accuracy is justified. Farther from the expansion point, the same low-degree polynomial need not remain accurate.

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.

Taylor's theorem expresses a sufficiently differentiable function as a finite Taylor polynomial plus a Taylor remainder. One convenient version assumes $f$ is $n+1$ times continuously differentiable on the segment between $a$ and $x$. Thenfor some intermediate point $\xi$, when $x\ne a$. Thus a bound $|f^{(n+1)}|\le M$ on that segment gives $|R_{n,a}(x)|\le M|x-a{|}^{n+1}/(n+1)!$. The intermediate point usually need not be computed: bounding the derivative throughout the segment is enough. This is why local approximation can yield reliable numerical bounds. A finite Taylor polynomial does not require an infinite Taylor series to represent the function.

Historically, the Royal Society holds Brook Taylor's letter of 26 July 1712 to John Machin. Its catalogue connects his work on Kepler's second law with the expansion later published in the 1715 Methodus incrementorum directa et inversa. This is a documented route from an orbital calculation to a general approximation method, rather than a claim that every earlier use of such expansions began with Taylor. See Royal Society archive, letter MS/82/2.

Nonnegative tensor-product surface basis factors yield nonnegative product weights. Their two partition of unity sums multiply to one. If the factors are $C^r$ and $C^s$, mixed derivatives through separate orders $r,s$ are continuous, so the resulting finite or locally finite surface representation is jointly $C^{\min (r,s)}$. Control coefficients can cancel derivative jumps and yield more smoothness; inherited continuity is a guaranteed lower bound, not a claim that every surface has exactly the basis's worst regularity.

Products of two univariate bases form a surface basis. Nonnegativity, partition of unity and coordinate-wise linear precision pass to the products by multiplication of the respective univariate identities. If the factors are $C^r$ and $C^s$, each mixed derivative through orders $r,s$ is the product of the corresponding derivatives, giving continuous mixed derivatives and joint $C^{\min (r,s)}$ regularity.

Consider a point mass of fixed mass $M>0$ released from rest in a stationary medium. Choose downward as positive, assume constant gravitational acceleration $g>0$, neglect buoyancy, and impose linear drag $F_d=-bv$ with constant linear friction coefficient $b>0$. Here $v$ is the signed downward velocity component; it equals speed during this descent. Newton's second law becomesThe terminal velocity balances the two forces, and the time constant $\tau$ controls how quickly that balance is approached:For the illustrative parameters $M=1\mathrm{kg}$, $b=2kg/s$ and $g=9.8m/{\mathrm{s}}^2$, the time constant is $0.5\mathrm{s}$ and the terminal velocity is $4.9m/s$ downward. At $t=1\mathrm{s}=2\tau$, the downward velocity is $4.9(1-e^{-2})m/s\simeq 4.23686m/s$ and the downward acceleration is $9.8e^{-2}m/{\mathrm{s}}^2\simeq 1.32629m/{\mathrm{s}}^2$. This is a toy linear drag model, not a claim that every $1\mathrm{kg}$ falling object has this resistance law.

For a proof outline, subtract the equilibrium value: $u=v-v_T$ obeys $du/dt=-u/\tau$. The exponential decay solution and the initial condition give $u=-v_T{e}^{-t/\tau }$; one derivative gives the acceleration. Every time constant multiplies the remaining gap $v_T-v$ by $e^{-1}$. The exact limit is reached only as $t\to \infty$. This makes the model useful for estimating a settling duration without mistaking vanishing acceleration for vanishing velocity.

Newton already analyzed vertical motion under uniform gravity and resistance proportional to velocity in Book II, Section I, Proposition III of the Principia. His geometric construction includes a greatest downward velocity, illustrating that a limiting rate can emerge from balancing a driving force and resistance. ^[ref]

The Normal convergence of the Weierstrass elliptic-function series allows its derivative to be written as an absolutely locally uniformly convergent sum of inverse cubes. For a period lattice $\Lambda$, this derivative sum is invariant under translation by any lattice vector, by reindexing. Consequently $\wp (z+\omega )-\wp (z)$ is constant; its apparent poles cancel. The Weierstrass elliptic function is even, so for a primitive lattice generator $\omega$, evaluation at $z=-\omega /2$ makes that constant zero. Applying this to two generators proves that this is a periodic function, namely $\wp$. This argument avoids separating the original convergent series into two divergent inverse-square sums.

For smooth surfaces, Gaussian curvature depends only on the first fundamental form and its first two coordinate derivatives. Consequently any smooth local isometry $f:M\to N$ satisfies $K_N(f(p))=K_M(p)$. The surprise is that $K$ is defined using the shape operator, which describes bending in space, yet its value can be recovered from measurements made entirely within the surface.

In his 1827 paper, Carl Friedrich Gauss expressed Gaussian curvature through the metric coefficients and their partial derivatives in Article 11, then stated its invariance under development in Article 12. Article 13 explicitly compares a plane with surfaces developable onto it, including cylindrical surfaces. The paper was presented on 8 October 1827 and printed in 1828; the Morehead–Hiltebeitel English translation appeared in 1902. ^[ref]

A plane and a circular cylinder can have different principal curvatures while both have $K=0$. Their matching first fundamental forms exhibit the actual local isometry. In contrast, no nonempty open patch of a sphere of radius $R$ can be flattened smoothly into a plane while preserving its first fundamental form: $1/R^2$ cannot become zero. Making the patch small reduces practical distortion, but never produces an exact local isometry. This is a geometric obstruction to a perfectly distance-preserving flat map of a spherical world, independent of where one places a seam.

The following proof outline connects the two descriptions of Gaussian curvature. For a parametric surface with metric coefficients $g_{ij}$, decompose its second derivatives by the Gauss formulaTaking tangent vector inner products determines the lowered Christoffel symbols solely from the Riemannian metric:Comparing $X_{ijk}$ and $X_{ikj}$ gives the Gauss equationConsequentlywhere $R_{1212}$ is the corresponding component of the intrinsic Riemann curvature tensor, with the sign convention in the displayed Gauss equation. The final expression uses only $g_{ij}$ and its first two derivatives.

With thermal diffusivity $\xi$, a Fourier mode of angular wavenumber $k$ decays at rate $\beta =\xi k^2$. To avoid confusing this $k$ with thermal conductivity, use $\alpha =\xi$ and angular wavenumber $K$ below: the same rate is $\beta =\alpha K^2$, with units ${\mathrm{s}}^{-1}$.

Consider an insulated bar $0\le x\le L$ with constant thermal diffusivity, no sources and no heat flux through either end or the sides. An original initial temperature profile and its exact evolution areThe cosine has zero temperature gradient at both ends, satisfying the insulated Neumann boundary conditions. Its spatial mean is zero, so conservation of energy leaves the mean temperature $\overline{T}$ unchanged while the nonuniform part disappears. A short derivation substitutes $T=\overline{T}+A(t)\cos (K_{n}x)$ into the heat equation: the second spatial derivative supplies $-K_n^2$, hence $A^{\prime }=-\alpha K_n^{2}A$. The exponential decay time is ${\tau }_n=1/(\alpha K_n^2)$; doubling angular wavenumber quarters this time. If cycles-per-length spatial frequency $f_x$ is used instead, $K=2\pi f_x$ and the rate is $4{\pi }^2\alpha f_x^2$.

For the two model materials in thermal diffusivity and $L=0.010\mathrm{m}$, ${\tau }_{1,A}\approx 5.07\mathrm{s}$ and ${\tau }_{1,B}\approx 20.3\mathrm{s}$. After $10\mathrm{s}$, the remaining amplitude fractions are approximately $0.139$ and $0.610$. A narrow hot-and-cold pattern disappears faster than a broad one because its temperature curvature, and hence the imbalance in conductive heat flux, is larger.