Trigonometric and Oscillatory Examples in Several Variables: Examples and Counterexamples

Given: The curve $f(t)=(\cos t,\sin t)$ on $[0,2\pi ]$.

The functions $\sin$ and $\cos$ are differentiable on $\mathbb{R}$, with $(\sin t{)}^{\mathrm{\prime }}=\cos t$ and $(\cos t{)}^{\mathrm{\prime }}=-\sin t$; also $\sin 0=0$ and $\cos 0=1$ (The derivatives of sine and cosine are cosine and minus sine).

Let $m\in \mathbb{N}$ with $m\ge 1$, let $A\subseteq \mathbb{R}$, and let $c\in A$ be a limit point of $A$. A map $g:A\to {\mathbb{R}}^m$ is differentiable at $c$ if and only if every component is differentiable there, and its derivative then has the component derivatives as coordinates (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

Let $m\in \mathbb{N}$ with $m\ge 1$, let $A$ be a subspace of a metric space, and let $g:A\to {\mathbb{R}}^m$. The map $g$ is continuous at a point of $A$ if and only if every component is continuous there (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions).

For a real-valued function on $A\subseteq \mathbb{R}$, differentiability at a limit point $c\in A$ implies continuity there (A function differentiable at $c$ is continuous at $c$).

For every real $t$, ${\sin }^{2}t+{\cos }^{2}t=1$ (Parity and the Pythagorean identity for sine and cosine).

Both sine and cosine have period $2\pi$ (The zero sets of sine and cosine and the least positive common period 2 pi).

On ${\mathbb{R}}^2$, $\parallel (u,v){\parallel }_2=\sqrt{u^2+v^2}$ (The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$).

The number $\pi$ is positive (Pi as twice the smallest positive zero of cosine).

Given: The universal equality claim in the Statement.

The curve $f(t)=(\cos t,\sin t)$ on $[0,2\pi ]$ is continuous on its interval and differentiable in its interior, its endpoint increment is zero, and $\parallel f^{\mathrm{\prime }}(t){\parallel }_2=1$ for every $t\in (0,2\pi )$; hence no $\xi \in (0,2\pi )$ satisfies the claimed equality (The circular curve defeats the equality form of the vector-valued mean value theorem).

Let $m\in \mathbb{N}$ with $m\ge 1$, let $a<b$ be real, and let $M\ge 0$ be real. If $f:[a,b]\to {\mathbb{R}}^m$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $\parallel f^{\mathrm{\prime }}(t){\parallel }_2\le M$ throughout $(a,b)$, then $\parallel f(b)-f(a){\parallel }_2\le M(b-a)$ (The mean value inequality: if $f:[a,b]\to {\mathbb{R}}^m$ is continuous and differentiable on $(a,b)$ with $\parallel f^{\mathrm{\prime }}{\parallel }_2\le M$, then $\parallel f(b)-f(a){\parallel }_2\le M(b-a)$).

Given: The function $f(x,y)=\sin (xy)$ on ${\mathbb{R}}^2$.

The functions $\sin$ and $\cos$ are differentiable on $\mathbb{R}$, with $(\sin t{)}^{\mathrm{\prime }}=\cos t$ and $(\cos t{)}^{\mathrm{\prime }}=-\sin t$ (The derivatives of sine and cosine are cosine and minus sine).

If real functions $g$ and $h$ are differentiable at the relevant points, then $(g\circ h{)}^{\mathrm{\prime }}=(g^{\mathrm{\prime }}\circ h)h^{\mathrm{\prime }}$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$).

If real functions $g$ and $h$ are differentiable at a point, then $(gh{)}^{\mathrm{\prime }}=g^{\mathrm{\prime }}h+g{h}^{\mathrm{\prime }}$; sums and scalar multiples obey the corresponding derivative rules (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).

A coordinate partial derivative is the derivative at zero of the corresponding coordinate-line restriction (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$).

Given: The function $f$ in the Example and $s=x^2+y^2$.

For every real $t$, $\mathrm{\mid }\sin t\mathrm{\mid }\le 1$ and $\mathrm{\mid }\cos t\mathrm{\mid }\le 1$ (Parity and the Pythagorean identity for sine and cosine).

The functions $\sin$ and $\cos$ are differentiable on $\mathbb{R}$, with $(\sin t{)}^{\mathrm{\prime }}=\cos t$ and $(\cos t{)}^{\mathrm{\prime }}=-\sin t$; also $\sin 0=0$ and $\cos 0=1$ (The derivatives of sine and cosine are cosine and minus sine).

If real functions $g$ and $h$ are differentiable at the relevant points, then $(g\circ h{)}^{\mathrm{\prime }}=(g^{\mathrm{\prime }}\circ h)h^{\mathrm{\prime }}$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$).

Products, sums, scalar multiples, and quotients with nonzero denominator obey their usual derivative rules (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).

A map is totally differentiable at the origin with derivative zero when $\mathrm{\mid }f(h)\mathrm{\mid }\mathrm{/}\parallel h{\parallel }_2\to 0$ as $h\to 0$ through nonzero vectors (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder).

The coordinate partial derivatives are the derivatives of the two coordinate-line restrictions (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$).

For $(x,y)\in {\mathbb{R}}^2$, $\parallel (x,y){\parallel }_2=\sqrt{x^2+y^2}$ (The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$).

Sine and cosine have period $2\pi$ (The zero sets of sine and cosine and the least positive common period 2 pi).

For every real $\epsilon >0$ there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$ (For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$).

Every nonnegative real $a$ has a unique nonnegative square root $a^{1\mathrm{/}2}$, positive when $a>0$ (Existence and uniqueness of $n$-th roots: a unique $a^{1\mathrm{/}n}\ge 0$ with $(a^{1\mathrm{/}n}{)}^n=a$).

The number $\pi$ is positive (Pi as twice the smallest positive zero of cosine).

Given: The function $F$ in the Example and $r=\parallel (x,y){\parallel }_2$.

For $(x,y)\in {\mathbb{R}}^2$, $\parallel (x,y){\parallel }_2=\sqrt{x^2+y^2}$ (The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$).

For every real $t$, $\mathrm{\mid }\sin t\mathrm{\mid }\le 1$ and ${\sin }^{2}t+{\cos }^{2}t=1$ (Parity and the Pythagorean identity for sine and cosine).

The functions $\sin$ and $\cos$ are differentiable on $\mathbb{R}$, with $(\sin t{)}^{\mathrm{\prime }}=\cos t$ and $(\cos t{)}^{\mathrm{\prime }}=-\sin t$; also $\sin 0=0$ and $\cos 0=1$ (The derivatives of sine and cosine are cosine and minus sine).

If real functions $g$ and $h$ are differentiable at the relevant points, then $(g\circ h{)}^{\mathrm{\prime }}=(g^{\mathrm{\prime }}\circ h)h^{\mathrm{\prime }}$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$).

Products, sums, scalar multiples, and quotients with nonzero denominator obey their usual derivative rules (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).

A map is totally differentiable at the origin with derivative zero when $\mathrm{\mid }F(h)\mathrm{\mid }\mathrm{/}\parallel h{\parallel }_2\to 0$ as $h\to 0$ through nonzero vectors (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder).

For a scalar function, the gradient is the vector of its coordinate partial derivatives (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

On $(0,\mathrm{\infty })$, $(u^{1\mathrm{/}2}{)}^{\mathrm{\prime }}=\frac{1}{2}{u}^{-1\mathrm{/}2}$ (Continuity and derivatives of positive-base real powers).

If $a>0$ and $q\in \mathbb{Q}$, then the real power $a^q$ agrees with the rational power; in particular this holds for $q=1\mathrm{/}2$ (The exponential definition of real powers agrees with the existing rational powers).

Every nonnegative real has a unique nonnegative square root (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$).

For every real $t$, $\sin (t+\pi )=-\sin t$ and $\cos (t+\pi )=-\cos t$ (Quarter-turn values and shifts by pi/2 and pi).

Both sine and cosine have period $2\pi$ (The zero sets of sine and cosine and the least positive common period 2 pi).

For every real $\epsilon >0$ there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$ (For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$).

The number $\pi$ is positive (Pi as twice the smallest positive zero of cosine).

If a map is totally differentiable at a point, then each partial derivative there equals the total derivative applied to the corresponding standard basis vector (A total derivative computes every directional derivative, and its matrix is the Jacobian).

Given: The compact square $K=[0,\pi {]}^2$ and the family $\mathcal{F}$ in the Example.

For a nonempty compact metric space $K$, a family $\mathcal{F}\subseteq C(K,\mathbb{R})$ is equicontinuous at $a$ when every $\epsilon >0$ admits one $\delta >0$ that works for every $f\in \mathcal{F}$ and every $x\in K$; it is uniformly bounded when one $M\ge 0$ bounds $\mathrm{\mid }f(x)\mathrm{\mid }$ for all $f$ and $x$ (Equicontinuity, pointwise boundedness, and uniform boundedness for families in $C(K,\mathbb{R})$).

For every real $t$, $\sin (-t)=-\sin t$ and $\mathrm{\mid }\sin t\mathrm{\mid }\le 1$ (Parity and the Pythagorean identity for sine and cosine).

For all reals $u,v$, $\mathrm{\mid }\sin u-\sin v\mathrm{\mid }\le \mathrm{\mid }u-v\mathrm{\mid }$ (Sine and cosine are $1$-Lipschitz on $\mathbb{R}$).

$\sin (\pi \mathrm{/}2)=1$ (Quarter-turn values and shifts by pi/2 and pi).

For every real $\epsilon >0$ there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$ (For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$).

The number $\pi$ is positive (Pi as twice the smallest positive zero of cosine).

Every closed box in ${\mathbb{R}}^2$ is compact in the Euclidean metric (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

For $(u,v)\in {\mathbb{R}}^2$, $\parallel (u,v){\parallel }_2=\sqrt{u^2+v^2}$ (The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$).

Given: The profile $f(x)=\sin x$ on $[0,\pi ]$ and its solid of revolution about the $x$-axis.

If $a\le b$ and $f:[a,b]\to [0,\mathrm{\infty })$ is continuous, then its solid of revolution about the $x$-axis is compact and Jordan measurable and has volume $\pi {\int }_a^{b}f(x{)}^{2}dx$ (The disc formula for the volume of a solid of revolution).

$\sin \pi =0$, and $\sin x>0$ for every $x$ with $0<x<\pi$; thus $\pi$ is the first positive zero of sine, in particular $\pi >0$ (Pi is the first positive zero of sine).

The functions $\sin$ and $\cos$ are differentiable on $\mathbb{R}$, with $(\sin x{)}^{\mathrm{\prime }}=\cos x$ and $(\cos x{)}^{\mathrm{\prime }}=-\sin x$; also $\sin 0=0$ and $\cos 0=1$ (The derivatives of sine and cosine are cosine and minus sine).

For a real-valued function on $A\subseteq \mathbb{R}$, differentiability at a limit point $c\in A$ implies continuity there (A function differentiable at $c$ is continuous at $c$).

For every real $x$, ${\sin }^{2}x=(1-\cos 2x)\mathrm{/}2$ (Double-angle and quadratic power-reduction identities).

Every continuous real-valued function on a nondegenerate closed interval is Riemann integrable there (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion).

The integral is linear on integrable functions (Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and ${\int }_a^b(\lambda f+\mu g)=\lambda {\int }_a^{b}f+\mu {\int }_a^{b}g$).

If $a<b$, every constant function on $[a,b]$ is integrable and ${\int }_a^{b}c=c(b-a)$ (If $m\le f\le M$ on $[a,b]$ then $m(b-a)\le L(f,P)\le \underset{‾}{{\int }_a^b}f\le \overline{{\int }_a^b}f\le U(f,P)\le M(b-a)$ for every partition $P$; in particular every constant function is integrable, with ${\int }_a^{b}c=c(b-a)$).

If $a<b$, $G:[a,b]\to \mathbb{R}$ is differentiable at every point of $[a,b]$, and $G^{\mathrm{\prime }}$ is integrable, then ${\int }_a^b{G}^{\mathrm{\prime }}=G(b)-G(a)$ (The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$).

If real functions $g$ and $h$ are differentiable at the relevant points, then $(g\circ h{)}^{\mathrm{\prime }}=(g^{\mathrm{\prime }}\circ h)h^{\mathrm{\prime }}$ (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$).

Sums and scalar multiples obey the usual derivative rules (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$).

Both sine and cosine have period $2\pi$ (The zero sets of sine and cosine and the least positive common period 2 pi).

Given: The radius function $r(x)=\sin x$ on $[0,\pi ]$ and the surface obtained by rotating its graph about the $x$-axis.

Under the hypotheses of the scalar surface-integral theorem for a surface of revolution, the rotated surface has area $2\pi {\int }_a^{b}r(s)\sqrt{1+r^{\mathrm{\prime }}(s{)}^2}ds$ (The surface of revolution has area $2\pi {\int }_a^{b}r(s)\sqrt{1+r^{\mathrm{\prime }}(s{)}^2}ds$).

Those hypotheses require $a<b$ and $r:[a,b]\to [0,\mathrm{\infty })$ to be $C^1$ on a neighbourhood of $[a,b]$, positive on $(a,b)$, and allowed to vanish only at the endpoints (Scalar surface integrals on a surface of revolution).

$\sin \pi =0$, and $\sin x>0$ for every $x$ with $0<x<\pi$; thus $\pi$ is the first positive zero of sine, in particular $\pi >0$ (Pi is the first positive zero of sine).

The functions $\sin$ and $\cos$ are differentiable on $\mathbb{R}$, with $(\sin x{)}^{\mathrm{\prime }}=\cos x$ and $(\cos x{)}^{\mathrm{\prime }}=-\sin x$; also $\sin 0=0$ and $\cos 0=1$ (The derivatives of sine and cosine are cosine and minus sine).

For a real-valued function on $A\subseteq \mathbb{R}$, differentiability at a limit point $c\in A$ implies continuity there (A function differentiable at $c$ is continuous at $c$).

The function $\sinh :\mathbb{R}\to \mathbb{R}$ is odd, strictly increasing, and onto; $\cosh$ is positive; ${\cosh }^{2}v-{\sinh }^{2}v=1$; and $(\sinh v{)}^{\mathrm{\prime }}=\cosh v$ (Addition formulas, identities, parity, and derivatives of the hyperbolic functions).