Real Analysis

Derives Proposition 7.3. From \(-\abs{x} \le x \le \abs{x}\) and the same for \(y\), adding gives \(-(\abs{x}+\abs{y}) \le x + y \le \abs{x} + \abs{y}\), which is the claim.

Derives Proposition 7.5. If \(a_n \to L\) and \(a_n \to L'\) with \(L \neq L'\), take \(\varepsilon = \abs{L - L'}/2 > 0\). For \(n\) beyond both thresholds, \(\abs{L - L'} \le \abs{L - a_n} + \abs{a_n - L'} < 2\varepsilon = \abs{L - L'}\), a contradiction.

Derives Proposition 7.6. Sum: given \(\varepsilon\), beyond the larger of the two thresholds for \(\varepsilon/2\), \(\abs{(a_n + b_n) - (A + B)} \le \abs{a_n - A} + \abs{b_n - B} < \varepsilon\). Product: convergent sequences are bounded (beyond \(N_1\) for \(\varepsilon = 1\) all terms lie within \(1\) of the limit, and finitely many remain), say \(\abs{a_n} \le M\); then

and each term is made smaller than \(\varepsilon/2\) beyond a suitable threshold. Quotient: it suffices to prove \(1/b_n \to 1/B\). Beyond some \(N_0\), \(\abs{b_n} > \abs{B}/2\) (take \(\varepsilon = \abs{B}/2\)); then \(\abs{1/b_n - 1/B} = \abs{B - b_n}/(\abs{b_n}\abs{B}) \le 2\abs{b_n - B}/\abs{B}^2\), which is controlled as before.

Derives Theorem 7.7. Let \(a_n \in [A_0, B_0]\) for all \(n\). Bisect the interval: at least one half contains \(a_n\) for infinitely many indices \(n\); call it \([A_1, B_1]\) and pick \(n_1\) with \(a_{n_1} \in [A_1, B_1]\). Iterating, we obtain nested intervals \([A_k, B_k]\) of length \((B_0 - A_0)/2^k\), each containing infinitely many terms, and indices \(n_1 < n_2 < \cdots\) with \(a_{n_k} \in [A_k, B_k]\). The sequence \((A_k)\) is nondecreasing and bounded above by \(B_0\), so by monotone convergence (Corollary A.46) \(A_k \to L\); since \(0 \le B_k - A_k = (B_0 - A_0)/2^k \to 0\) (Archimedean property), \(B_k \to L\) as well, and \(A_k \le a_{n_k} \le B_k\) squeezes \(a_{n_k} \to L\).

Derives Theorem 7.8. Necessity: if \(a_n \to L\), then beyond the threshold for \(\varepsilon/2\), \(\abs{a_m - a_n} \le \abs{a_m - L} + \abs{L - a_n} < \varepsilon\). Sufficiency is Corollary A.47, proven in Appendix A.7 from monotone convergence.

Derives Proposition 7.18. (\(\Rightarrow\)) The two-sided \(\delta\) works for each side separately. (\(\Leftarrow\)) Given \(\varepsilon\), let \(\delta_{-}\) and \(\delta_{+}\) be the one-sided thresholds and \(\delta = \min(\delta_-, \delta_+)\). Any \(x\) with \(0 < \abs{x - x_0} < \delta\) lies on one of the two sides and satisfies the corresponding estimate, so \(\abs{f(x) - L} < \varepsilon\).

Derives Proposition 7.22. (\(\Rightarrow\)) Given \(\varepsilon\), choose \(\delta\) from continuity, then \(N\) so that \(n \ge N \implies \abs{x_n - x_0} < \delta\); then \(\abs{f(x_n) - f(x_0)} < \varepsilon\). (\(\Leftarrow\)) If \(f\) is not continuous at \(x_0\), some \(\varepsilon_0\) admits no \(\delta\): for each \(n\) there is \(x_n\) with \(\abs{x_n - x_0} < 1/n\) but \(\abs{f(x_n) - f(x_0)} \ge \varepsilon_0\). Then \(x_n \to x_0\) while \(f(x_n) \not\to f(x_0)\).

Derives Theorem 7.23. Let \(S = \set{x \in [a,b] \mid f(x) < 0}\); \(S\) is nonempty (\(a \in S\)) and bounded above by \(b\), so \(c = \sup S\) exists (Axiom 7.1). We claim \(f(c) = 0\). If \(f(c) < 0\), then \(c < b\), and continuity provides \(\delta > 0\) with \(f < 0\) on \((c - \delta, c + \delta) \cap [a,b]\) (take \(\varepsilon = -f(c)\)); points just above \(c\) would then lie in \(S\), contradicting that \(c\) bounds \(S\). If \(f(c) > 0\), continuity gives \(f > 0\) on \((c - \delta, c]\) for some \(\delta > 0\); then \(c - \delta\) is already an upper bound of \(S\), contradicting leastness. Hence \(f(c) = 0\), and \(c \neq a, b\) because \(f(a) < 0 < f(b)\).

Derives Theorem 7.24. Bounded: if not, pick \(x_n \in [a,b]\) with \(\abs{f(x_n)} > n\). By Bolzano–Weierstrass (Theorem 7.7) a subsequence \(x_{n_k} \to c \in [a,b]\); continuity forces \(f(x_{n_k}) \to f(c)\), but \(\abs{f(x_{n_k})} > n_k \to \infty\) — contradiction. Attained: let \(M = \sup f([a,b])\), which now exists; pick \(x_n\) with \(f(x_n) > M - 1/n\). A convergent subsequence \(x_{n_k} \to c\) gives \(f(c) = \lim f(x_{n_k}) = M\). The infimum is the supremum of \(-f\).

Derives Theorem 7.25. If not, some \(\varepsilon_0\) admits points \(x_n, y_n\) with \(\abs{x_n - y_n} < 1/n\) but \(\abs{f(x_n) - f(y_n)} \ge \varepsilon_0\). Extract \(x_{n_k} \to c\) (Theorem 7.7); then \(y_{n_k} \to c\) too, and continuity gives \(f(x_{n_k}) - f(y_{n_k}) \to f(c) - f(c) = 0\), contradicting \(\abs{f(x_{n_k}) - f(y_{n_k})} \ge \varepsilon_0\).

Derives Proposition 7.27. \(f(x) - f(x_0) = \dfrac{f(x)-f(x_0)}{x-x_0}\,(x - x_0) \longrightarrow f'(x_0)\cdot 0 = 0\) as \(x \to x_0\), by the algebra of limits.

Derivation. Derives Proposition 7.28. From Equation (7.14) we have

Derives Proposition 7.29. Both difference quotients split termwise, and the algebra of limits (Proposition 7.6) distributes the limit over the pieces.

Derivation. Derives Proposition 7.30. We have

where the last step uses Equation (7.14) and the continuity of \(g\) (Proposition 7.27).

Derivation. Derives Proposition 7.31. Define, for \(y\) in the domain of \(g\) and \(u_0 = f(x)\),

Differentiability of \(g\) at \(u_0\) says precisely that \(\varphi\) is continuous at \(u_0\), and by construction \(g(y) - g(u_0) = \varphi(y)\,(y - u_0)\) for all \(y\) — including \(y = u_0\), which is the case the naive quotient argument cannot handle. Substituting \(y = f(x + h)\),

As \(h \to 0\): \(f(x+h) \to f(x)\) by Proposition 7.27, so \(\varphi(f(x+h)) \to \varphi(f(x)) = g'(f(x))\) by continuity of \(\varphi\), while the second factor tends to \(f'(x)\). The algebra of limits gives Equation (7.17).

Derives Proposition 7.32. For \(y \neq y_0\) write \(x = f^{-1}(y)\); continuity and strict monotonicity make \(f^{-1}\) continuous (its one-sided limits exist by monotonicity and cannot jump without leaving a gap in the image, forbidden by Theorem 7.23), so \(y \to y_0\) forces \(x \to x_0\). Then

Derives Lemma 7.33. Say the extremum is a maximum. For \(h > 0\) small, \(\left[f(c+h)-f(c)\right]/h \le 0\), so the right limit gives \(f'(c) \le 0\); for \(h < 0\), the quotient is \(\ge 0\), so \(f'(c) \ge 0\). Hence \(f'(c) = 0\).

Derives Theorem 7.34. By the extreme value theorem (Theorem 7.24) \(f\) attains a maximum and a minimum on \([a,b]\). If both are attained at the endpoints, \(f\) is constant (max \(=\) min) and any interior \(c\) works. Otherwise one extremum is interior, and Lemma 7.33 finishes.

Derives Theorem 7.35. Apply Rolle to \(\phi(x) = f(x) - f(a) - \dfrac{f(b)-f(a)}{b-a}\,(x - a)\), which is continuous on \([a,b]\), differentiable inside, and vanishes at both endpoints; \(\phi'(c) = 0\) is Equation (7.19).

Derives Corollary 7.36. For \(x < y\) in the interval, Theorem 7.35 on \([x,y]\) gives \(f(y) - f(x) = f'(c)(y - x)\) with \(c \in (x,y)\) — an interior point — and the right side vanishes in the first case and is positive in the second.

Derives Theorem 7.37. Rolle applied to \(\psi(x) = \left[f(b)-f(a)\right]g(x) - \left[g(b)-g(a)\right]f(x)\) (which takes the value \(f(b)g(a) - g(b)f(a)\) at both ends) gives the first claim. For the second, take \(x \neq x_0\) and apply it on the interval with endpoints \(x_0, x\): since \(f(x_0) = g(x_0) = 0\),

As \(x \to x_0\), also \(c_x \to x_0\), and the right-hand side tends to \(L\).

Derives Theorem 7.38. Fix \(x\) and let \(M\) be defined by \(f(x) = \sum_{k=0}^{n}\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k + M\,(x-x_0)^{n+1}\). Consider

Then \(g(x) = 0\) and \(g(x_0) = 0\) by the choice of \(M\), so Rolle gives \(\xi\) between \(x_0\) and \(x\) with \(g'(\xi) = 0\). Differentiating the sum in \(t\) telescopes: all terms cancel pairwise except the last, leaving

Setting \(t = \xi\) and dividing by \((x-\xi)^n \neq 0\) yields \(M = f^{(n+1)}(\xi)/(n+1)!\), which is Equation (7.21).

Derives Theorem 7.40. Given \(\varepsilon > 0\), Heine–Cantor (Theorem 7.25) provides \(\delta\) with \(\abs{f(x)-f(y)} < \varepsilon/(b-a)\) whenever \(\abs{x-y} < \delta\). For any partition with all steps \(x_i - x_{i-1} < \delta\), the extreme value theorem gives points where \(m_i, M_i\) are attained, at distance \(< \delta\), so \(M_i - m_i < \varepsilon/(b-a)\) and

Upper and lower extrema squeezed within every \(\varepsilon\) coincide.

Derives Theorem 7.42. For \(h > 0\) (the case \(h<0\) is symmetric), additivity gives

On \([x, x+h]\) let \(m_h, M_h\) be the (attained) minimum and maximum of \(f\); monotonicity of the integral squeezes \(m_h \le \frac1h \int_x^{x+h} f \le M_h\). As \(h \to 0\), continuity of \(f\) at \(x\) drives \(m_h, M_h \to f(x)\), and the difference quotient converges to \(f(x)\).

Derives Theorem 7.43. \(F(x) = \int_a^x f\) is an antiderivative by Theorem 7.42, so \(G = F + C\) for a constant \(C\) (Definition 7.41). Then \(G(b) - G(a) = F(b) - F(a) = \int_a^b f - 0\).

Derives Corollary 7.44. Both sides of Equation (7.27) equal \(F(g(b)) - F(g(a))\) for an antiderivative \(F\) of \(f\), by Theorem 7.43 and the chain rule \(\left(F\circ g\right)' = f(g)\,g'\) (Proposition 7.31). Equation Equation (7.28) is Theorem 7.43 applied to the Leibniz rule \((fg)' = f'g + fg'\) (Proposition 7.30).

Derives Proposition 7.46. \((1-q)\,s_N = 1 - q^{N+1}\), so \(s_N = (1 - q^{N+1})/(1-q)\). Since \(\abs{q}<1\), \(q^{N+1} \to 0\) (its absolute value is decreasing and bounded below, so it converges by monotone convergence, and the limit \(\ell\) satisfies \(\ell = \abs{q}\,\ell\), forcing \(\ell = 0\)).

Derives Proposition 7.47. The partial sums of \(\sum a_n\) are nondecreasing and bounded above by \(\sum b_n\); monotone convergence (Corollary A.46) concludes. For the second claim, apply the Cauchy criterion (Theorem 7.8): for \(M > N\), \(\abs{s_M - s_N} = \abs{\sum_{N+1}^{M} a_n} \le \sum_{N+1}^{M} \abs{a_n}\), and the right side is the Cauchy difference of the convergent series \(\sum\abs{a_n}\).

Derives Proposition 7.48. Pick \(q\) with \(r < q < 1\): beyond some \(N\), \(\abs{a_{n+1}} \le q\abs{a_n}\), so \(\abs{a_{N+k}} \le q^{k}\abs{a_N}\) and comparison with the geometric series (Propositions 7.46 and 7.47) gives absolute convergence. In the divergent case \(\abs{a_n}\) is eventually nondecreasing and nonzero, so \(a_n \not\to 0\), while convergence of a series forces its terms to \(0\) (Cauchy criterion with \(M = N+1\)).

Derives Proposition 7.49. Write \(A^{*} = \sum_n \abs{a_n}\) and \(B^{*} = \sum_n \abs{b_n}\). For any \(N\),

since every product \(\abs{a_k}\abs{b_j}\) with \(k + j \le N\) appears exactly once in the middle double sum and all products are nonnegative. The partial sums of \(\sum\abs{c_n}\) are nondecreasing and bounded, hence convergent (Corollary A.46): the Cauchy product converges absolutely. For its value, compare \(C_N = \sum_{n=0}^{N} c_n\) with the product of partial sums \(A_N B_N\): expanding both, \(A_N B_N - C_N\) consists exactly of the products \(a_j b_k\) with \(j, k \le N\) but \(j + k > N\). Every such pair has \(j > N/2\) or \(k > N/2\) (double counting only enlarges the bound), so

because each tail is the difference between \(A^{*}\) (respectively \(B^{*}\)) and one of its own partial sums, and those partial sums converge to it. Since \(A_N B_N \to AB\) by the algebra of limits (Proposition 7.6), also \(C_N \to AB\).

Derives Theorem 7.50. Let \(S = \set{\rho \ge 0 \mid (\abs{c_n}\rho^n) \text{ is bounded}}\) and \(R = \sup S\) (allowing \(\infty\)). If \(\abs{x - x_0} < R\), pick \(\rho \in S\) with \(\abs{x-x_0} < \rho\) and a bound \(\abs{c_n}\rho^n \le M\); then \(\abs{c_n (x-x_0)^n} \le M\,(\abs{x-x_0}/\rho)^n\), a convergent geometric majorant, so comparison gives absolute convergence. If \(\abs{x - x_0} > R\), the terms \(c_n (x-x_0)^n\) are unbounded (else \(\abs{x-x_0} \in S\)), so they cannot tend to \(0\) and the series diverges.

Derives Theorem 7.51. Translate so that \(x_0 = 0\). Same radius. Let \(\abs{x} < R\). As in the proof of Theorem 7.50, pick \(\rho\) in the set \(S\) of that proof with \(\abs{x} < \rho\) (possible because \(R = \sup S > \abs{x}\)) and a bound \(\abs{c_n}\rho^{n} \le M\); then

and \(\sum_n n\,t^{n-1}\) converges for \(0 < t < 1\) by the ratio test (Proposition 7.48: the ratio of consecutive terms is \(\tfrac{n+1}{n}\,t \to t < 1\)), and trivially at \(t = 0\), where only the \(n = 1\) term survives. Comparison (Proposition 7.47) gives absolute convergence of the derived series whenever \(\abs{x} < R\). Conversely \(\abs{c_n x^{n}} \le \abs{x}\cdot n\abs{c_n}\abs{x}^{n-1}\) for \(n \ge 1\), so wherever the derived series converges absolutely the original does too: the two radii coincide. Applying the statement to the derived series itself, \(\sum_{n\ge2} n(n-1)\,c_n\,x^{n-2}\) also has radius \(R\).

The difference quotient. Fix \(\abs{x} < R\), choose \(r\) with \(\abs{x} < r < R\), and let \(y \neq x\) with \(\abs{y} \le r\). Subtracting the two absolutely convergent series term by term and dividing by \(y - x\) (Proposition 7.6 on partial sums),

The factorization \(y^{n} - x^{n} = (y-x)\sum_{k=0}^{n-1} y^{k}x^{n-1-k}\) turns the bracket into \(\sum_{k=0}^{n-1}\left(y^{k} - x^{k}\right)x^{n-1-k}\), and the same factorization once more bounds \(\abs{y^{k} - x^{k}} \le k\,r^{k-1}\abs{y - x}\), so

Multiplying by \(\abs{c_n}\) and summing,

with \(K < \infty\) because \(r < R\) (the twice-derived series above). Letting \(y \to x\), the difference quotient converges to \(\sum_n n\,c_n\,x^{n-1}\), which is Equation (7.30). Differentiability implies continuity (Proposition 7.27), and iterating on the derived series — again a power series of radius \(R\) — yields derivatives of every order.

Derives Proposition 7.52. The even partial sums are nonincreasing, \(S_{2k+2} - S_{2k} = -b_{2k+1} + b_{2k+2} \le 0\), and bounded below by \(S_1\) (since \(S_{2k+1} \le S_{2k+2} \le S_{2k}\), shown by the same computation applied once more); the odd partial sums are nondecreasing and bounded above by \(S_0\) by the mirror argument. Both are monotone and bounded, hence convergent (Corollary A.46), to limits \(S_{\mathrm{e}}\) and \(S_{\mathrm{o}}\) respectively; since \(S_{2k} - S_{2k+1} = b_{2k+1} \to 0\), \(S_{\mathrm{e}} = S_{\mathrm{o}} =: S\). Monotonicity of each subsequence gives \(S_1 \le S \le S_0\).

Derives Lemma 7.54. The tail \(\ee-s_{n}=\sum_{k=n+1}^{\infty}1/k!\) has positive terms, so it is at least its first term \(1/(n+1)!\); that is the left inequality. For the right one, put \(k=n+1+j\) with \(j\ge0\) and factor out \(1/(n+1)!\):

the empty product at \(j=0\) being \(1\). Each of the \(j\) factors obeys \(n+i\ge n+2>n+1\), so the product is at most \((n+1)^{-j}\), strictly so once \(j\ge1\). The comparison series is geometric with ratio \(1/(n+1)\in(0,1)\), hence summable (Proposition 7.46), and

Derivation. Derives Corollary 7.55. Take \(n=12\) in Lemma 7.54. Over the common denominator \(12!=479001600\) the partial sum is a ratio of integers obtained by finitely many additions, with no limit process anywhere:

The error bound is \(1/(12\cdot12!)=1/5748019200\), itself smaller than \(1.74\times 10^{-10}\). Carrying out the two divisions to sixteen decimals — rounding the lower endpoint down and the upper endpoint up, so that the enclosure is certain — Equation (7.32) gives

The endpoints agree in their first nine decimals, so every real number between them — \(\ee\) among them — begins \(2.718281828\).

Derives Proposition 7.56. Suppose \(\ee=p/q\) with \(p,q\) positive integers, and set \(n=\max\{q,2\}\). Consider

Every \(n!/k!\) with \(k\le n\) is an integer, so the sum is an integer; and \(n!\,\ee=n!\,p/q\) is an integer because \(q\le n\) makes \(q\) a divisor of \(n!\). Hence \(x\in\Z\). Multiplying Equation (7.32) by \(n!>0\), however,

so \(x\) is an integer lying strictly between \(0\) and \(1\). No such integer exists, and the assumption fails.

Derives Proposition 7.57. By the binomial theorem (Equation (5.3)),

using \(\binom{n}{k}/n^{k} = \frac{1}{k!}\prod_{j=0}^{k-1}(n-j)/n = \frac{1}{k!}\prod_{j=0}^{k-1}(1-j/n)\). Each factor \(1-j/n \in (0,1]\), so every term of Equation (7.38) is at most \(1/k!\); summing,

the last step because a partial sum of a convergent series of positive terms is bounded by the full sum. This gives \(\limsup_n (1+1/n)^n \le \ee\). Conversely, fix \(N\); for \(n \ge N\), truncating Equation (7.38) at \(k=N\) (dropping the remaining nonnegative terms) gives

For each fixed \(k \le N\), the product \(\to 1\) as \(n \to \infty\) (a finite product of terms each \(\to 1\), by Proposition 7.6), so the right side \(\to \sum_{k=0}^{N} 1/k!\) as \(n \to \infty\). Hence \(\liminf_n (1+1/n)^n \ge \sum_{k=0}^{N} 1/k!\) for every \(N\), and letting \(N \to \infty\), \(\liminf_n(1+1/n)^n \ge \ee\). The limsup and liminf bounds coincide, so the limit exists and equals \(\ee\).

Derives Lemma 7.59. Derivatives. Termwise differentiation (Theorem 7.51) of Equation (7.40) gives

and, re-indexing \(n = m + 1\) in the derived series of \(\cos\),

Pythagorean identity. Both series converge absolutely at every \(x\), so each square is a Cauchy product (Proposition 7.49). Squaring the cosine series, the terms \(a_j = (-1)^{j}x^{2j}/(2j)!\) collect at fixed \(j + k = n\) into

multiplying and dividing by \((2n)!\) in the last step. The same computation squares the sine series (terms \((-1)^{j}x^{2j+1}/(2j+1)!\), product powers \(x^{2n+2}\), then re-indexed so the power reads \(x^{2n}\)):

Adding the two, the constant term is \(1\) and the coefficient of \(x^{2n}\) for \(n \ge 1\) is

by the binomial theorem (Equation (5.3)): the two brackets are the even and the odd halves of \(\sum_{i=0}^{2n}(-1)^{i}\binom{2n}{i}\). Hence \(\sin^{2}x + \cos^{2}x = 1\). (Once the derivatives are known the identity also follows in one line — \((\sin^{2} + \cos^{2})' = 2\sin\cos - 2\cos\sin = 0\), so the function is constant (Corollary 7.36) with value \(1\) at \(x = 0\); the Cauchy-product computation above is that statement read coefficient by coefficient.)

Derives Lemma 7.60. At \(x=2\) the terms of Equation (7.40) have magnitude \(t_n = 2^{2n}/(2n)!\); for \(n \ge 1\),

so \((t_n)_{n\ge1}\) is nonincreasing (and \(t_n\to0\)). By Proposition 7.52 applied to the tail starting at \(n=2\), the sum \(S = \sum_{n=2}^{\infty}(-1)^{n}t_n = t_2 - t_3 + t_4 - \cdots\) satisfies \(0 \le S \le t_2 = 16/24 = 2/3\). Since \(\cos 2 = t_0 - t_1 + S = 1 - 2 + S = -1 + S\),

in particular \(\cos 2 < 0\).

Derives Proposition 7.62. On \([0,z)\), \(\cos > 0\) by minimality of \(z\) in Definition 7.61, and \(\sin' = \cos\) (Lemma 7.59), so \(\sin\) is strictly increasing on \([0,z]\) (Corollary 7.36); from \(\sin 0 = 0\) it is positive on \((0,z]\), and \(\sin^{2}z = 1 - \cos^{2}z = 1\) (Lemma 7.59 again, with \(\cos z = 0\)) forces \(\sin z = 1\). The path \(t \mapsto (\cos t, \sin t)\), \(t \in [0,z]\), therefore lies on the unit circle by the Pythagorean identity, starts at \((1,0)\), ends at \((0,1)\), and traverses the first-quadrant arc exactly once: \(\sin\) is a continuous strictly increasing bijection of \([0,z]\) onto \([0,1]\) (Theorem 7.23), and on that arc the height \(\sin t\) determines the point, since \(\cos t \ge 0\). The speed of the path is

(both parts of Lemma 7.59), so its arc length is \(\int_{0}^{z} 1 \,\dd t = z\). The reflections \((x,y) \mapsto (-x,y)\) and \((x,y) \mapsto (x,-y)\) map the unit circle to itself and preserve arc length, so the four quadrantal arcs have the same length \(z\), and the full circumference is \(4z = 2\pi\).

Derives Theorem 7.68. Reduction to one component. With \(L\) as in Equation (7.46), the \(j\)-th component of \(\vect{F}(\vect{a}+\vect{h})-\vect{F}(\vect{a})-L(\vect{h})\) is \(R_j(\vect{h})=F_j(\vect{a}+\vect{h})-F_j(\vect{a}) -\nabla F_j(\vect{a})\cdot\vect{h}\), and the Euclidean length of a vector is at most the sum of the absolute values of its components. It therefore suffices to prove that \(\abs{R_j(\vect{h})}/\abs{\vect{h}}\longrightarrow 0\) for a single scalar \(C^{1}\) function \(f=F_j\).

A staircase of coordinate steps. Since \(A\) is open, fix \(r>0\) with \(B_r(\vect{a})\subseteq A\) (Definition 6.26) and let \(\abs{\vect{h}}<r\). Put \(\vect{p}_0=\vect{a}\) and

so that \(\vect{p}_m=\vect{a}+\vect{h}\). Every point of the segment joining \(\vect{p}_{k-1}\) to \(\vect{p}_k\) has the form \(\vect{a}+\sum_{i<k}h_i\vect{e}_i+t\,h_k\vect{e}_k\) with \(t\in[0,1]\), and its distance to \(\vect{a}\) is at most \(\abs{\vect{h}}<r\): the whole staircase lies inside \(B_r(\vect{a})\). Telescoping,

The mean value theorem on each step. The \(k\)-th bracket moves only the \(k\)-th coordinate, so it is the increment over \([0,1]\) of the one-variable function \(g_k(t)=f(\vect{p}_{k-1}+t\,h_k\vect{e}_k)\). Assume \(h_k\neq 0\). The difference quotient of \(g_k\) at \(t\) with increment \(s\) becomes, on substituting \(u=s\,h_k\), exactly \(h_k\) times the quotient appearing in Equation (7.44), so \(g_k'(t)=h_k\,\pp_k f(\vect{p}_{k-1}+t\,h_k\vect{e}_k)\) wherever the partial derivative \(\pp_k f\) exists — that is, throughout \(B_r(\vect{a})\). Both endpoints of the segment lie at distance \(\abs{\vect{h}}<r\) from \(\vect{a}\), hence in the interior of the ball, so \(g_k\) is defined and differentiable on an open interval containing \([0,1]\) and is in particular continuous there (Proposition 7.27). By Theorem 7.35 there is \(\theta_k\in(0,1)\) with

and \(\abs{\vect{\xi}_k-\vect{a}}\le\abs{\vect{h}}\). If \(h_k=0\) both sides of Equation (7.48) vanish and \(\vect{\xi}_k=\vect{p}_{k-1}\) serves.

The estimate. Subtracting \(\nabla f(\vect{a})\cdot\vect{h}=\sum_{k}h_k\,\pp_k f(\vect{a})\) from Equation (7.47) and inserting Equation (7.48),

so that, since \(\abs{h_k}\le\abs{\vect{h}}\) for every \(k\),

Given \(\varepsilon>0\), continuity of the \(m\) partial derivatives at \(\vect{a}\) (Definition 7.66) provides \(\delta\in(0,r)\) such that \(\abs{\vect{y}-\vect{a}}<\delta\) implies \(\abs{\pp_k f(\vect{y})-\pp_k f(\vect{a})}<\varepsilon/m\) for every \(k\). For \(\abs{\vect{h}}<\delta\) each \(\vect{\xi}_k\) obeys \(\abs{\vect{\xi}_k-\vect{a}}\le\abs{\vect{h}}<\delta\), so the right-hand side of Equation (7.49) is smaller than \(\varepsilon\). This is Equation (7.45).

Derives Proposition 7.72. Write \(L=\dd\vect{G}(\vect{a})\) and \(M=\dd\vect{F}(\vect{b})\). A linear map between Euclidean spaces is bounded: expanding in the canonical bases and applying the Cauchy–Schwarz inequality (Linear Algebra and Representation Theory) row by row produces constants \(C_L\), \(C_M\) with \(\abs{L\vect{v}}\le C_L\abs{\vect{v}}\) and \(\abs{M\vect{w}}\le C_M\abs{\vect{w}}\). By Definition 7.67,

where \(\abs{\vect{\rho}(\vect{h})}/\abs{\vect{h}}\longrightarrow 0\), and where \(\abs{\vect{\sigma}(\vect{k})} =\varepsilon(\vect{k})\,\abs{\vect{k}}\) with \(\varepsilon(\vect{k})\longrightarrow 0\) as \(\vect{k}\longrightarrow\vect{0}\); setting \(\varepsilon(\vect{0})=0\) is consistent, since \(\vect{\sigma}(\vect{0})=\vect{0}\). Put \(\vect{k}(\vect{h})=L\vect{h}+\vect{\rho}(\vect{h})\), so that

The first remainder obeys \(\abs{M\vect{\rho}(\vect{h})}\le C_M\abs{\vect{\rho}(\vect{h})}\) and is therefore \(o(\abs{\vect{h}})\). For the second, taking \(\abs{\vect{h}}\) small enough that \(\abs{\vect{\rho}(\vect{h})}\le\abs{\vect{h}}\) gives \(\abs{\vect{k}(\vect{h})}\le\left(C_L+1\right)\abs{\vect{h}}\), whence

and \(\vect{k}(\vect{h})\longrightarrow\vect{0}\) with \(\vect{h}\), so this remainder is \(o(\abs{\vect{h}})\) as well. Hence \(ML\) satisfies Equation (7.45) for \(\vect{F}\circ\vect{G}\) and is its differential, by the uniqueness established in Definition 7.67. Reading off the matrix of \(ML\) in the canonical bases, the rows of \(L\) and of \(M\) being the gradients of the components — which mere differentiability already secures, by the identification in Definition 7.67 — is Equation (7.52).

Derives Proposition 7.73. Fix \((x,y)\) and consider the second difference \(\Delta(h,k) = f(x+h, y+k) - f(x+h,y) - f(x,y+k) + f(x,y)\). Applying the mean value theorem (Theorem 7.35) in \(x\) to \(u(t) = f(t, y+k) - f(t, y)\) gives \(\Delta = \left[\pp_x f(\xi, y+k) - \pp_x f(\xi, y)\right] h\) for some \(\xi\), and a second application in \(y\) gives \(\Delta = \pp_y \pp_x f(\xi, \eta)\, h k\). Symmetrically, \(\Delta = \pp_x \pp_y f(\xi', \eta')\, h k\). Dividing by \(hk\) and letting \((h,k) \to 0\), continuity of the second partials forces \(\pp_y \pp_x f(x,y) = \pp_x \pp_y f(x,y)\).

Derives Theorem 7.74. Since \(A\) is open and the segment is compact, there is \(\varepsilon>0\) with \(\vect{a}+t\vect{h}\in A\) for every \(t\in(-\varepsilon,1+\varepsilon)\); put \(g(t)=f(\vect{a}+t\vect{h})\) there. The map \(t\longmapsto\vect{a}+t\vect{h}\) is differentiable with differential \(\vect{h}\), so the chain rule Equation (7.52) gives

the second by applying the same rule to each \(\pp_i f\), which is \(C^{1}\) by hypothesis. Hence \(g\) is twice differentiable on an interval containing \([0,1]\), and Theorem 7.38 at order \(n=1\) supplies \(\theta\in(0,1)\) with \(g(1)=g(0)+g'(0)+\tfrac{1}{2}g''(\theta)\), which is Equation (7.54).

For Equation (7.55), subtract the two quadratic terms: the difference is

whose modulus is at most \(\tfrac{1}{2}\abs{\vect{h}}^{2}\sum_{i,j} \abs{H_{ij}(\vect{a}+\theta\vect{h})-H_{ij}(\vect{a})}\), since \(\abs{h_i}\le\abs{\vect{h}}\) for every \(i\). As \(\vect{h}\longrightarrow\vect{0}\) the point \(\vect{a}+\theta\vect{h}\) tends to \(\vect{a}\) whatever value \(\theta\in(0,1)\) takes, so continuity of the second partial derivatives sends that sum to zero. The difference is therefore \(o(\abs{\vect{h}}^{2})\).

Derives Proposition 7.76. Fix \(\vect{u}\in U\). Because \(U\) is a cone, \(\mu\longmapsto f(\mu\vect{u})\) is defined for every \(\mu>0\), and the chain rule Equation (7.52) makes it differentiable with

If Equation (7.57) holds, differentiate it in \(\mu\): the left-hand side is Equation (7.59) and the right-hand side is \(k\mu^{k-1}f(\vect{u})\). Setting \(\mu=1\) leaves Equation (7.58).

Conversely, assume Equation (7.58) throughout \(U\) and put \(\psi(\mu)=\mu^{-k}f(\mu\vect{u})\) for \(\mu>0\). By the product rule (Proposition 7.30) and Equation (7.59),

the bracket vanishing because Equation (7.58) holds at the point \(\mu\vect{u}\in U\). A function with vanishing derivative on an interval is constant (Corollary 7.36), so \(\psi(\mu)=\psi(1)=f(\vect{u})\) for every \(\mu>0\), which is Equation (7.57).

Derives Theorem 7.77. Fix \(t\in I\) and choose \(\eta>0\) with \(J=[t-\eta,t+\eta]\subset I\). The set \(D\times J\) is closed and bounded in \(\R^{N+1}\), hence compact (Theorem 6.12), so \(\pp f/\pp t\) is uniformly continuous on it (Theorem 7.25). Both \(f(\cdot,s)\) and \(\pp_t f(\cdot,s)\) are continuous on \(D\), hence integrable there (Remark 7.96); write \(G(t)=\int_{D}\pp_t f(\vect{x},t)\,\dd V\).

Let \(0<\abs{h}\le\eta\). For each fixed \(\vect{x}\in D\) the one-variable function \(s\longmapsto f(\vect{x},s)\) is differentiable on \(J\), so the mean value theorem (Theorem 7.35) supplies \(\theta=\theta(\vect{x},h)\in(0,1)\) with

Subtracting \(\pp_t f(\vect{x},t)\), integrating over \(D\) and bounding the integral of a function by the volume times its supremum,

Given \(\varepsilon>0\), uniform continuity provides \(\delta\in(0,\eta]\) such that \(\abs{s-s'}<\delta\) implies \(\abs{\pp_t f(\vect{x},s)-\pp_t f(\vect{x},s')}<\varepsilon/(V+1)\) for every \(\vect{x}\in D\). Since \(\abs{(t+\theta h)-t}\le\abs{h}\), taking \(\abs{h}<\delta\) makes the right-hand side of Equation (7.62) smaller than \(\varepsilon\); this is Equation (7.60). The same bound with \(t+\theta h\) replaced by an arbitrary \(t'\in J\) gives \(\abs{G(t)-G(t')}\le V\sup_{D}\abs{\pp_t f(\cdot,t) -\pp_t f(\cdot,t')}\), so \(G\) is continuous.

Derives Corollary 7.78. Put \(\Phi(u,v,t)=\int_{u}^{v}f(x,t)\,\dd x\) on \([c,d]\times[c,d]\times I\). The fundamental theorem of calculus (Theorem 7.42), together with the sign rule for exchanging the limits of integration, gives \(\pp_v\Phi=f(v,t)\) and \(\pp_u\Phi=-f(u,t)\); Theorem 7.77 applied on the interval with endpoints \(u\) and \(v\) gives \(\pp_t\Phi=\int_{u}^{v}\pp_t f(x,t)\,\dd x\). All three are continuous in \((u,v,t)\): the first two because \(f\) is, the third by the last clause of Theorem 7.77 together with the estimate \(\abs{\int_{u}^{v}g-\int_{u'}^{v'}g} \le\left(\abs{u-u'}+\abs{v-v'}\right)\sup\abs{g}\) applied to \(g=\pp_t f\). Hence \(\Phi\) is \(C^{1}\) and therefore differentiable (Theorem 7.68), and the chain rule Equation (7.52) applied to \(t\longmapsto\Phi\left(\alpha(t),\beta(t),t\right)\) is Equation (7.63).

Derives Proposition 7.79. The cofactor form. By Equation (5.19) the determinant is a polynomial in the \(n^{2}\) entries, and each of its terms carries exactly one entry from each column. Differentiating that sum of products by the product rule (Proposition 7.30),

Collecting the terms that multiply one fixed \(\dd A^{i}{}_{j}/\dd t\) leaves precisely the coefficient of \(A^{i}{}_{j}\) in Equation (5.19), that is \(\pp\det A/\pp A^{i}{}_{j}=C^{i}{}_{j}\), which is Equation (7.64). Note that \(C^{i}{}_{j}\) contains no entry of the \(j\)-th column at all.

The trace form. Since \(\det\) is linear in the \(j\)-th column and \(C^{i}{}_{j}\) is free of that column, expanding along it gives \(\sum_{i}A^{i}{}_{j}C^{i}{}_{j}=\det A\) for each \(j\); and for \(k\neq j\) the sum \(\sum_{i}A^{i}{}_{k}C^{i}{}_{j}\) is that same expansion for the matrix whose \(j\)-th column has been replaced by its \(k\)-th, a matrix with two equal columns, whose determinant therefore vanishes. These are the two properties of Equation (5.19) on which Equation (5.16) also rests, and together they say

If \(\det A\neq0\), Equation (7.66) states that the matrix with entries \(C^{i}{}_{j}/\det A\), read with its indices exchanged, is a left inverse of \(A\) and hence its inverse: \(\left(A^{-1}\right)^{j}{}_{i}=C^{i}{}_{j}/\det A\). Substituting in Equation (7.64),

because the \(j\)-th diagonal entry of \(A^{-1}\dd A/\dd t\) is \(\sum_{i}\left(A^{-1}\right)^{j}{}_{i}\dd A^{i}{}_{j}/\dd t\). This is Equation (7.65), and the logarithmic form follows from it by the chain rule (Proposition 7.31).

Derives Theorem 7.83. Replacing \(\vect{F}\) by \(\vect{F}-\vect{F}(\vect{p})\) — a translation of the target, which changes neither \(D\vect{F}\) nor its rank — we may assume \(\vect{F}(\vect{p})=\vect{0}\).

Straightening the source. Since \(D\vect{F}(\vect{p})\) has rank \(r\), some \(r\times r\) minor of it is non-zero; relabelling the coordinates of \(\R^{N}\) and of \(\R^{N'}\), which are linear changes and hence \(C^{\infty}\), we may take it to be the minor built from the first \(r\) rows and the first \(r\) columns. Define

Its derivative at \(\vect{p}\) is block triangular, carrying that invertible \(r\times r\) minor in the upper left and the identity in the lower right, so \(\det D\varphi(\vect{p})\neq0\) and Corollary 7.81 makes \(\varphi\) a \(C^{k}\) diffeomorphism of a neighbourhood of \(\vect{p}\) onto a neighbourhood of \(\varphi(\vect{p})=\vect{0}\). Shrink the latter to an open box \(B=B'\times B''\) with \(B'\subseteq\R^{r}\) and \(B''\subseteq\R^{N-r}\), and shrink the former to its preimage.

The tail depends only on the head. Put \(\vect{G}=\vect{F}\circ\varphi^{-1}\) on \(B\), of class \(C^{k}\). By construction the first \(r\) components of \(\vect{G}\) are the first \(r\) coordinates, so

and the derivative has the block form

The chain rule Equation (7.52) gives \(D\vect{G}=D\vect{F}\cdot D\varphi^{-1}\) with \(D\varphi^{-1}\) invertible, so \(D\vect{G}\) has rank \(r\) at every point of \(B\). Its first \(r\) columns are already independent by Equation (7.74), so every column of \(D_{\vect{v}''}\vect{g}\) must be a combination of them; but those columns vanish in the first \(r\) rows, so each is a combination with zero coefficients, that is \(D_{\vect{v}''}\vect{g}\equiv0\) on \(B\). Because \(B''\) is a box, any two of its points are joined inside \(B\) by a chain of coordinate steps, along each of which \(\vect{g}\) is constant (Corollary 7.36); hence \(\vect{g}\) does not depend on \(\vect{v}''\), and we write \(\vect{g}=\vect{g}(\vect{v}')\).

Straightening the target. On \(B'\times\R^{N'-r}\) define

It is \(C^{k}\), and it is a bijection of \(B'\times\R^{N'-r}\) onto itself with the \(C^{k}\) inverse \(\left(\vect{w}',\vect{w}''\right)\longmapsto \left(\vect{w}',\vect{w}''+\vect{g}(\vect{w}')\right)\), hence a \(C^{k}\) diffeomorphism. Composing with Equation (7.73),

which is Equation (7.71). The final clause is that statement read backwards: in the coordinates \(\chi\) the image is where the last \(N'-r\) coordinates vanish.

Derives Lemma 7.84. Adapted coordinates. Fix \(\vect{p}\in\Sigma\). The \(J\times n\) matrix of the gradients \(\nabla\phi_j(\vect{p})\) has rank \(J\), so \(J\) of its columns are independent; relabelling the coordinates, take them to be the first \(J\). Let \(i_1<\cdots<i_{n-J}\) be the remaining indices and put

Expanding \(\det D\vect{\Psi}(\vect{p})\) along its last \(n-J\) rows, each of which is a vector of the canonical basis, leaves — up to sign — exactly the non-zero \(J\times J\) minor just chosen, so \(\vect{\Psi}\) is a smooth diffeomorphism of a neighbourhood of \(\vect{p}\) onto a neighbourhood of \(\vect{\Psi}(\vect{p})=\vect{0}\) (Corollary 7.81), which we shrink to an open box \(B=B'\times B''\), \(B'\subseteq\R^{J}\), \(B''\subseteq\R^{n-J}\). Write \(\vect{y}=(\vect{y}',\vect{y}'')=\vect{\Psi}(\vect{x})\) and \(\tilde{f}=f\circ\vect{\Psi}^{-1}\), smooth on \(B\). By construction \(\vect{y}'=\left(\phi_1,\ldots,\phi_J\right)\), so \(\Sigma\) meets the chart in \(\set{\vect{y}'=\vect{0}}\) and \(\tilde{f}\) vanishes there.

Integrating along the segment. For \(\vect{y}\in B\) the segment from \((\vect{0},\vect{y}'')\) to \((\vect{y}',\vect{y}'')\) lies in the box \(B\), so the fundamental theorem of calculus (Theorem 7.43) and the chain rule Equation (7.52) give

where the integral computes \(\tilde{f}(\vect{y}',\vect{y}'')-\tilde{f}(\vect{0},\vect{y}'')\) and the subtracted term is zero because \(\tilde{f}\) vanishes on \(\set{\vect{y}'=\vect{0}}\). Each \(\tilde{c}_j\) is smooth: its integrand and every partial derivative of the integrand are continuous on \(B\times[0,1]\), so Theorem 7.77 may be applied to it as often as one pleases. Setting \(c^{j}=\tilde{c}_j\circ\vect{\Psi}\) and recalling \(y'^{\,j}=\phi_j(\vect{x})\) turns Equation (7.79) into Equation (7.77).

Derives Proposition 7.85. Write \(\vect{G}=(g_1,\ldots,g_J)\). Its derivative at \(\vect{x}_\star\) has rank \(J\), so \(J\) of its columns are independent; relabelling the coordinates, split \(\vect{x}=(\vect{x}',\vect{y})\) with \(\vect{y}\in\R^{J}\) carrying those columns, so that \(\det\left[D_{\vect{y}}\vect{G}(\vect{x}_\star)\right]\neq0\). By Theorem 7.80 there are neighbourhoods \(W\ni\vect{x}'_\star\) and \(V\ni\vect{y}_\star\) and a \(C^{1}\) map \(\vect{h}:W\longrightarrow V\) whose graph is the constraint set inside \(W\times V\), with the derivative Equation (7.69).

The function \(\Xi(\vect{x}')=f\left(\vect{x}', \vect{h}(\vect{x}')\right)\) therefore has an ordinary, unconstrained local extremum at \(\vect{x}'_\star\); applying Lemma 7.33 to each one-variable function \(t\longmapsto \Xi(\vect{x}'_\star+t\vect{e}_i)\) gives \(\nabla\Xi(\vect{x}'_\star)=\vect{0}\), and the chain rule Equation (7.52) turns that into

everything evaluated at \(\vect{x}_\star\), the second equality being Equation (7.69). Now define the row vector \(\lambda=D_{\vect{y}}f\left[D_{\vect{y}}\vect{G}\right]^{-1}\), which exists and is unique because \(D_{\vect{y}}\vect{G}\) is invertible; unwinding, \(\lambda\) is the unique solution of \(\pp f/\pp y_j=\sum_\alpha\lambda^{\alpha}\pp g_\alpha/\pp y_j\), which is Equation (7.80) in the \(\vect{y}\) block. Substituting it into Equation (7.81),

which is Equation (7.80) in the \(\vect{x}'\) block. Uniqueness over all \(n\) components follows from the linear independence of the \(\nabla g_\alpha(\vect{x}_\star)\): two solutions would differ by a vanishing combination of them.

Derives Proposition 7.88. Apply Theorem 7.80 to the scalar equation \(\pp f/\pp a(\vect{x},a)=0\) at \((\vect{x}_0,a_0)\), where the derivative with respect to \(a\) is \(\pp^{2}f/\pp a^{2}\neq0\): it returns a neighbourhood of \(\vect{x}_0\) and a \(C^{1}\) function \(a(\vect{x})\) with \(a(\vect{x}_0)=a_0\) and

identically there, and no other root of \(\pp f/\pp a\) nearby — so every point of \(E\) in that neighbourhood has \(a(\vect{x})\) as its witness and satisfies \(\Psi(\vect{x})=0\). By the chain rule Equation (7.52),

whose second term vanishes identically by Equation (7.84); at \(\vect{x}_0\) this is Equation (7.83). That same vector is the gradient at \(\vect{x}_0\) of \(\vect{x}\longmapsto f(\vect{x},a_0)\), the defining function of \(S_{a_0}\). Two level sets through one point whose defining functions have the same non-zero gradient there have the same tangent plane (Definition 7.64), which is the assertion.

Derives Lemma 7.89. Both sides change sign under \(j\leftrightarrow k\) and under \(l\leftrightarrow m\), and both vanish when \(j=k\) or \(l=m\); so only the cases with \(j\neq k\) and \(l\neq m\) need checking. On the left the sum over \(i\) has at most one surviving term, the one whose \(i\) differs from both \(j\) and \(k\), and it survives only if \(i\) also differs from \(l\) and \(m\) — that is, only if \(\set{l,m}\) is the same pair as \(\set{j,k}\). If \((l,m)=(j,k)\) that term is \(\left(\epsilon_{ijk}\right)^{2}=1\) and the right-hand side is \(1\cdot1-0=1\); if \((l,m)=(k,j)\) the term is \(\epsilon_{ijk}\epsilon_{ikj}=-1\) and the right-hand side is \(0-1\cdot1=-1\); and if \(\set{l,m}\neq\set{j,k}\) the left-hand side vanishes, while on the right \(\delta_{jl}\delta_{km}=1\) would force \((l,m)=(j,k)\) and \(\delta_{jm}\delta_{kl}=1\) would force \((l,m)=(k,j)\), so both products vanish too. This exhausts the cases.

Derives Proposition 7.90. Throughout, the product rule Proposition 7.30 is applied to each Cartesian component and Equation (7.96) is used to contract the two symbols that appear whenever two cross products meet.

Equations (7.97) and (7.98). Directly, \(\pp_i(\psi A_i)=\psi\,\pp_i A_i+A_i\,\pp_i\psi\), which is the first; and \(\epsilon_{ijk}\pp_j(\psi A_k) =\psi\,\epsilon_{ijk}\pp_j A_k+\epsilon_{ijk}(\pp_j\psi)A_k\), whose second term is \(\left(\nabla\psi\times\vect{A}\right)_i\) by Equation (7.95).

Equation (7.99). \(\pp_i\left(\epsilon_{ijk}A_jB_k\right) =\epsilon_{ijk}\left(\pp_iA_j\right)B_k +\epsilon_{ijk}A_j\left(\pp_iB_k\right)\). In the first term relabel \(\epsilon_{ijk}=\epsilon_{kij}\), so that it is \(B_k\left(\nabla\times\vect{A}\right)_k\); in the second, \(\epsilon_{ijk}=-\epsilon_{jik}\), so that it is \(-A_j\left(\nabla\times\vect{B}\right)_j\).

Equation (7.100). Using Equation (7.95) twice and then Equation (7.96) with the summed index moved to the front of both symbols,

and expanding the two derivatives by the product rule gives \(A_i\,\pp_mB_m+B_m\,\pp_mA_i-B_i\,\pp_lA_l-A_l\,\pp_lB_i\), which is Equation (7.100) component by component.

Equation (7.101). By the same two steps,

and interchanging \(\vect{A}\) and \(\vect{B}\), \(\left[\vect{B}\times\left(\nabla\times\vect{A}\right)\right]_i =B_j\,\pp_iA_j-B_j\,\pp_jA_i\). Adding the two and using \(\pp_i\left(A_jB_j\right)=A_j\,\pp_iB_j+B_j\,\pp_iA_j\) rearranges to Equation (7.101). Equation (7.102) is the case \(\vect{B}=\vect{A}\), in which the two advective terms coincide and \(\pp_i(A_jA_j)=\pp_i\abs{\vect{A}}^{2}\), so that the whole identity may be divided by two.

Derives Proposition 7.91. The first two are the contraction of an object antisymmetric in two indices with one symmetric in them. By Proposition 7.73 the second partial derivatives of a \(C^{2}\) field commute, so \(\pp_j\pp_k\psi\) is symmetric in \((j,k)\) while \(\epsilon_{ijk}\) is antisymmetric, and \(\left(\nabla\times\nabla\psi\right)_i=\epsilon_{ijk}\pp_j\pp_k\psi\) vanishes: interchanging the summation labels \(j\) and \(k\) turns the sum into minus itself. The same argument on \(\nabla\cdot\left(\nabla\times\vect{A}\right) =\epsilon_{ijk}\,\pp_i\pp_jA_k\) gives Equation (7.104). For the third, by Equation (7.95) and then Equation (7.96),

the last step again by Proposition 7.73, and this is Equation (7.105).

Derives Theorem 7.99. We derive the identity for a simple region (Definition 7.95) and then for any region that can be cut into finitely many of them; what that leaves uncovered is stated in Remark 7.102.

Step 1: the \(P\) term on a vertically simple region. Let \(D\) be as in Equation (7.110). Since \(\pp_y P\) is continuous, Equation (7.112) and the fundamental theorem of calculus (Theorem 7.43), applied in \(y\) at fixed \(x\), give

Traversed counterclockwise, \(\pp D\) consists of the lower graph \(x\longmapsto(x,\varphi_1(x))\) run from \(x=a\) to \(x=b\), the vertical segment at \(x=b\), the upper graph \(x\longmapsto(x,\varphi_2(x))\) run back from \(x=b\) to \(x=a\), and the vertical segment at \(x=a\). On the two vertical pieces \(x\) is constant, so \(x'(t)=0\) and they contribute nothing to Equation (7.107). Hence, using Equation (7.116) in the last step,

Step 2: the \(Q\) term on a horizontally simple region. With \(D=\set{(x,y) \mid c\le y\le d,\ \psi_1(y)\le x\le\psi_2(y)}\) the same two moves give

The counterclockwise boundary now runs up the right graph \(y\longmapsto(\psi_2(y),y)\) from \(y=c\) to \(y=d\), which is the sense that keeps the interior on the left, and down the left graph \(y\longmapsto(\psi_1(y),y)\) from \(y=d\) to \(y=c\); the two horizontal pieces have \(y'(t)=0\) and contribute nothing to \(\oint Q\,\dd y\). Therefore

Step 3: simple regions. On a region that is both vertically and horizontally simple, both steps apply, and adding Equation (7.117) to Equation (7.118) is exactly Equation (7.115).

Step 4: decomposition. Let \(D=\bigcup_{k=1}^{N}D_k\) with every \(D_k\) simple and any two of them meeting at most along parts of their boundaries. Summing Equation (7.115) over \(k\), the area integrals add to \(\iint_{D}\left(\pp_x Q-\pp_y P\right)\) by additivity (Remark 7.96). On the boundary side, an arc interior to \(D\) belongs to exactly two of the \(\pp D_k\), whose regions lie on opposite sides of it, so it is traversed once in each direction; the two line integrals then cancel, since reversing the parametrization reverses the sign of Equation (7.107). What survives is the integral over the arcs making up \(\pp D\), traversed counterclockwise.

Derives Theorem 7.100. It suffices to treat a surface that is a graph over a plane region to which Theorem 7.99 applies: a piecewise-smooth oriented surface can be cut into finitely many such pieces — after a permutation of the coordinate axes for each — and the line integrals along the interior cuts cancel in pairs exactly as in Step 4 of the derivation of Theorem 7.99.

So let \(S=\set{(x,y,g(x,y))\in\R^{3} \mid (x,y)\in D}\) with \(g\) of class \(C^{2}\) on a neighbourhood of \(D\), oriented by the upward normal. The parametrization \(\vect{r}(x,y)=(x,y,g(x,y))\) has

whose third component is positive, so Equation (7.108) with this parametrization is the chosen orientation; and \(\pp S\) is the lift of \(\pp D\), traversed so that \(\pp D\) runs counterclockwise.

The line integral. Parametrize \(\pp D\) by \(t\longmapsto(x(t),y(t))\). Its lift has \(z(t)=g(x(t),y(t))\), so the chain rule (Proposition 7.72) gives \(z'=\pp_x g\,x'+\pp_y g\,y'\). Writing \(\tilde{A}_i(x,y)=A_i(x,y,g(x,y))\) and collecting the coefficients of \(x'\) and \(y'\) in Equation (7.106),

Both \(P\) and \(Q\) are \(C^{1}\) on \(D\), because \(\vect{A}\) is \(C^{1}\) and \(g\) is \(C^{2}\).

Green's theorem. Apply Theorem 7.99 to Equation (7.121). The chain rule (Equation (7.52), applied to the composition of \((x,y)\longmapsto(x,y,g(x,y))\) with \(A_i\)) gives \(\pp_x\tilde{A}_i=\pp_x A_i+\pp_z A_i\,\pp_x g\) and \(\pp_y\tilde{A}_i=\pp_y A_i+\pp_z A_i\,\pp_y g\), every partial derivative of \(A_i\) being evaluated on \(S\), so that

On subtraction the two second-derivative terms cancel by Clairaut–Schwarz (Proposition 7.73), which is the one place where \(g\) is required to be \(C^{2}\), and so do the two terms carrying \(\pp_z A_z\,\pp_x g\,\pp_y g\). What is left is

the second line being just Equation (7.87) read component by component. Integrating Equation (7.122) over \(D\) and comparing with Equation (7.109) and Equation (7.120),

which is Equation (7.119).

Derives Theorem 7.101. Again we derive the identity for a simple region (Definition 7.95) and extend it by decomposition.

The \(z\) component. Let \(V\) be as in Equation (7.111). Because \(\pp_z A_z\) is continuous, the three-dimensional form of Equation (7.112) and the fundamental theorem of calculus (Theorem 7.43), applied in \(z\) at fixed \((x,y)\), give

The boundary \(\pp V\) splits into the upper cap \(z=\chi_2\), the lower cap \(z=\chi_1\), and a lateral part on which the outward normal is horizontal — possibly empty, as for a ball, whose two caps meet along the equator. On the upper cap the parametrization \(\vect{r}(x,y)=(x,y,\chi_2(x,y))\) has \(\pp_x\vect{r}\times\pp_y\vect{r} =(-\pp_x\chi_2,\,-\pp_y\chi_2,\,1)\), whose third component is positive; this is therefore the outward orientation, and Equation (7.109) gives \(\iint A_z\,\hat{\vect{z}}\cdot\dd\vect{S} =\iint_{D_z}A_z(x,y,\chi_2)\,\dd x\,\dd y\). On the lower cap the outward normal points downwards, so \(\dd\vect{S}=(\pp_x\chi_1,\,\pp_y\chi_1,\,-1)\,\dd x\,\dd y\) and the same computation returns \(-\iint_{D_z}A_z(x,y,\chi_1)\,\dd x\,\dd y\). On the lateral part \(\hat{\vect{z}}\cdot\dd\vect{S}=0\). Adding the three contributions and comparing with Equation (7.124),

The other two components. Because \(V\) is simple in all three coordinate directions, the identical argument with the axes permuted gives the analogues of Equation (7.125) for \(A_x\hat{\vect{x}}\) and \(A_y\hat{\vect{y}}\). Adding the three identities, and using \(\vect{A}=A_x\hat{\vect{x}}+A_y\hat{\vect{y}}+A_z\hat{\vect{z}}\) together with Equation (7.86), yields Equation (7.123).

Decomposition. If \(V=\bigcup_{k}V_k\) with every \(V_k\) simple and any two meeting at most along parts of their boundaries, summing the identity over \(k\) adds the volume integrals. On the surface side an interior face belongs to exactly two of the \(\pp V_k\), which assign it opposite outward normals, so those two fluxes cancel; the flux through \(\pp V\) is what remains.

Derives Proposition 7.104. (1) Parametrize the loop by \(\vect{r}(t)\), \(t \in [0,1]\), with \(\vect{r}(0) = \vect{r}(1)\). By the chain rule (Proposition 7.72), \(\dv{}{t} f(\vect{r}(t)) = \nabla f(\vect{r}(t))\cdot\vect{r}'(t)\), so

by the fundamental theorem of calculus (Theorem 7.43). (2) Componentwise, e.g.\ \((\nabla\times\nabla f)_z = \pp_x\pp_y f - \pp_y\pp_x f = 0\) by Clairaut–Schwarz (Proposition 7.73); the hypothesis \(\vect{F} \in C^1\) makes \(f\) twice continuously differentiable. (3) Fix \(\vect{a} \in A\) and define \(f(\vect{p}) = \int_{\gamma}\vect{F}\cdot\dd\vect{l}\) along any path \(\gamma \subset A\) from \(\vect{a}\) to \(\vect{p}\). Well-definedness is the point: two paths bound (by simple connectedness) a surface swept by a homotopy inside \(A\), and Stokes' theorem (Theorem 7.100) with \(\nabla\times\vect{F} = \vect{0}\) makes the two line integrals equal. Differentiating along a coordinate step then gives \(\pp_i f = F_i\), i.e. \(\nabla f = \vect{F}\).

Derives Lemma 7.107. Let \(\varepsilon>0\). For each \(j\) choose, by Definition 7.106, a countable family of boxes covering \(E_j\) of total volume less than \(\varepsilon2^{-j}\). The union of these families is a countable family — a countable union of countable families is countable, by Proposition 3.67 — it covers \(\bigcup_jE_j\), and its total volume is at most \(\sum_{j\ge1}\varepsilon2^{-j}=\varepsilon\) by the geometric series (Proposition 7.46). A single point is null, being contained in a cube of arbitrarily small volume, so a countable set is a countable union of null sets and the first part applies; and \(\Q^{N}\) is countable by Corollary 3.68 and Proposition 3.67.

Derives Proposition 7.109. One threshold. Let \(0\le s<t\) and suppose \(\mathcal{H}^{s}(E)<\infty\). For any family covering \(E\) with diameters at most \(\delta\),

so \(\mathcal{H}^{t}_{\delta}(E)\le\delta^{\,t-s} \mathcal{H}^{s}_{\delta}(E)\le\delta^{\,t-s}\mathcal{H}^{s}(E)\), which tends to \(0\) as \(\delta\longrightarrow0^{+}\): hence \(\mathcal{H}^{t}(E)=0\) for every \(t>s\). Consequently the set \(Z=\set{s\ge0\mid\mathcal{H}^{s}(E)=0}\) contains, with any of its points, every larger one, and contains every \(s\) at which \(\mathcal{H}^{s}(E)\) is merely finite together with everything above it. Put \(d=\inf Z\), which exists by Axiom 7.1 once \(Z\) is known to be non-empty — the next paragraph shows \((N,\infty)\subseteq Z\). If \(s>d\) there is \(t\in Z\) with \(t<s\), so \(\mathcal{H}^{s}(E)=0\). If \(s<d\) then \(\mathcal{H}^{s}(E)\) cannot be finite: were it finite, every \(t>s\) would lie in \(Z\) and \(\inf Z\) would be at most \(s<d\). So \(\mathcal{H}^{s}(E)=+\infty\), which is Equation (7.134); and those two clauses determine \(d\) as \(\inf Z\), so it is unique.

The bound \(d\le N\). It is enough to show \(\mathcal{H}^{s}(E)=0\) for every \(s>N\). First, \(\mathcal{H}^{s}_{\delta}\) is countably subadditive — covers of the pieces concatenate into a cover of the union — so \(\mathcal{H}^{s}_{\delta}\left(\bigcup_jE_j\right) \le\sum_j\mathcal{H}^{s}_{\delta}(E_j) \le\sum_j\mathcal{H}^{s}(E_j)\), and letting \(\delta\longrightarrow0^{+}\) shows \(\mathcal{H}^{s}\) is countably subadditive too. Since \(\R^{N}\) is the union of countably many unit lattice cubes, it is enough to treat \(E\) inside one such cube \(Q\). Split \(Q\) into \(n^{N}\) closed subcubes of edge \(1/n\), hence of diameter \(\sqrt{N}/n\): they cover \(E\), and

whenever \(s>N\), while their diameters tend to \(0\). Hence \(\mathcal{H}^{s}_{\delta}(E)=0\) for every \(\delta>0\), so \(\mathcal{H}^{s}(E)=0\).

Null sets. Suppose \(\mathcal{H}^{N}(E)=0\) and let \(\varepsilon>0\). Choose a family \(\set{U_k}\) covering \(E\) with \(\sum_k\left(\mathrm{diam}\,U_k\right)^{N}<\varepsilon\). A set of diameter \(t\) has each of its \(N\) coordinate projections contained in an interval of length \(t\), hence lies in a closed cube of edge \(t\) and volume \(t^{N}\). Replacing each \(U_k\) by such a cube covers \(E\) by countably many boxes of total volume less than \(\varepsilon\), which is Equation (7.131). If \(\dim_{\mathrm{H}}E<N\) then \(\mathcal{H}^{N}(E)=0\) by Equation (7.134), so \(E\) is null.

Derives Proposition 7.111. The second inequality is the definition of a lower and an upper limit. For the first, let \(s>\underline{\dim}_{\mathrm{B}}E\) and pick \(s'\) with \(\underline{\dim}_{\mathrm{B}}E<s'<s\). By Equation (7.136) there is a sequence \(\delta_1>\delta_2>\cdots\longrightarrow0\) along which \(\ln N_{\delta_j}(E)/\ln(1/\delta_j)<s'\), that is \(N_{\delta_j}(E)<\delta_j^{-s'}\). Covering \(E\) by that many sets of diameter at most \(\delta_j\),

and since \(\mathcal{H}^{s}(E)\) is the limit of \(\mathcal{H}^{s}_{\delta}(E)\) along any sequence \(\delta\longrightarrow0^{+}\) (Equation (7.133)), it follows that \(\mathcal{H}^{s}(E)=0\) and hence \(\dim_{\mathrm{H}}E\le s\) by Equation (7.134). Letting \(s\) decrease to \(\underline{\dim}_{\mathrm{B}}E\) gives Equation (7.137).

For the example, let \(E=\Q\cap[0,1]\). It is countable, so for \(s>0\) and \(\varepsilon\in(0,\delta)\) covering its \(k\)-th point by an interval of length \(\varepsilon2^{-k}\) gives \(\mathcal{H}^{s}_{\delta}(E)\le\varepsilon^{s} \sum_{k\ge1}2^{-ks}=\varepsilon^{s}/(2^{s}-1)\), which tends to \(0\) with \(\varepsilon\); hence \(\mathcal{H}^{s}(E)=0\) for every \(s>0\) and \(\dim_{\mathrm{H}}E=0\) (consistently with Lemma 7.107, which already makes \(E\) null). But any cover of \(E\) by sets of diameter at most \(\delta\) may be replaced by the cover by their closures, of the same diameters, which then covers the closure \([0,1]\) of \(E\); so \(N_\delta(E)\ge1/\delta\), and Equation (7.136) gives \(\underline{\dim}_{\mathrm{B}}E\ge1\), while \(E\subseteq[0,1]\) forces \(\overline{\dim}_{\mathrm{B}}E\le1\).

Derives Proposition 7.113. Each \(\delta_\sigma\) is positive with unit integral (the Gaussian integral; computed in Probability and Statistics), so

Split the domain at \(\abs{x - x_0} = \eta\): on the inner part continuity makes \(\abs{f - f(x_0)} < \varepsilon\) and the weight integrates to at most \(1\); on the outer part \(\abs{f - f(x_0)} \le 2\sup\abs{f}\) while the Gaussian tail mass tends to \(0\) as \(\sigma \to 0\) for fixed \(\eta\). Both contributions are eventually below \(\varepsilon\). The scaling rule follows by substituting \(u = ax\) in \(\int f(x)\,\delta(ax)\,\dd x\) (Corollary 7.44), which gives \(f(0)/\abs{a}\) (the sign of \(a\) flips the orientation, whence the absolute value); and \(\int f(x)\,x\,\delta(x)\,\dd x = 0\cdot f(0) = 0\) for every test function, which is the meaning of \(x\,\delta(x) = 0\).