Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Theorem 4.4. Under the Vandiver conjecture, the Kummer duality gives rise\nto a compact discrete duality\n\n\\[\n\\operatorname{Gal}{\\left( \\Omega /{\\Omega }_{E}\\right) }^{ + }\\text{dual to}{C}_{\\infty }^{ - }\n\\]\n\nand also\n\n\\[\n\\operatorname{Gal}{\\left( \\Omega /{\\Omega }^{\\mathrm{{nr}}}\\right) }^{ + ... | Proof. By Theorems 2.2,2.3 and Lemma 1 of \\( §2 \\) we know that \\( \\Omega = {\\Omega }_{A} \\) and that\n\n\\[\n\\operatorname{Gal}\\left( {\\Omega /{\\Omega }_{E}}\\right) \\text{is dual to}{C}_{\\infty }\\text{.}\n\\]\n\nTaking eigenspaces for complex conjugation yields the first assertion. As to the second, we k... | Yes |
Theorem 1.1. Let \( 1 \leq k, j \leq p - 2 \) . (i) \( {\varphi }_{k}\left( {\eta }_{k}\right) = - k{\;\operatorname{mod}\;p} \) . (ii) \( {\varphi }_{k}\left( {\eta }_{j}\right) = 0 \) if \( k \neq j \) . | Proof. The second assertion is a special case of \( \mathbf{K}\mathbf{4} \) (ii). As to the first, \[ {\varphi }_{k}\left( {\eta }_{k}\right) = {\varphi }_{k}\left( {1 - {\pi }^{k}}\right) \] An associated power series of \( 1 - {\pi }^{k} \) is \( f\left( X\right) = 1 - {X}^{k} \), and \[ {f}^{\prime }/f\left( X\right... | Yes |
Theorem 1.2. Let \( u \) be as above. Then\n\n\[ \n{t}_{k} \equiv - \frac{1}{k}{\varphi }_{k}\left( u\right) {\;\operatorname{mod}\;p}.\n\] | Proof. Immediate, from \( \mathbf{K}\mathbf{4} \) and the fact that \( {\varphi }_{k} \) is a \( {\mathbf{Z}}_{p} \) -morphism. | No |
Theorem 1.3. Let \( j, k = 1,\ldots, p - 2 \) . Then:\n\n(i) \( {\varphi }_{k}\left( {\eta }_{j}\right) = 0 \) if \( k \neq j \) .\n\n(ii) \( {\varphi }_{k}\left( {\eta }_{k}\right) = - k{\;\operatorname{mod}\;p} \) . | Proof. Taking the logarithmic derivative formally, we have:\n\n\[ \frac{d\log }{d{w}_{0}}{\eta }_{k} = \frac{d\log }{d{w}_{0}}\mathop{\prod }\limits_{\sigma }{\left( 1 - {\chi }^{k}\left( \sigma \right) {w}_{0}^{k}\right) }^{-{\chi }^{k}\left( \sigma \right) } \]\n\n\[ = \mathop{\sum }\limits_{\sigma } - {\chi }^{-k}\l... | Yes |
Theorem 1.4. Let\n\n\[ \n u\left( {\mathcal{A},\mathcal{N}}\right) = {\eta }_{1}^{{t}_{1}}\cdots {\eta }_{p - 2}^{{t}_{p - 2}}{\;\operatorname{mod}\;{w}_{0}^{p - 1}}. \n\]\n\nThen\n\n\[ \n {t}_{k} = \frac{1}{{k}^{2}}{B}_{k}\sum {n}_{i}{a}_{i}^{k}{\;\operatorname{mod}\;p}. \n\]\n\nwhere \( {B}_{k} \) is the Bernoulli nu... | Proof. Let \( A \) be the formal multiplicative group, \( B \) the special Lubin-Tate group associated with it. The power series\n\n\[ \n {g}_{{\mathbf{G}}_{a}}\left( Z\right) = {e}^{aZ} - 1 \n\]\n\ncorresponds to the power series \( {g}_{B}\left( W\right) \) such that\n\n\[ \n {g}_{B}\left( {w}_{0}\right) = {\zeta }^{... | Yes |
Lemma 2. U has no \( {\mathbf{Z}}_{p} \) -torsion. | Proof. Otherwise there exists a fixed power \( {p}^{r} \) and an element \( u = \lim {u}_{n} \) such that \( {u}_{n}^{{p}^{r}} = 1 \) for all \( n \) . Then \( {u}_{n} \) is a root of unity, and if \( {u}_{n} \neq 1 \) for some \( n \) , then the order of \( {u}_{m} \) becomes arbitrarily large as \( m \) becomes large... | No |
Theorem 2.1. For each character \( \chi \neq 1,{\varkappa }_{0} \) of \( {G}_{0} \) there is a \( \Lambda \) -isomorphism\n\n\[U\left( \chi \right) \approx \Lambda\]\n\nIn other words, \( U\left( \chi \right) \) is free of dimension 1 over \( \Lambda \) . | The proof will occupy the rest of this section, and will result from a sequence of lemmas. A \ | No |
Lemma 3. If \( \chi \neq 1 \) then we have isomorphisms\n\n\[ \operatorname{Gal}\left( {{K}_{n}^{\mathrm{{ab}}}/{K}_{\infty }}\right) \left( \chi \right) \approx \operatorname{Gal}{\left( {K}_{\infty }^{\mathrm{{ab}}}/{K}_{\infty }\right) }_{\left( n\right) }\left( \chi \right) \]\n\n\[ \approx \operatorname{Gal}\left(... | Proof. This is clear from the fact that \( {K}_{n}^{\mathrm{{ab}}} \) is the maximal abelian extension of \( {K}_{n} \) contained in \( {K}_{\infty }^{\mathrm{{ab}}} \), together with the exact sequence\n\n\[ 0 \rightarrow \operatorname{Gal}\left( {{K}_{n}^{\mathrm{{ab}}}/{K}_{\infty }}\right) \rightarrow \operatorname... | Yes |
Lemma 4. Let \( M \) be a finitely generated \( \Lambda \) -module such that\n\n\[ M/\left( {{\gamma }^{{p}^{n}} - 1}\right) M \] \nis free over \( {\mathbf{Z}}_{p} \) of rank \( {p}^{n} \) for all \( n \) . Then \( M \) is quasi-isomorphic to \( \Lambda \) . | Proof. Obvious from the structure theorem in Chapter 5. | No |
Lemma 5. In the exact sequence, we have \( B = 0 \), for \( \chi \neq 1,{\varkappa }_{0} \) . | Proof. From the exact sequence\n\n\[ 0 \rightarrow U\left( \chi \right) \rightarrow \Lambda \rightarrow B \rightarrow 0 \]\n\nwe get the exact (cohomology) sequence\n\n\[ 0 \rightarrow U{\left( \chi \right) }^{\left( n\right) } \rightarrow {\Lambda }^{\left( n\right) } \rightarrow {B}^{\left( n\right) } \rightarrow U{\... | Yes |
Lemma 1. Given \( \chi = {\varkappa }_{0}^{k} \neq 1,{\varkappa }_{0} \) there exists \( \lambda \in {\mu }_{p - 1} \) such that if we let \( b = \lambda - 1 \), and\n\n\[ \n{\xi }_{0} = {\xi }_{0}^{\left( \lambda \right) } = \omega {\left( b\right) }^{-1}\left( {b - {x}_{0}}\right) \n\]\n\nthen:\n\n(i) \( {\varphi }_{... | Proof. We shall check below that for a suitable choice of \( \lambda \) (depending on \( k \) ) the Kummer-Takagi exponent given by Theorem 1.2 is \( ≢ 0{\;\operatorname{mod}\;p} \) . Then \( {\xi }_{0}\left( \chi \right) \) generates \( {U}_{0}\left( \chi \right) /{U}_{0}{\left( \chi \right) }^{p} \), and hence genera... | Yes |
Theorem 3.1. Let \( \chi \neq 1,{\varkappa }_{0} \) . We can choose \( \lambda \in {\mu }_{p - 1} \) such that the element\n\n\[ \xi \left( \chi \right) = {\xi }^{\left( \lambda \right) }\left( \xi \right) \]\n\ngenerates \( U\left( \chi \right) \) over \( \Lambda \), i.e.,\n\n\[ U\left( \chi \right) = \Lambda \cdot \x... | Proof. We know from Theorem 2.2 that\n\n\[ {U}_{0}\left( \chi \right) = U\left( \chi \right) /\left( {\gamma - 1}\right) U\left( \chi \right) \]\n\nand so by Lemma \( 1, e\left( \chi \right) \cdot \xi \) generates \( U\left( \chi \right) {\;\operatorname{mod}\;{\mathfrak{m}}_{\Lambda }} \cdot U\left( \chi \right) \) . ... | Yes |
To every element \( u \in U \) there is a unique power series \( f \in {\mathbf{Z}}_{p}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) such that\n\n\[ \n{f}_{u}\left( {x}_{n}\right) = {u}_{n} \n\]\n\nThis power series satisfies \( {f}_{u}\left( X\right) \equiv 1{\;\operatorname{mod}\;\left( {p, X}\right) } \)... | We first note that uniqueness is obvious since a power series has only a finite number of zeros (Weierstrass preparation theorem).\n\nThe proof of existence will proceed via several steps, which also develop systematically other properties of these series. First:\n\nCW 0.\n\[ \n{f}_{\xi }\left( X\right) = \frac{1}{\ome... | No |
Theorem 4.2. Given a congruence class \( \alpha {\;\operatorname{mod}\;p} - 1 \), there exists a power series \( {h}_{\alpha } \) such that for any \( k \equiv \alpha {\;\operatorname{mod}\;p} - 1 \), we have\n\n\[ \left( {1 - {p}^{k - 1}}\right) {\varphi }_{k}\left( \xi \right) = {h}_{\alpha }\left( {x{\left( \gamma \... | Proof. Let\n\n\[ {f}_{1}\left( X\right) = D\log {f}_{\xi }\left( X\right) = \left( {1 + X}\right) {f}_{\xi }^{\prime }/{f}_{\xi }\left( X\right) . \]\n\nThen by Meas 6 of Chapter 4,\n\n\[ {\varphi }_{k}\left( \xi \right) = {D}^{k - 1}{f}_{1}\left( 0\right) = {\int }_{{\mathbf{Z}}_{p}}{x}^{k - 1}d{\mu }_{{f}_{1}}\left( ... | Yes |
Theorem 5.1. For each even character \( \chi \neq 1 \) we have\n\n\[ \n{V}_{n}\left( \chi \right) = {\mathrm{Z}}_{p}\left\lbrack {G}_{n}\right\rbrack {v}_{n}\left( \chi \right) \]\n\nand hence\n\n\[ \nV\left( \chi \right) = \Lambda \cdot v\left( \chi \right) \]\n | Proof. Immediate from Theorem 3.2 of Chapter 6. | No |
Theorem 5.2. Let \( \chi = {\varkappa }_{0}^{\alpha } \) be an even character \( \neq 1 \) . We have an isomorphism\n\n\[ U\left( \chi \right) /V\left( \chi \right) \approx \Lambda /{g}_{\chi }\Lambda \]\n\nThe power series \( {g}_{\chi } \) (determined up to a unit in \( \Lambda \) ) can be selected such that it is eq... | Proof. We have\n\n\[ {\varphi }_{k}\left( v\right) = {\varphi }_{k}\left( {v\left( \chi \right) }\right) \]\n\nby \( \mathbf{{CW}}\mathbf{4} \)\n\n\[ = {g}_{\chi }\left( {u{\left( \gamma \right) }^{k} - 1}\right) {\varphi }_{k}\left( {\xi \left( \chi \right) }\right) \]\n\nby CW 5\n\n\[ = {g}_{\chi }\left( {x{\left( \g... | Yes |
Theorem 1.1. To each Frobenius power series \( f \) in \( {\mathcal{F}}_{\pi } \) there exists a unique formal group \( {F}_{f} \) (defined over \( \mathfrak{o} \) ) such that \( f \) is an endomorphism of \( {F}_{f} \) . | The proof of this theorem will follow from a general lemma, as will the fact that the formal group \( {F}_{f} \) then admits \( \mathfrak{o} \) in a natural way as a ring of endomorphisms commuting with \( f \) . | No |
Theorem 1.2. The association \( a \mapsto {a}_{f} \) is an injective ring homomorphism of o into \( \operatorname{End}\left( {F}_{f}\right) \), such that\n\n\[{\pi }_{f} = f\text{.}\]\n\nMore generally, the association \( a \mapsto {a}_{f, g} \) is an injective additive homomorphism of \( \mathfrak{o} \) into \( \opera... | Proof. In each case, one checks immediately that both the left-hand side and right-hand side of the desired identity are solutions of the type given in the Lemma, whose solution is unique.\n\nIt is clear that if \( f, g \in {\mathcal{F}}_{\pi } \) then the element \( {1}_{f, g} \) is an isomorphism between \( {F}_{g} \... | Yes |
Theorem 2.1. (i) The group \( {A}_{{\pi }^{n}} \) is a free 1-dimensional module over \( \mathfrak{o}/{\pi }^{n}\mathfrak{o} \) . (ii) \( K\left( {A}_{{\pi }^{n}}\right) \) is abelian over \( K \), totally ramified, and we have a natural isomorphism \[ \varkappa : \operatorname{Gal}\left( {K\left( {A}_{{\pi }^{n}}\righ... | Proof. Let \( \left( {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right) \) be a sequence with \( {x}_{k} \in {A}_{{\pi }^{k}} \), such that \( {x}_{1} \neq 0 \) and \( {\pi }_{f}\left( {x}_{k}\right) = {x}_{k - 1} \) . Without loss of generality we may assume \[ f\left( X\right) = {X}^{q} + {\pi X}. \] For \( k > 1 \) we see tha... | Yes |
Theorem 2.2. The prime \( \pi \) is a norm from every extension \( K\left( {A}_{{\pi }^{n}}\right) \) . | Proof. Consider first the bottom level of the tower \( K\left( {A}_{\pi }\right) \) over \( K \), obtained from the equation\n\n\[ \n{X}^{q - 1} + \pi = 0.\n\]\n\nLet \( \alpha \) be a root. Then\n\n\[ \n{\left( -1\right) }^{q - 1}N\left( \alpha \right) = \pi \n\]\n\nIf \( q \) is odd then \( \pi \) is the norm of \( \... | Yes |
Theorem 2.3. Let \( B \) be the special Lubin-Tate group associated with the prime \( \pi \) and the Frobenius polynomial \( {X}^{q} + {\pi X} \) . Let \( \zeta \in {\mu }_{q - 1} \) . Then:\n\n(i)\n\n\[ \left\lbrack \zeta \right\rbrack \left( X\right) = {\zeta X}\text{.}\]\n\n(ii) If \( F\left( {X, Y}\right) \) is the... | Proof. Let \( {X}_{n} \) be a generator of \( {B}_{{\pi }^{n + 1}} \) such that\n\n\[ \left\lbrack \pi \right\rbrack \left( {x}_{n}\right) = {x}_{n - 1} \]\n\nSince \( {x}_{0} \) is a root of \( {X}^{q - 1} + \pi = 0 \) it follows by a trivial recursion that the irreducible polynomial for \( {x}_{n} \) over \( K \) is ... | Yes |
Theorem 4.1. The field \( {K}^{\left( \pi \right) }{K}_{\mathrm{{nr}}} \) is independent of \( \pi \) . Let \( a \in {K}^{ * } \) . Write\n\n\[ a = u{\pi }^{m} \]\n\nfor some unit \( u \), and some integer \( m \) . Let \( {r}_{\pi }\left( a\right) \) be the automorphism of \( {K}^{\left( \pi \right) }{K}_{\mathrm{{nr}... | Proof. Let \( L \) be the completion of \( {K}_{\mathrm{{nr}}} \) as in the preceding section. Let \( A \) be the Lubin-Tate formal groups associated with the prime \( \pi \), and let \( {A}^{\prime } \) be associated with the prime \( {\pi }^{\prime } \) . Since \( A \) and \( {A}^{\prime } \) are isomorphic over \( L... | Yes |
Lemma 1. We have \( {N}_{n}^{-1}\left( {\pi }^{\mathbf{Z}}\right) = \mathop{\bigcap }\limits_{{m \geq n}}{N}_{m, n}{K}_{m}^{ * } \) . | Proof. Suppose \( {N}_{n}\alpha \in {\pi }^{\mathbf{Z}} \) . Then\n\n\[ \left( {\alpha ,{K}_{m}/{K}_{n}}\right) = \left( {{N}_{n}\alpha ,{K}_{m}/K}\right) \in \left( {{\pi }^{\mathbf{Z}},{K}_{m}/K}\right) = 1 \]\n\n because \( \pi \) is a norm from each extension \( {K}_{m} \) by Theorem 2.2. Hence \( \alpha \) is a no... | Yes |
Lemma 2. Assume \( k > {q}^{n} \) . Then \( A\left( {\pi {\pi }_{n}^{k}{\mathfrak{o}}_{n}}\right) = \left\lbrack \pi \right\rbrack A\left( {{\pi }_{n}^{k}{\mathfrak{o}}_{n}}\right) \) . | Proof. The inclusion \( \supset \) is obvious. We prove the reverse inclusion. Let \( z = \pi {\pi }_{n}^{k}t \) with \( t \in {\mathfrak{o}}_{n} \) . We must solve\n\n\[ \n{x}^{q} + {\pi x} = z\text{ with }x = {\pi }_{n}^{k}y\text{ and }y \in {\mathfrak{o}}_{n}.\n\]\n\nThis is equivalent to\n\n\[ \n{\pi }_{n}^{qk}{y}^... | Yes |
Theorem 5.1. Let \( {w}_{i} \) be elements of \( {\mathfrak{p}}_{n} \) for \( i = 1,\ldots ,{q}^{n + 1} \), such that \( {\operatorname{ord}}_{{\mathfrak{p}}_{n}}{w}_{i} = i \) . Then these elements generate \( A\left( {\mathfrak{p}}_{n}\right) {\;\operatorname{mod}\;\left\lbrack \pi \right\rbrack }A\left( {\mathfrak{p... | Proof. Since \( X{ + }_{A}Y \equiv X + Y{\;\operatorname{mod}\;\deg }2 \), given \( x \in {\mathfrak{p}}_{n} \) we can find \( {a}_{1} \in \mathfrak{o} \) such that\n\n\[ x - {}_{A}\left\lbrack {a}_{1}\right\rbrack {w}_{1} \equiv 0{\;\operatorname{mod}\;{\mathfrak{p}}_{n}^{2}}, \]\n\nbecause \( \left\lbrack {a}_{1}\rig... | Yes |
Theorem 5.3. Assume that \( K = {\mathbf{Q}}_{p} \), and that \( K\left( {A}_{\pi }\right) \) does not contain the \( p \) th roots of unity. Then the local pairing\n\n\[ \nA\left( {\mathfrak{p}}_{n}\right) /\left\lbrack {\pi }^{n + 1}\right\rbrack A\left( {\mathfrak{p}}_{n}\right) \times {K}_{n}^{ * }/{K}_{n}^{*{p}^{n... | Proof. In Theorem 5.2 we have determined the order of \n\n\[ \nA\left( {\mathfrak{p}}_{n}\right) /\left\lbrack {\pi }^{n + 1}\right\rbrack A\left( {\mathfrak{p}}_{n}\right) \]\n\nIt is a standard exercise of local algebraic number theory [L 1], Chapter II, §3 to determine that \n\n\[ \n\text{order of}{K}_{n}^{ * }/{K}_... | No |
Lemma 1. The limit\n\n\[ \lambda \left( X\right) = \lim \frac{1}{{\pi }^{n}}{\pi }_{A}^{n}\left( X\right) \]\n\nexists, and gives a formal isomorphism of the Lubin-Tate formal group \( A \) with the additive group \( {\mathbf{G}}_{a} \) . | Proof of the lemma. We look at the difference\n\n\[ \frac{1}{{\pi }^{n + r}}{\pi }_{A}^{n + r}\left( X\right) - \frac{1}{{\pi }^{n}}{\pi }_{A}^{n}\left( X\right) = \frac{1}{{\pi }^{n + r}}\left( {{\pi }_{A}^{r} \circ {\pi }_{A}^{n}\left( X\right) - {\pi }^{r}{\pi }_{A}^{n}\left( X\right) }\right) .\n\nLet\n\n\[ {\pi }_... | Yes |
Lemma 2. The \( \log {\lambda }_{A} \) commutes with the action of \( \mathfrak{o} \), that is,\n\n\[ \n{\lambda }_{A}\left( {{a}_{A}\left( X\right) }\right) = a{\lambda }_{A}\left( X\right) \;\text{ for }a \in {\mathfrak{o}}_{K}.\n\] | Proof. The function \( X \mapsto {\lambda }_{A}\left( {{a}_{A}\left( X\right) }\right) \) is an additive formal power series. such that\n\n\[ \n{\lambda }_{A}\left( {{a}_{A}\left( X\right) }\right) \equiv {aX}{\;\operatorname{mod}\;\deg }2.\n\]\n\nThe uniqueness of the logarithm shows that this function is \( a{\lambda... | Yes |
Lemma 3. (i) Let \( {\lambda }^{\prime } \) denote the formal derivative \( {d\lambda }/{dX} \) . Then \( {\lambda }_{A}^{\prime }\left( X\right) \) has coefficients in \( \mathfrak{o} \) . | Proof. For (i) we differentiate with respect to \( Y \) the relation\n\n\[ \n{\lambda }_{A}\left( {F\left( {X, Y}\right) }\right) = {\lambda }_{A}\left( X\right) + {\lambda }_{A}\left( Y\right) \n\]\n\nand get\n\n\[ \n{\lambda }_{A}^{\prime }\left( {F\left( {X, Y}\right) }\right) {D}_{2}F\left( {X, Y}\right) = {\lambda... | Yes |
Lemma 4. Let \( {e}_{A}\left( Z\right) \) be the power series (with coefficients in \( K \) ) which is the inverse of \( {\lambda }_{A}\left( X\right) \) . Let \( D \) be the disc in \( {\mathfrak{m}}_{{K}^{\mathrm{a}}} \) consisting of those elements \( z \) such that\n\n\[ \n{\operatorname{ord}}_{\pi }z > \frac{1}{q ... | Proof. Let \( y \in D, y \neq 0 \) . Define\n\n\[ \n{\lambda }_{y}\left( X\right) = \frac{1}{y}{\lambda }_{A}\left( {yX}\right) \n\]\n\nThen for \( i > 0 \),\n\n\[ \n\operatorname{ord}\frac{1}{y}\frac{{y}^{{q}^{i}}}{{\pi }^{i}} = \left( {{q}^{i} - 1}\right) \operatorname{ord}y - i > \frac{{q}^{i} - 1}{q - 1} - i. \n\]\... | Yes |
Lemma 5. The kernel of \( {\lambda }_{A} \) in the maximal ideal of the algebraic closure of \( K \) is precisely \( {A}_{\text{tor }} \), the group of torsion points on \( A \), or in other words, the group \( {A}^{\left( \pi \right) } \) . | Proof. A point \( x \) is a torsion point if and only if \( \left\lbrack {\pi }^{n}\right\rbrack x \) is a torsion point for some positive integer \( n \), or for every large positive integer \( n \) . But \( \left\lbrack {\pi }^{n}\right\rbrack x \) approaches 0, and for large \( n \), lies in the neighborhood of 0 wh... | Yes |
Theorem 1.1. Suppose \( \alpha \in {\mathfrak{o}}_{n} \) and \( \alpha \equiv 1{\;\operatorname{mod}\;{\mathfrak{p}}_{n}} \) . Then\n\n\[ \n{N}_{n}\alpha \equiv 1{\;\operatorname{mod}\;{\pi }^{n + 1}} \n\]\n\nand\n\n\[ \n{\left\langle {x}_{n},\alpha \right\rangle }_{n} = \left\lbrack {\frac{1}{{\pi }^{n + 1}}\left( {{N... | Proof. By the formalism of the norm residue symbol, we know that\n\n\[ \n1 = \left( {\alpha ,{K}_{n}/{K}_{n}}\right) = \left( {{N}_{n}\alpha ,{K}_{n}/K}\right) . \n\]\n\nHence \( \left\lbrack {{N}_{n}\alpha }\right\rbrack {x}_{n} = {x}_{n} \) by the Lubin-Tate theory, so the first assertion is clear.\n\nWe choose \( t ... | Yes |
Corollary 1. Let \( \\alpha \\equiv 1{\\;\\operatorname{mod}\\;{\\mathfrak{p}}_{n}} \) . Assume that \( K \) is unramified over \( {\\mathbf{Q}}_{p} \) . Then\n\n\[ \n\\left\\langle {{x}_{n},\\alpha }\\right\\rangle = \\left\\lbrack {-\\frac{1}{{\\pi }^{n + 1}}{T}_{n}\\left( {\\log \\alpha }\\right) }\\right\\rbrack {x... | Proof. Since \( \\pi \) is unramified, we can write\n\n\[ \n{N}_{n}{\\alpha }^{-1} = 1 + z \n\]\n\nwhere \( z \\equiv 0{\\;\\operatorname{mod}\\;{p}^{n + 1}} \) . Since \( p \\neq 2 \) it follows that\n\n\[ \n\\log {N}_{n}{\\alpha }^{-1} \\equiv z{\\;\\operatorname{mod}\\;{z}^{2}}. \n\]\n\nHence\n\n\[ \n- {T}_{n}\\left... | Yes |
Corollary 2. Let \( A = {\mathbf{G}}_{m} \) be the formal multiplicative group. Let \( \zeta \) be a primitive \( {p}^{n + 1} \) th root of unity, and let \( \alpha \equiv 1{\;\operatorname{mod}\;{p}_{n}} \) . Then \[ \left( {\zeta ,\alpha }\right) = {\zeta }^{-\left( {1/{p}^{n + 1}}\right) {T}_{n}\left( {\log \alpha }... | Proof. Special case of Corollary 1. | Yes |
Theorem 1.2. Let \( x \in A\left( {\mathfrak{p}}_{n}\right) \) and \( \alpha \in {K}_{n}^{ * } \) . Suppose \( \alpha = {N}_{m, n}{\alpha }_{m} \) for some \( {\alpha }_{m} \in {K}_{m}^{ * } \) . Under either one of the conditions (i),(ii), the symbol \( {\left\lbrack x,{\alpha }_{m}\right\rbrack }_{m} \) has value in ... | \[ \langle x,\alpha {\rangle }_{n}^{A} = {\left\lbrack x,{\alpha }_{m}\right\rbrack }_{m}^{A}\left( {x}_{n}\right) \] | No |
Lemma 3.1. The symbol \( {\left\lbrack x,\alpha \right\rbrack }_{m} \) is well defined \( {\;\operatorname{mod}\;{\pi }^{n + 1}} \) in each of the following cases:\n\n(i) \( x \in A\left( {\mathfrak{p}}_{n}\right) \) and \( m \geq {2n} + 1 \) ;\n\n(ii) \( x \in A\left( {\mathfrak{p}}_{n}^{2{q}^{n}}\right) \) and \( m \... | Proof. By DL 2 we know that \( {\delta }_{m}\left( \alpha \right) \) is well defined mod \( {\mathfrak{D}}_{0}{\mathfrak{o}}_{m} \) . Hence the symbol is defined \( {\;\operatorname{mod}\;{\pi }^{n + 1}} \) if\n\n\[ \n{T}_{m}\left( {\frac{1}{\pi }\lambda \left( x\right) {\mathfrak{D}}_{0}{\mathfrak{v}}_{m}}\right) \sub... | Yes |
Lemma 3.2. Let \( k \geq m \geq n \) . Let \( {\alpha }_{m} \in {K}_{m}^{ * } \) and \( {\alpha }_{k} \in {K}_{k}^{ * } \) be such that\n\n\[{\alpha }_{m} = {N}_{k, m}{\alpha }_{k}\]\n\nThen\n\n\[{\left\lbrack x,{\alpha }_{k}\right\rbrack }_{k} \equiv {\left\lbrack x,{\alpha }_{m}\right\rbrack }_{m}{\;\operatorname{mod... | Proof. We have:\n\n\[{\left\lbrack x,{\alpha }_{m}\right\rbrack }_{m} = {\left\lbrack x,{N}_{k, m}{\alpha }_{k}\right\rbrack }_{m}\]\n\n\[\equiv \frac{1}{\pi }{T}_{m}\left( {\lambda \left( x\right) {\delta }_{m}\left( {{N}_{k, m}{\alpha }_{k}}\right) }\right) {\;\operatorname{mod}\;{\pi }^{n + 1}}\]\n\n\[\equiv \frac{1... | Yes |
Theorem 5.1. We have the equality\n\n\[ \n{\left\\langle x,{x}_{n}\right\\rangle }_{n} = {\left\\lbrack x,{x}_{m}\right\\rbrack }_{m}^{A}\left( {x}_{n}\right) \n\]\n\nunder either of the following conditions:\n\n(i) \( x \in A\left( {\\mathfrak{p}}_{n}\right) \) and \( m \geq {2n} + 1 \)\n\n(ii) \( x \in A\left( {{\\ma... | Proof. By LS 5 of \( §5 \) in the preceding chapter, we have for \( m \geq n \)\n\n\[ \n{\left\\langle x,{x}_{n}\right\\rangle }_{n} = {\left\\langle \left\\lbrack {\\pi }^{m - n}\right\\rbrack x,{x}_{m}\right\\rangle }_{m}.\n\]\n\nThis shifts the burden of the proof to level \( m \), and \( {\left\\lbrack {\\pi }^{m -... | Yes |
Theorem 5.2. Assume that the Frobenius power series associated with the Lubin-Tate group \( A \) has the form\n\n\[ f\left( X\right) = {X}^{q} + \cdots + {\pi X} \]\n\ni.e., is a polynomial of degree \( q \) with leading coefficient 1 . Define more precisely\n\n\[ \left\lbrack {x, - {x}_{n}}\right\rbrack = \frac{1}{{\p... | Proof. First observe that the elements \( - {x}_{m} \) form a vector\n\n\[ \left( {-{x}_{0}, - {x}_{1},\ldots }\right) \in T\left( {K}_{\infty }^{ * }\right) \]\n\ni.e., each is the norm of the successive one. Instead of using Lemma 3.2, however, which relied on DL-5, we may now use directly the more precise relation D... | Yes |
Take \( A = {\mathbf{G}}_{m} \) to be the formal multiplicative group. Then it satisfies the hypothesis of Theorem 5.2, and we obtain another reciprocity law of Artin-Hasse: | \[ \left( {x, - {x}_{n}}\right) = {\zeta }^{\left\lbrack x, - {x}_{n}\right\rbrack } \] where \[ \left\lbrack {x, - {x}_{n}}\right\rbrack = \frac{1}{{p}^{n + 1}}{T}_{n}\left( {\frac{\zeta }{{x}_{n}}\log \left( {1 + x}\right) }\right) \] \[ {x}_{n} = \zeta - 1 \] and \( \zeta \) is a primitive \( {p}^{n + 1} \) th root ... | Yes |
We contend that:\n\n\[ \left\langle {{x}_{0}^{i},{x}_{0}}\right\rangle = 0 \\text{if} i \\text{is an integer prime to} p \\text{, or} i > p \\text{.} \]\n\n\[ \left\langle {{x}_{0}^{p},{x}_{0}}\right\rangle = {x}_{0} \] | Proof. For the first statement, we have by multiplicativity:\n\n\[ 0 = \left\langle {{x}_{0}^{i},{x}_{0}^{i}}\right\rangle = \left\lbrack i\right\rbrack \left\langle {{x}_{0}^{i},{x}_{0}}\right\rangle \]\n\nIf \( i \) is prime to \( p \), this proves our assertion because \( {A}_{p} \) is a \( p \) -group. If \( i > p ... | Yes |
Let \( j \geq 1 \) . Then\n\n\[ \n{\left\langle {x}_{n}^{i},\varepsilon {x}_{n}^{j} - 1\right\rangle }_{n} = \left\lbrack {-j}\right\rbrack \mathop{\sum }\limits_{{r = 1}}^{\infty }{\left\langle {\varepsilon }^{r}{x}_{n}^{i + {rj}},{x}_{n}\right\rangle }_{n}.\n\] | Proof. Let \( F \) be the group law on \( A \) . Since\n\n\[ \nF\left( {X, Y}\right) \equiv X + Y{\;\operatorname{mod}\;X}Y \]\n\nwe obtain for \( x, y \in {\mathfrak{p}}_{n} \),\n\n\[ \nx\left\lbrack +\right\rbrack y \equiv x + y{\;\operatorname{mod}\;x}y\text{ and }x\left\lbrack -\right\rbrack y \equiv x - y{\;\opera... | Yes |
Lemma 6.3. Assume that \( B \) is the special Lubin-Tate group with Frobenius power series \( {X}^{q} + {\pi X} \) . Let \( {w}_{0} \in {B}_{\pi } \) . Then for \( i \geq 2 \) and \( j \geq 1 \) we have the same formula with \( n = 0 \) as in the previous lemma, namely \[ {\left\langle {w}_{0}^{i},{w}_{0}^{j} - 1\right... | Proof. By Theorem 2.3(ii) of Chapter 8, we know that the group law on \( B \) satisfies \[ F\left( {X, Y}\right) = X + Y + \text{ terms of degree } \geq q. \] If \( x, y \in {\mathfrak{p}}_{0}{}^{2} \) it follows that \[ x + y \equiv x\left\lbrack +\right\rbrack y{\;\operatorname{mod}\;{\mathfrak{p}}_{0}^{2q}}. \] But ... | Yes |
Theorem 7.1. Let \( x \in A\left( {\mathfrak{p}}_{n}\right) \) and let\n\n\[ \alpha = \left( {{\alpha }_{n},{\alpha }_{n + 1},\ldots ,{\alpha }_{k}}\right) \;\text{ with }{\alpha }_{m} = {N}_{k, m}{\alpha }_{k}. \]\n\nAssume that\n\n\[ k \geq \left\lbrack {n/2}\right\rbrack + 2\left( {n + 1}\right) \;\text{if}p\text{is... | Proof. The theorem has already been proved when \( \alpha \) is a power of \( {x}_{n} \) or when \( \alpha \in {\mu }_{q - 1} \) . We may therefore assume that \( \alpha \) is a unit \( \equiv 1{\;\operatorname{mod}\;{\mathfrak{p}}_{n}} \) . We reduce the theorem to the result of the preceding section in exactly the sa... | Yes |
Theorem 7.2. Assume \( p \neq 2 \) . Let \( x \in A\left( {\mathfrak{p}}_{0}^{2}\right) \) and \( \alpha \in {K}_{0}^{ * } \) . Then\n\n\[{\left\lbrack x,\alpha \right\rbrack }_{0} = \frac{1}{\pi }{T}_{0}\left( {\lambda \left( x\right) {\delta }_{0}\left( \alpha \right) }\right)\]\n\nis well defined \( {\;\operatorname... | Proof. Since \( \lambda \left( {x}_{0}\right) = 0 \) it follows that \( \lambda \left( {\mathfrak{p}}_{0}\right) = \lambda \left( {\mathfrak{p}}_{0}^{2}\right) = {\mathfrak{p}}_{0}^{2} \) . This shows that \( {\left\lbrack x,\alpha \right\rbrack }_{0} \) is well defined \( {\;\operatorname{mod}\;\pi } \) because \( {\d... | Yes |
Problem. Show that this rectangle is inscribed in the ellipse \( U \leq E \) . | The general solution of our equations is \( {x}_{1} = {A}_{1}\sin \left( {t + {\varphi }_{1}}\right) ,{x}_{2} = \) \( {A}_{2}\sin \left( {{\omega t} + {\varphi }_{2}}\right) \) ; a moving point independently performs an oscillation with frequency 1 and amplitude \( {A}_{1} \) along the horizontal and an oscillation wit... | No |
Problem 1. Show that the angle \( \Phi \) between the pericenter and apocenter is equal to the semiperiod of an oscillation in the one-dimensional system with potential energy \( W\left( x\right) = U\left( {M/x}\right) + \left( {{x}^{2}/2}\right) \) . | Hint. The substitution \( x = M/r \) gives\n\n\[ \Phi = {\int }_{{x}_{\min }}^{{x}_{\max }}\frac{dx}{\sqrt{2\left( {E - W}\right) }} \] | No |
Problem 2. Find the angle \( \Phi \) for an orbit close to the circle of radius \( r \) . | Answer. \( \Phi \approx {\Phi }_{\text{cir }} = \pi \left( {M/{r}^{2}\sqrt{{V}^{\prime \prime }\left( r\right) }}\right) = \pi \sqrt{{U}^{\prime }/\left( {3{U}^{\prime } + r{U}^{\prime \prime }}\right) } \) . | Yes |
Problem 3. For which values of \( U \) is the magnitude of \( {\Phi }_{\text{cir }} \) independent of the radius \( r \) ? | Answer. \( U\left( r\right) = a{r}^{\alpha }\left( {\alpha \geq - 2,\alpha \neq 0}\right) \) and \( U\left( r\right) = b\log r \) . It follows that \( {\Phi }_{\text{cir }} = \pi /\sqrt{\alpha + 2} \) (the logarithmic case corresponds to \( \alpha = 0) \) . For example, for \( \alpha = 2 \) we have \( {\Phi }_{\text{ci... | Yes |
Problem 4. Let in the situation of problem \( {3U}\left( r\right) \rightarrow \infty \) as \( r \rightarrow \infty \) . Find \( \mathop{\lim }\limits_{{E \rightarrow \infty }}\Phi \left( {E, M}\right) \) . | Answer. \( \pi /2 \) . | No |
Problem 5. Let \( U\left( r\right) = - k{r}^{-\beta },0 < \beta < 2 \) . Find \( {\Phi }_{0} = \mathop{\lim }\limits_{{E \rightarrow - 0}}\Phi \) . | Answer. \( {\Phi }_{0} = {\int }_{0}^{1}{dx}/\sqrt{{x}^{\beta } - {x}^{2}} = \pi /\left( {2 - \beta }\right) \) . Note that \( {\Phi }_{0} \) does not depend on \( M \) . | Yes |
Problem 6. Find all central fields in which bounded orbits exist and are all closed. | Answer. \( U = a{r}^{2} \) or \( U = - k/r \) . Solution. If all bounded orbits are closed, then, in particular, \( {\Phi }_{\text{cir }} = \) \( {2\pi }\left( {m/n}\right) = \) const. According to Problem 3, \( U = a{r}^{\alpha }\left( {\alpha \geq - 2}\right) \), or \( U = b\ln r \) \( \left( {\alpha = 0}\right) \) .... | Yes |
Corollary 1 (The law of conservation of angular momentum). If the system is closed, then \( \mathbf{M} = \) const. | We denote the sum of the moments of the external forces by \( \mathbf{N} = \) \( \mathop{\sum }\limits_{{i = 1}}^{n}\left\lbrack {{\mathbf{r}}_{i},{\mathbf{F}}_{i}^{\prime }}\right\rbrack \)\n\nThen, by the theorem above, \( d\mathbf{M}/{dt} = \mathbf{N} \), from which we have | No |
Example 2. We consider planar motion in a central field in polar coordinates \( {q}_{1} = r,{q}_{2} = \varphi \) . From the relation \( \dot{\mathbf{r}} = \dot{r}{\mathbf{e}}_{r} + \dot{\varphi }r{\mathbf{e}}_{\varphi } \) we find the kinetic energy \( T = \frac{1}{2}m{\dot{\mathbf{r}}}^{2} = \frac{1}{2}m\left( {{\dot{... | The generalized momenta will be \( \mathbf{p} = \partial L/\partial \dot{\mathbf{q}} \), i.e., \[ {p}_{1} = m\dot{r}\;{p}_{2} = m{r}^{2}\dot{\varphi } \] The first Lagrange equation \( {\dot{p}}_{1} = \partial L/\partial {q}_{1} \) takes the form \[ m\ddot{r} = {mr}{\dot{\varphi }}^{2} - \frac{\partial U}{\partial r} \... | Yes |
Theorem. The Legendre transformation is involutive, i.e., its square is the identity: if under the Legendre transformation \( f \) is taken to \( g \), then the Legendre transform of \( g \) will again be \( f \) . | Proof. In order to apply the Legendre transform to \( g \), with variable \( p \), we must by definition look at a new independent variable (which we will call \( x \) ), construct the function\n\n\[ G\left( {x, p}\right) = {xp} - g\left( p\right) \]\n\nand find the point \( p\left( x\right) \) at which \( G \) attains... | Yes |
Corollary 1. \( {dH}/{dt} = \partial H/\partial t \) . In particular, for a system whose hamiltonian function does not depend explicitly on time \( \left( {\partial H/\partial t = 0}\right) \), the law of conservation of the hamiltonian function holds: \( H\left( {\mathbf{p}\left( t\right) ,\mathbf{q}\left( t\right) }\... | Proof. We consider the variation in \( H \) along the trajectory \( H\left( {\mathbf{p}\left( t\right) ,\mathbf{q}\left( t\right), t}\right) \) . Then, by Hamilton's equations,\n\n\[ \frac{dH}{dt} = \frac{\partial H}{\partial \mathbf{p}}\left( {-\frac{\partial H}{\partial \mathbf{q}}}\right) + \frac{\partial H}{\partia... | Yes |
Let \( {q}_{1} \) be a cyclic coordinate. Then \( {p}_{1} \) is a first integral. In this case the variation of the remaining coordinates with time is the same as in a system with the \( n - 1 \) independent coordinates \( {q}_{2},\ldots ,{q}_{n} \) and with hamiltonian function\n\n\[ H\left( {{p}_{2},\ldots ,{p}_{n},{... | Proof. We set \( {\mathbf{p}}^{\prime } = \left( {{p}_{2},\ldots ,{p}_{n}}\right) \) and \( {\mathbf{q}}^{\prime } = \left( {{q}_{2},\ldots ,{q}_{n}}\right) \) . Then Hamilton’s equations take the form\n\n\[ \frac{d}{dt}{\mathbf{q}}^{\prime } = \frac{\partial H}{\partial {\mathbf{p}}^{\prime }}\;\frac{d}{dt}{q}_{1} = \... | Yes |
Corollary 3. Every closed system with two degrees of freedom \( \left( {n = 2}\right) \) which has a cyclic coordinate is integrable. | Proof. In this case the system for \( {p}^{\prime } \) and \( {q}^{\prime } \) is one-dimensional and is immediately integrated by means of the integral \( H\left( {{p}^{\prime },{q}^{\prime }}\right) = c \) . | Yes |
Theorem 2. If div \( \mathbf{f} \equiv 0 \), then \( {g}^{t} \) preserves volume: \( v\left( t\right) = v\left( 0\right) \) . | C Proof | No |
Lemma 1. \( {\left. \left( dv/dt\right) \right| }_{t = 0} = {\int }_{D\left( 0\right) }\operatorname{div}\mathbf{f}{dx}\;\left( {{dx} = d{x}_{1}\cdots d{x}_{n}}\right) \) . | Proof. For any \( t \), the formula for changing variables in a multiple integral gives\n\n\[ v\left( t\right) = {\int }_{D\left( 0\right) }\det \frac{\partial {g}^{t}\mathbf{x}}{\partial \mathbf{x}}{dx} \]\n\nCalculating \( \partial {g}^{t}\mathbf{x}/\partial \mathbf{x} \) by formula (1), we find\n\n\[ \frac{\partial ... | Yes |
Lemma 2. For any matrix \( A = \left( {a}_{ij}\right) \) , \[ \det \left( {E + {At}}\right) = 1 + t\operatorname{tr}A + \mathrm{O}\left( {t}^{2}\right) ,\;t \rightarrow 0, \] where \( \operatorname{tr}A = \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{ii} \) is the trace of \( A \) (the sum of the diagonal elements). | (The proof of Lemma 2 is obtained by a direct expansion of the determinant: we get 1 and \( n \) terms in \( t \) ; the remaining terms involve \( {t}^{2},{t}^{3} \), etc.) | No |
Problem 1. Suppose that a particle moves in the field of the uniform helical line \( x = \cos \varphi \) , \( y = \sin \varphi, z = {c\varphi } \) . Find the law of conservation corresponding to this helical symmetry. | Answer. In any system which admits helical motions leaving our helical line fixed, the quantity \( I = c{P}_{3} + {M}_{3} \) is conserved. | Yes |
Problem 4. Extend Noether's theorem to non-autonomous lagrangian systems. | Hint. Let \( {M}_{1} = M \times \mathbb{R} \) be the extended configuration space (the direct product of the configuration manifold \( M \) with the time axis \( \mathbb{R} \) ).\n\nDefine a function \( {L}_{1} : T{M}_{1} \rightarrow \mathbb{R} \) by\n\n\[ L\frac{dt}{d\tau } \]\n\ni.e., in local coordinates \( \mathbf{... | No |
Consider the system of two identical mathematical pendulums of length \( {l}_{1} = {l}_{2} = 1 \) and mass \( {m}_{1} = {m}_{2} = 1 \) in a gravitational field with \( g = 1 \). Suppose that the pendulums are connected by a weightless spring whose length is equal to the distance between the points of suspension (Figure... | Set \[ {Q}_{1} = \frac{{q}_{1} + {q}_{2}}{\sqrt{2}}\text{ and }{Q}_{2} = \frac{{q}_{1} - {q}_{2}}{\sqrt{2}}. \] Then \[ {q}_{1} = \frac{{Q}_{1} + {Q}_{2}}{\sqrt{2}}\text{ and }{q}_{2} = \frac{{Q}_{1} - {Q}_{2}}{\sqrt{2}} \] and both forms are reduced to principal axes: \[ T = \frac{1}{2}\left( {{\dot{Q}}_{1}^{2} + {\do... | Yes |
Theorem 2. If the ellipsoid \( E \) with semi-axes \( {a}_{1} \geq {a}_{2} \geq \cdots \geq {a}_{n} \) contains the ellipsoid \( {E}^{\prime } \) with semi-axes \( {a}_{1}^{\prime } \geq {a}_{2}^{\prime } \geq \cdots \geq {a}_{n}^{\prime } \), both ellipses having the same center, then the semi-axes of the inside ellip... | \[ {a}_{1} \geq {a}_{1}^{\prime },{a}_{2} \geq {a}_{2}^{\prime },\ldots ,{a}_{n} \geq {a}_{n}^{\prime }.\] | Yes |
Theorem 3. The characteristic frequencies of the system with a constraint separate the characteristic frequencies of the original system (Figure 92):\n\n\[ \n{\omega }_{1} \leq {\omega }_{1}^{\prime } \leq {\omega }_{2} \leq {\omega }_{2}^{\prime } \leq \cdots \leq {\omega }_{n - 1} \leq {\omega }_{n - 1}^{\prime } \le... | By Lemma 2 this theorem is equivalent to the following geometric proposition. | No |
Theorem 4. Consider the cross-section of the n-dimensional ellipsoid \( E = \) \( \{ \mathbf{q} : \left( {B\mathbf{q},\mathbf{q}}\right) = 1\} \; \) with semi-axes \( \;{a}_{1} \geq {a}_{2} \geq \cdots \geq {a}_{n}\; \) by a hyperplane \( \;{\mathbb{R}}^{n - 1} \) through its center. Then the semi-axes of this \( \left... | \[ {a}_{1} \geq {a}_{1}^{\prime } \geq {a}_{2} \geq {a}_{2}^{\prime } \geq \cdots \geq {a}_{n - 1} \geq {a}_{n - 1}^{\prime } \geq {a}_{n}. \] | Yes |
Theorem 5. The smallest semi-axis of any cross-section of the ellipsoid \( E \) with semi-axes \( {a}_{1} \geq {a}_{2} \geq \cdots \geq {a}_{n} \) by a subspace \( {\mathbb{R}}^{k} \) is less than or equal to \( {a}_{k} \) : | \[ {a}_{k} = \mathop{\max }\limits_{\left\{ {\mathbb{R}}^{k}\right\} }\mathop{\min }\limits_{{\mathbf{x} \in {\mathbb{R}}^{k} \cap E}}\parallel \mathbf{x}\parallel \] (the upper bound is attained on the subspace spanned by the semi-axes \( \left. {{a}_{1} \geq {a}_{2} \geq \cdots \geq {a}_{k}}\right) \). Proof. \( {}^{... | Yes |
A pendulum in a periodically varying gravitational field (for example, the moon) is described by Hill's equation: | \[ \ddot{q} = - {\omega }^{2}\left( t\right) q\;\omega \left( {t + T}\right) = \omega \left( t\right) \] | No |
Lemma 1. The operator \( A \) is skew-symmetric: \( {A}^{t} + A = 0 \) . | Proof. Since \( B : K \rightarrow k \) is an orthogonal operator from one euclidean space to another, its transpose is its inverse: \( {B}^{t} = {B}^{-1} : k \rightarrow K \) . By differentiating the relationship \( B{B}^{t} = E \) with respect to \( t \), we get\n\n\[ \dot{B}{B}^{t} + B{\dot{B}}^{t} = 0\;\dot{B}{B}^{-... | Yes |
Lemma 2. Every skew-symmetric operator \( A \) on a three-dimensional oriented euclidean space is the operator of vector multiplication by a fixed vector: | Proof. The skew-symmetric operators from \( {\mathbb{R}}^{3} \) to \( {\mathbb{R}}^{3} \) form a linear space. Its dimension is 3, since a skew-symmetric \( 3 \times 3 \) matrix is determined by its three elements below the diagonal.\n\nThe operator of vector multiplication by \( \mathbf{\omega } \) is linear and skew-... | Yes |
Theorem. Motion in a rotating coordinate system takes place as if three additional inertial forces acted on every moving point \( \mathbf{Q} \) of mass \( m \) :\n\n1. the inertial force of rotation: \( m\left\lbrack {\dot{\mathbf{\Omega }},\mathbf{Q}}\right\rbrack \) ,\n\n2. the Coriolis force: \( {2m}\left\lbrack {\m... | Proof of the theorem. We notice that for any vector \( \mathbf{X} \in K \) we have \( \dot{B}\mathbf{X} = B\left\lbrack {\mathbf{\Omega },\mathbf{X}}\right\rbrack \) . In fact, by Section 26, \( \dot{B}\mathbf{X} = \left\lbrack {\mathbf{\omega },\mathbf{x}}\right\rbrack = \left\lbrack {B\mathbf{\Omega }, B\mathbf{X}}\r... | Yes |
Show that for every 2 -form \( {\omega }^{2} \) on \( {\mathbb{R}}^{n} \) we have\n\n\[ \n{\omega }^{2}\left( {\xi ,\xi }\right) = 0,\;\forall \xi \in {\mathbb{R}}^{n}.\n\] | Solution. By skew symmetry, \( {\omega }^{2}\left( {\xi ,\xi }\right) = - {\omega }^{2}\left( {\xi ,\xi }\right) \). | Yes |
Show that this space is finite-dimensional, and find its dimension. | ANSWER. \( n\left( {n - 1}\right) /2 \) ; a basis is shown below. | No |
Show that this vector space is finite-dimensional and find its dimension. | ANSWER. \( \left( \begin{array}{l} n \\ k \end{array}\right) \) : a basis is shown below. | No |
Show that the mapping\n\n\\[ \n\\left( {{\\omega }_{1},{\\omega }_{2}}\\right) \\rightarrow {\\omega }_{1} \\land {\\omega }_{2} \n\\]\n\nis bilinear and skew symmetric: | \\[ \n{\\omega }_{1} \\land {\\omega }_{2} = - {\\omega }_{2} \\land {\\omega }_{1} \n\\]\n\n\\[ \n\\left( {{\\lambda }^{\\prime }{\\omega }_{1}^{\\prime } + {\\lambda }^{\\prime \\prime }{\\omega }_{1}^{\\prime \\prime }}\\right) \\land {\\omega }_{2} = {\\lambda }^{\\prime }{\\omega }_{1}^{\\prime } \\land {\\omega }... | No |
Show that every 2-form on the \( n \) -dimensional space with coordinates \( {x}_{1},\ldots ,{x}_{n} \) can be uniquely represented in the form\n\n\[ \n{\omega }^{2} = \mathop{\sum }\limits_{{i < j}}{a}_{ij}{x}_{i} \land {x}_{j} \n\] | Hint. Let \( {\mathbf{e}}_{i} \) be the \( i \) -th basis vector, i.e., \( {x}_{i}\left( {\mathbf{e}}_{i}\right) = 1,{x}_{j}\left( {\mathbf{e}}_{i}\right) = 0 \) for \( i \neq j \) . Look at the value of the form \( {\omega }^{2} \) on the pair \( {\mathbf{e}}_{i},{\mathbf{e}}_{j} \) . Then\n\n\[ \n{a}_{ij} = {\omega }... | No |
Problem 13. Show that every \( k \) -form on \( {\mathbb{R}}^{n} \) can be uniquely represented as a linear combination of basic forms: | \[ {\omega }^{k} = \mathop{\sum }\limits_{{1 \leq {i}_{1} < \cdots < {i}_{k} \leq n}}{a}_{{i}_{1},\ldots ,{i}_{k}}{x}_{{i}_{1}} \land \cdots \land {x}_{{i}_{k}}. \] Hint. \( {a}_{{i}_{1},\ldots ,{i}_{k}} = {\omega }^{k}\left( {{\mathbf{e}}_{{i}_{1}},\ldots ,{\mathbf{e}}_{{i}_{k}}}\right) \) . | Yes |
Show that the product of monomials is associative: | \[ \left( {{\omega }^{k} \land {\omega }^{l}}\right) \land {\omega }^{m} = {\omega }^{k} \land \left( {{\omega }^{l} \land {\omega }^{m}}\right) \] | Yes |
Problem 3. Find the exterior square of the 2 -form \( {\omega }^{2} \). | \[ {\omega }^{2} \land {\omega }^{2} = - 2\mathop{\sum }\limits_{{i > j}}{p}_{i} \land {p}_{j} \land {q}_{i} \land {q}_{j} \] | Yes |
Problem 4. Find the exterior \( k \) -th power of \( {\omega }^{2} \) . | \[ \underset{k}{\underbrace{{\omega }^{2} \land {\omega }^{2} \land \cdots \land {\omega }^{2}}} = \pm k!\mathop{\sum }\limits_{{{i}_{1} < \cdots < {i}_{k}}}{p}_{{i}_{1}} \land \cdots \land {p}_{{i}_{k}} \land {q}_{{i}_{1}} \land \cdots \land {q}_{{i}_{k}}. \] | Yes |
Problem 5. Show that the maps \( \mathbf{A} \rightarrow {\omega }_{\mathbf{A}}^{1} \) and \( \mathbf{A} \rightarrow {\omega }_{\mathbf{A}}^{2} \) establish isomorphisms of the linear space \( {\mathbb{R}}^{3} \) of vectors \( \mathbf{A} \) with the linear spaces of 1 -forms on \( {\mathbb{R}}^{3} \) and 2 -forms on \( ... | \[ {\omega }_{\mathbf{A}}^{1} = {A}_{1}{x}_{1} + {A}_{2}{x}_{2} + {A}_{3}{x}_{3} \] and \[ {\omega }_{A}^{2} = {A}_{1}{x}_{2} \land {x}_{3} + {A}_{2}{x}_{3} \land {x}_{1} + {A}_{3}{x}_{1} \land {x}_{2}. \] | Yes |
Show that, under the isomorphisms established above, the exterior product of a 1 -form and a 2 -form becomes the scalar product of vectors in \( {\mathbb{R}}^{3} \) : | \[ {\omega }_{\mathbf{A}}^{1} \land {\omega }_{\mathbf{B}}^{2} = \left( {\mathbf{A},\mathbf{B}}\right) {x}_{1} \land {x}_{2} \land {x}_{3} \] | Yes |
Problem 1. Let \( \xi \) be the velocity vector of the plane curve \( x\left( t\right) = \cos t, y\left( t\right) = \sin t \) at \( t = 0 \) . Calculate the values of the differentials \( {dx} \) and \( {dy} \) of the functions \( x \) and \( y \) on the vector \( \xi \) (Figure 141). | \[ {\left. dx\right| }_{\left( 1,0\right) }\left( \xi \right) = 0,{\left. dy\right| }_{\left( 1,0\right) }\left( \xi \right) = 1 \] | Yes |
Problem 4. Calculate the value of the forms \( {\omega }_{1} = d{x}_{1},{\omega }_{2} = {x}_{1}d{x}_{2} \), and \( {\omega }_{3} = d{r}^{2}\left( {{r}^{2} = {x}_{1}^{2} + {x}_{2}^{2}}\right) \) on the vectors \( {\xi }_{1},{\xi }_{2} \), and \( {\xi }_{3} \). | <table><thead><tr><th></th><th>\( {\xi }_{1} \)</th><th>\( {\xi }_{2} \)</th><th>\( {\xi }_{3} \)</th></tr></thead><tr><td>\( {\omega }_{1} \)</td><td>0</td><td>\( - 1 \)</td><td>1</td></tr><tr><td>\( {\omega }_{2} \)</td><td>0</td><td>\( - 2 \)</td><td>\( - 2 \)</td></tr><tr><td>\( {\omega }_{3} \)</td><td>0</td><td>\... | Yes |
Problem 7. Show that the \( k \) -forms on \( M \) form a vector space (infinite-dimensional if \( k \) does not exceed the dimension of \( M \) ). | Differential forms can be multiplied by functions as well as by numbers. Therefore, the set of \( {C}^{\infty } \) differential \( k \) -forms has a natural structure as a module over the ring of infinitely differentiable real functions on \( M \) . | No |
Problem 8. Calculate the value of the forms \( {\omega }_{1} = d{x}_{1} \land d{x}_{2},{\omega }_{2} = {x}_{1}d{x}_{1} \land d{x}_{2} - {x}_{2}d{x}_{2} \land \) \( d{x}_{1} \), and \( {\omega }_{3} = {rdr} \land {d\varphi } \) (where \( {x}_{1} = r\cos \varphi \) and \( {x}_{2} = r\sin \varphi \) ) on the pairs of vect... | <table><thead><tr><th></th><th>\( \left( {{\xi }_{1},{\eta }_{1}}\right) \)</th><th>\( \left( {{\xi }_{2},{\eta }_{2}}\right) \)</th><th>\( \left( {{\xi }_{3},{\eta }_{3}}\right) \)</th></tr></thead><tr><td>\( {\omega }_{1} \)</td><td>1</td><td>1</td><td>\( - 1 \)</td></tr><tr><td>\( {\omega }_{2} \)</td><td>2</td><td>... | Yes |
Problem 9. Calculate the value of the forms \( {\omega }_{1} = d{x}_{2} \land d{x}_{3},{\omega }_{2} = {x}_{1}d{x}_{3} \land d{x}_{2} \), and \( {\omega }_{3} = d{x}_{3} \land d{r}^{2}\left( {{r}^{2} = {x}_{1}^{2} + {x}_{2}^{2} + {x}_{3}^{2}}\right) \), on the pair of vectors \( \xi = \left( {1,1,1}\right) ,\mathbf{\et... | ANSWER. \( {\omega }_{1} = 1,{\omega }_{2} = - 2,{\omega }_{3} = - 8 \) . | No |
Problem 11. Given the form written in the \( x \) -coordinates (i.e., the \( {X}_{i} \) ) and the change of variables formulas \( \mathbf{x} = \mathbf{x}\left( \mathbf{y}\right) \), write the form in \( y \) -coordinates, i.e., find \( Y \) . | Solution. We have \( d{x}_{i} = \left( {\partial {x}_{i}/\partial {y}_{1}}\right) d{y}_{1} + \left( {\partial {x}_{i}/\partial {y}_{2}}\right) d{y}_{2} + \left( {\partial {x}_{i}/\partial {y}_{3}}\right) d{y}_{3} \) . Therefore,\n\n\[ d{x}_{2} \land d{x}_{3} = \left( {\frac{\partial {x}_{2}}{\partial {y}_{1}}d{y}_{1} +... | Yes |
Problem 12. Find \( {E}_{1},{E}_{2} \), and \( {E}_{3} \) for cartesian coordinates \( x, y, z \), for cylindrical coordinates \( r,\varphi, z \) and for spherical coordinates \( R,\varphi ,\theta \) in the euclidean space \( {\mathbb{R}}^{3} \) (Figure 144). | \[ d{s}^{2} = d{x}^{2} + d{y}^{2} + d{z}^{2} = d{r}^{2} + {r}^{2}d{\varphi }^{2} + d{z}^{2} = d{R}^{2} + {R}^{2}{\cos }^{2}{\theta d}{\varphi }^{2} + {R}^{2}d{\theta }^{2}. \] | No |
Problem 13. Find the values of the forms \( d{x}_{1}, d{x}_{2} \), and \( d{x}_{3} \) on the vectors \( {\mathbf{e}}_{1},{\mathbf{e}}_{2} \), and \( {\mathbf{e}}_{3} \) . | Answer. \( d{x}_{i}\left( {\mathbf{e}}_{i}\right) = 1/\sqrt{{E}_{i}} \), the rest are zero. In particular, for cartesian coordinates \( {dx}\left( {\mathbf{e}}_{x}\right) = \) \( {dy}\left( {\mathbf{e}}_{y}\right) = {dz}\left( {\mathbf{e}}_{z}\right) = 1 \) ; for cylindrical coordinates \( {dr}\left( {\mathbf{e}}_{r}\r... | Yes |
Problem 14. Calculate \( \left\lbrack {{\mathbf{e}}_{1},{\mathbf{e}}_{2}}\right\rbrack ,\left( {{\mathbf{e}}_{R},{\mathbf{e}}_{0}}\right) \), and \( \left( {{\mathbf{e}}_{z},{\mathbf{e}}_{x},{\mathbf{e}}_{y}}\right) \) . | ANSWER. \( {\mathbf{e}}_{3},0,1 \) . | Yes |
Problem 15. Given the components of the vector field \( \mathbf{A} \), find the decompositions of the 1-form \( {\omega }_{\mathbf{A}}^{1} \) and the 2 -form \( {\omega }_{\mathbf{A}}^{2} \) . | Solution. We have \( {\omega }_{\mathbf{A}}^{1}\left( {\mathbf{e}}_{1}\right) = \left( {\mathbf{A},{\mathbf{e}}_{1}}\right) = {A}_{1} \) . Also, \( \left( {{a}_{1}d{x}_{1} + {a}_{2}d{x}_{2} + {a}_{3}d{x}_{3}}\right) \left( {\mathbf{e}}_{1}\right) = \) \( {a}_{1}d{x}_{1}\left( {\mathbf{e}}_{1}\right) = {a}_{1}/\sqrt{{E}... | Yes |
Problem 16. Find the components of the gradient of a function in the basis \( {\mathbf{e}}_{1},{\mathbf{e}}_{2},{\mathbf{e}}_{3} \) . | Solution. We have \( {df} = \left( {\partial f/\partial {x}_{1}}\right) d{x}_{1} + \left( {\partial f/\partial {x}_{2}}\right) d{x}_{2} + \left( {\partial f/\partial {x}_{3}}\right) d{x}_{3} \) . By the problem above\n\n\[ \operatorname{grad}f = \frac{1}{\sqrt{{E}_{1}}}\frac{\partial f}{\partial {x}_{1}}{\mathbf{e}}_{1... | Yes |
Problem 4. Show that the map \( {f}^{ * } \) preserves operations on forms: | \[ {f}^{ * }\left( {{\lambda }_{1}{\omega }_{1} + {\lambda }_{2}{\omega }_{2}}\right) = {\lambda }_{1}{f}^{ * }\left( {\omega }_{1}\right) + {\lambda }_{2}{f}^{ * }\left( {\omega }_{2}\right) ,\] \[ {f}^{ * }\left( {{\omega }_{1} \land {\omega }_{2}}\right) = \left( {{f}^{ * }{\omega }_{1}}\right) \land \left( {{f}^{ *... | Yes |
Problem 6. Let \( {D}_{1} \) and \( {D}_{2} \) be two compact, convex polyhedra in the oriented \( k \) -dimensional space \( {\mathbb{R}}^{k} \) and \( f : {D}_{1} \rightarrow {D}_{2} \) a differentiable map which is an orientation-preserving diffeomorphism \( {}^{55} \) of the interior of \( {D}_{1} \) onto the inter... | Hint. This is the change of variables theorem for a multiple integral: \[ {\int }_{{D}_{1}}\frac{\partial \left( {{y}_{1},\ldots ,{y}_{n}}\right) }{\partial \left( {{x}_{1},\ldots ,{x}_{n}}\right) }\varphi \left( {y\left( x\right) }\right) d{x}_{1}\cdots d{x}_{n} = {\int }_{{D}_{2}}\varphi \left( y\right) d{y}_{1}\cdot... | No |
Problem 8. Show that, under a change of orientation, the integral changes sign: | \[ {\int }_{-\sigma }\omega = - {\int }_{\sigma }\omega \] | Yes |
Problem 10. Show that the boundary of the boundary of any chain is zero: \( \partial \partial {c}_{k} = 0 \) . | Hint. By the linearity of \( \partial \) it is enough to show that \( \partial \partial D = 0 \) for a convex polyhedron \( D \) . It remains to verify that every \( \left( {k - 2}\right) \) -dimensional face of \( D \) appears in \( \partial \partial D \) twice, with opposite signs. It is enough to prove this for \( k... | No |
Problem 11. Show that the integral depends linearly on the form: | \[ {\int }_{{c}_{k}}{\omega }_{1}^{k} + {\omega }_{2}^{k} = {\int }_{{c}_{k}}{\omega }_{1}^{k} + {\int }_{{c}_{k}}{\omega }_{2}^{k} \] | Yes |
Let \( M \) be the plane \( \{ \left( {p, q}\right) \} ,{\omega }^{1} \) the form \( {pdq} \), and \( {c}_{1} \) the chain consisting of one cell \( \sigma \) with multiplicity 1 :\n\n\[ \left\lbrack {0 \leq t \leq {2\pi }}\right\rbrack \overset{f}{ \rightarrow }\left( {p = \cos t, q = \sin t}\right) . \]\n\nThen \( {\... | In general, if a chain \( {c}_{1} \) represents the boundary of a region \( G \) (Figure 154), then \( {\int }_{{c}_{1}}{pdq} \) is equal to the area of \( G \) with sign + or - depending on whether the pair of vectors (outward normal, oriented boundary vector) has the same or opposite orientation as the pair ( \( p \)... | No |
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