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Example 3.3 (Forward price of a stock that pays no dividends): If \( B = \) \( {S}_{1}\left( T\right) \), the first stock pays no dividends, and \( \sigma \left( \cdot \right) \) satisfies the Novikov condition (1.5.17) with \( n = 1 \), then (3.3) and (3.5) yield | \[ f\left( t\right) = \frac{{S}_{1}\left( t\right) /{S}_{0}\left( t\right) }{{E}_{0}\left\lbrack {1/{S}_{0}\left( T\right) \mid \mathcal{F}\left( t\right) }\right\rbrack },\;0 \leq t \leq T. \] | Yes |
Example 3.4 (Forward price of a stock with nonrandom dividend rate): If \( B = {S}_{1}\left( T\right) \), the dividend rate process \( {\delta }_{1}\left( \cdot \right) \) is nonrandom and the processes \( {\sigma }_{11}\left( \cdot \right) ,\ldots ,{\sigma }_{1N}\left( \cdot \right) \) are uniformly bounded, then the ... | To see this, observe from (1.5.16) that the process \( \frac{{S}_{1}\left( t\right) }{{S}_{0}\left( t\right) }\exp \left\{ {{\int }_{0}^{t}{\delta }_{1}\left( u\right) {du}}\right\} \) is a \( {P}_{0} \) -martingale, so the numerator of (3.4) is\n\n\[ {E}_{0}\left\lbrack {{S}_{1}\left( T\right) /{S}_{0}\left( T\right) ... | Yes |
Example 3.5 (Forward price of a stock with nonrandom dividend payments, when the money market is nonrandom): If \( B = {S}_{1}\left( T\right) \), if the dividend-payment \( \rho \left( \cdot \right) \triangleq {\delta }_{1}\left( \cdot \right) {S}_{1}\left( \cdot \right) \) and money-market prices \( {S}_{0}\left( \cdo... | \[
\frac{{S}_{1}\left( t\right) }{{S}_{0}\left( t\right) } + {\int }_{0}^{t}\frac{\rho \left( u\right) }{{S}_{0}\left( u\right) }{du},\;0 \leq t \leq T,
\]
is a martingale under \( {P}_{0} \) . From (3.4) we have
\[
f\left( t\right) = {S}_{0}\left( T\right) \left\lbrack {\frac{{S}_{1}\left( t\right) }{{S}_{0}\left( t... | Yes |
Corollary 3.9 (Forward-futures spread): Under the conditions of Theorem 3.7, we have\n\n\\[ \nf\\left( t\\right) = \\varphi \\left( t\\right) + \\frac{{\\operatorname{Cov}}_{0}\\left\\lbrack {B,1/{S}_{0}\\left( T\\right) \\mid \\mathcal{F}\\left( t\\right) }\\right\\rbrack }{{E}_{0}\\left\\lbrack {1/{S}_{0}\\left( T\\r... | Proof. Because\n\n\\[ \n{\\operatorname{Cov}}_{0}\\left\\lbrack {X, Y \\mid \\mathcal{F}\\left( t\\right) }\\right\\rbrack = {E}_{0}\\left\\lbrack {{XY} \\mid \\mathcal{F}\\left( t\\right) }\\right\\rbrack - {E}_{0}\\left\\lbrack {X \\mid \\mathcal{F}\\left( t\\right) }\\right\\rbrack \\cdot {E}_{0}\\left\\lbrack {Y \\... | Yes |
Example 4.1 (European call option): A European call option on the first stock in our market is the ECC given by \( C\left( t\right) = 0,\;0 \leq t < T \) and \( C\left( T\right) = {\left( {S}_{1}\left( T\right) - q\right) }^{ + } \) . The nonrandom constant \( q > 0 \) is called the strike price, and \( T \) is the exp... | With \( \varphi \left( x\right) \triangleq {\left( {x}_{1} - q\right) }^{ + } \), the Gaussian integration in (4.6) can be carried out explicitly, to yield\n\n\[ \n{u}^{ECC}\left( {s,{x}_{1};q}\right) = \left\{ \begin{array}{ll} {x}_{1}{e}^{-{\delta }_{1}s}\Phi \left( {{\rho }_{ + }\left( {s,{x}_{1};q}\right) }\right) ... | Yes |
The European put option confers to its holder the right to sell a stock at a future time at a prespecified price. We model a put on the first stock as the ECC with \( C\left( t\right) = 0 \) for \( 0 \leq t < T \) and \( C\left( T\right) = {\left( q - {S}_{1}\left( T\right) \right) }^{ + } \) . Because \( {\left( q - {... | First note from Remark 1.5.11 that\n\n\[ \n{e}^{-\left( {r - {\delta }_{1}}\right) t}{S}_{1}\left( t\right) = {S}_{1}\left( 0\right) + {\int }_{0}^{t}{e}^{-\left( {r - {\delta }_{1}}\right) u}\left\lbrack {d{S}_{1}\left( u\right) - \left( {r - {\delta }_{1}}\right) {S}_{1}\left( u\right) {du}}\right\rbrack \n\]\n\n\[ \... | Yes |
Consider an ECC of the form \( C\left( t\right) = 0 \) for \( 0 \leq t < T \) and \( C\left( T\right) = G\left( \omega \right) \) , where \( G : C\left( \left\lbrack {0, T}\right\rbrack \right) \rightarrow \mathbb{R} \) is a functional satisfying under \( {P}_{0} \) the conditions (E.4)-(E.6) of Appendix E. Then from t... | \[ {V}^{ECC}\left( t\right) = {e}^{-r\left( {T - t}\right) }{E}_{0}\left\lbrack {G\left( {W}_{0}\right) \mid \mathcal{F}\left( t\right) }\right\rbrack \] \[ = {e}^{-r\left( {T - t}\right) }{E}_{0}G\left( {W}_{0}\right) \] \[ + {e}^{-r\left( {T - t}\right) }{\int }_{0}^{t}{E}_{0}\left\lbrack {\partial G\left( {{W}_{0};(... | Yes |
Theorem 6.7 (McKean (1965)): Under the assumption (6.7), the value process for a perpetual American call option is given by\n\n\[ \n{V}^{AC}\left( {t;\infty }\right) = g\left( {S\left( t\right) }\right) ,\;0 \leq t < \infty ,\n\]\n\nwhere the function \( g \) is\n\n\[ \ng\left( x\right) = \left\{ \begin{array}{ll} \lef... | Proof. Itô's rule for convex functions (e.g., Karatzas and Shreve (1991), Theorem 3.6.22 and Problem 3.6.7(i)) implies\n\n\[ \nd\left( {{e}^{-{rt}}g\left( {S\left( t\right) }\right) }\right) = {e}^{-{rt}}S\left( t\right) {g}^{\prime }\left( {S\left( t\right) }\right) {\sigma d}{W}_{0}\left( t\right) - {e}^{-{rt}}\left(... | Yes |
Theorem 7.3 (Maximization of the expected utility from consumption): Let Assumptions 2.1 and 7.1 hold, let \( x \in \left( {{\mathcal{X}}_{1}\left( \infty \right) ,\infty }\right) \) be given, and define\n\n\[ \n{c}_{1}\left( t\right) \triangleq {I}_{1}\left( {t,{\mathcal{Y}}_{1}\left( x\right) {H}_{0}\left( t\right) }... | It is now not difficult to see that the value function \( {V}_{1} \) is given by (cf. (6.21))\n\n\[ \n{V}_{1}\left( x\right) = \left\{ \begin{array}{ll} {G}_{1}\left( {{\mathcal{Y}}_{1}\left( x\right) }\right) , & x > {\mathcal{X}}_{1}\left( \infty \right) , \\ E{\int }_{0}^{T}{U}_{1}\left( {t,\bar{c}\left( t\right) }\... | Yes |
Example 7.5 (Subsistence consumption): Suppose\n\n\[ \n{U}_{1}\left( c\right) = \left\{ \begin{array}{ll} \log \left( {c - \bar{c}}\right) , & \bar{c} < c < \infty , \\ - \infty , & - \infty < c \leq \bar{c}, \end{array}\right.\n\]\n\nwhere \( \bar{c} \) is a positive constant that consumption must exceed at all times.... | If \( {S}_{0}\left( \cdot \right) \) is deterministic, we can derive the optimal portfolio explicitly. Under this condition,\n\n\[ \n{X}_{1}\left( t\right) = \frac{T - t}{T{H}_{0}\left( t\right) }\left( {x - \bar{c}{h}_{1}}\right) + \bar{c}{S}_{0}\left( t\right) \left( {{\int }_{t}^{T}\frac{du}{{S}_{0}\left( u\right) }... | Yes |
Example 7.9 (Portfolio insurance): Suppose\n\n\\[ \n{U}_{2}\\left( x\\right) = \\left\\{ \\begin{array}{ll} \\log \\left( {x - \\bar{x}}\\right) , & \\bar{x} < x < \\infty , \\\\ - \\infty , & - \\infty < x \\leq \\bar{x}, \\end{array}\\right.\n\\]\n\nwhere \\( \\bar{x} \\) is a positive constant below which terminal w... | As in Example 7.5, we can derive the optimal portfolio explicitly when \\( {S}_{0}\\left( \\cdot \\right) \\) is deterministic. Under this condition,\n\n\\[ \n{M}_{2}\\left( t\\right) \\triangleq {H}_{0}\\left( t\\right) {X}_{2}\\left( t\\right) = x - \\bar{x}{h}_{2} + \\frac{\\bar{x}{Z}_{0}\\left( t\\right) }{{S}_{0}\... | Yes |
Theorem 8.11 (Hamilton-Jacobi-Bellman equation): Under Assumptions 8.1 and 8.2, the value function \( V\\left( {t, x}\\right) \) of (8.29),(8.30) is of class \( {C}^{1,2} \) on the set \( D \) of (8.10), continuous on the set \( \\{ \\left( {t, x}\\right) \\in \) \( \\left\\lbrack {0, T}\\right\\rbrack \\times \\left( ... | Proof. Differentiating (8.9) and (8.29) and using the formula (8.35), we obtain for \( \\left( {t, x}\\right) \\in D \) ,\n\n\[ {\\mathcal{X}}_{t}\\left( {t,\\mathcal{Y}\\left( {t, x}\\right) }\\right) + {\\mathcal{X}}_{y}\\left( {t,\\mathcal{Y}\\left( {t, x}\\right) }\\right) {\\mathcal{Y}}_{t}\\left( {t, x}\\right) =... | Yes |
Theorem 8.12 (Convex dual of \( V\left( {t, \cdot }\right) \) ): Let Assumptions 8.1 and 8.2 hold. Then, for each \( t \in \left\lbrack {0, T}\right\rbrack \), the function \( V\left( {t, \cdot }\right) \) satisfies all the conditions of Definition 4.1, and\n\n\[ \mathcal{X}\left( {t,\infty }\right) = \inf \{ x \in \ma... | Proof. All the claims (8.40)-(8.43) made here for fixed \( t \in \lbrack 0, T) \) are contained in Theorem 6.11, taking \( T \) in that theorem to be \( T - t \) here. When \( t = T,\left( {8.40}\right) - \left( {8.43}\right) \) and (8.45) follow directly from the definitions.\n\nEquation (8.42), Lemma 8.4, and Lemma 8... | Yes |
Consider the case that \( r\left( \cdot \right) = \) \( r > 0,\theta \left( \cdot \right) = \theta \neq 0 \), and \( \sigma \left( \cdot \right) = \sigma \) are constants, and \( A\left( \cdot \right) \equiv 0 \). Set \( \gamma = \frac{1}{2}\parallel \theta {\parallel }^{2} > 0 \). Assume that\n\n\[ \n{U}_{1}\left( {t,... | The functions of Theorem 8.12 can be computed explicitly, following Karatzas, Lehoczky, and Shreve (1987), as follows. Denote by \( {\lambda }_{ + } \) and \( {\lambda }_{ - } \) the respective positive and negative roots of the quadratic equation \( \gamma {\lambda }^{2} - \) \( \left( {r - \alpha - \gamma }\right) \l... | Yes |
In a continuous-time capital asset pricing model with an underlying \( N \) -dimensional Markov state process, the risk premia of assets can be computed theoretically from their covariances with a set of \( N + 1 \) mutual funds. | Breeden (1979) shows that rather than using the set of all covariances, one can in principle compute risk premia from the covariance of assets with the consumption process of an optimally behaving investor. Like the simple mean-variance capital asset pricing model, this consumption-based capital asset pricing model doe... | No |
Theorem 6.4 (Uniqueness of the equilibrium market): Assume that (6.4) holds. Then the equilibrium money market process \( {S}_{0}\left( \cdot \right) \), the state price density process \( {H}_{0}\left( \cdot \right) \), and the market price of risk process \( \theta \left( \cdot \right) \), are uniquely determined, as... | Proof. The uniqueness of \( {H}_{0}\left( \cdot \right) \) follows from Corollary 5.4, Theorem 6.1, and the initial condition \( {H}_{0}\left( 0\right) = 1 \) . The uniqueness of \( {\widehat{c}}_{1}\left( \cdot \right) ,\ldots ,{\widehat{c}}_{K}\left( \cdot \right) \) also follows from Theorem 6.1. The semimartingale ... | Yes |
Example 7.1 (Logarithmic utility with subsistence consumption): Let \( {U}_{k}\left( c\right) = \log \left( {c - {\bar{c}}_{k}}\right) \), for \( c > {c}_{k}, k = 1,\ldots, K \), where each \( {\bar{c}}_{k} \) is a nonnegative constant. Then | \[ {U}^{\prime }\left( {c;\Lambda }\right) = \mathcal{H}\left( {c;\Lambda }\right) = \frac{1}{c - \bar{c}}\mathop{\sum }\limits_{{k = 1}}^{K}{\lambda }_{k},\;c > \bar{c}. \] We normalize \( \Lambda \) by setting \( \mathop{\sum }\limits_{{k = 1}}^{K}{\lambda }_{k} = \epsilon \left( 0\right) - \bar{c} \), a strictly pos... | Yes |
Let \( {U}_{k}\left( c\right) = \) \( \frac{1}{p}{\left( c - {\bar{c}}_{k}\right) }^{p} \) for \( c > {\bar{c}}_{k}, k = 1,\ldots, K \), where \( p < 1, p \neq 0 \), and each \( {\bar{c}}_{k} \) is a nonnegative constant. Then | \[ {U}^{\prime }\left( {c;\underset{ \sim }{\Lambda }}\right) = \mathcal{H}\left( {c;\underset{ \sim }{\Lambda }}\right) = {\left\lbrack \frac{\mathop{\sum }\limits_{{k = 1}}^{K}{\lambda }_{k}^{\frac{1}{1 - p}}}{c - \bar{c}}\right\rbrack }^{1 - p},\;c > \bar{c}. \] We normalize \( \Lambda \) by setting \( \mathop{\sum ... | Yes |
Example 7.5 (Constant aggregate endowment): If the aggregate endowment \( \epsilon > \bar{c} \) is constant, then the unique vector \( \Lambda \) satisfying the normalization \( \mathcal{H}\left( {\epsilon ;\Lambda }\right) = 1 \) is \[ \underset{ \sim }{\Lambda } = \left( {\frac{1}{{U}_{1}^{\prime }\left( {\widehat{c}... | Constant aggregate endowment implies \( \nu \left( \cdot \right) \equiv 0,\xi \left( \cdot \right) \equiv 0,\rho \left( \cdot \right) \equiv 0 \) in (2.2), and the local time of \( \epsilon \left( \cdot \right) \) at every point is zero. Therefore, the equilibrium market coefficients (6.20)-(6.22) are \[ r\left( t\righ... | Yes |
Example 7.6 \( \\left( {K = 2,{U}_{1}\\left( c\\right) = \\log c,{U}_{2}\\left( c\\right) = \\sqrt{c}\\text{.}}\\right) : \) In this case, we have\n\n\[ \n{U}^{\\prime }\\left( {c;\\Lambda }\\right) = \\mathcal{H}\\left( {c;\\Lambda }\\right) = \\frac{{\\lambda }_{1}}{2c}\\left\\lbrack {1 + \\sqrt{1 + c{\\left( \\frac{... | The positive constants \( {\\lambda }_{1} \) and \( {\\lambda }_{2} \) are uniquely determined by (5.17) with \( k = 1 \) :\n\n\[ \n2{\\int }_{0}^{T}{e}^{-{\\int }_{0}^{t}\\beta \\left( u\\right) {du}}{dt} \n\]\n\n\[ \n= E{\\int }_{0}^{T}{e}^{-{\\int }_{0}^{t}\\beta \\left( u\\right) {du}}\\left\\lbrack {1 + \\sqrt{1 +... | Yes |
Example 7.8 (Ergodic aggregate endowment): Let us suppose that each agent \( k \) has utility function \( {U}_{k} \) with \( {\bar{c}}_{k} = 0 \) and \( {U}_{k}^{\prime }\left( 0\right) = \infty \), so \( \bar{c} = 0 \) . Let us further suppose that the aggregate endowment process \( \epsilon \left( \cdot \right) \) is... | Then the diffusion process \( \epsilon \left( \cdot \right) \) is ergodic with invariant measure \( m\left( {dc}\right) /m\left( \mathcal{I}\right) \) (cf. Proposition 5.5.22 and Exercise 5.5.40 in Karatzas and Shreve (1991)). | Yes |
Example 7.3 (European call option): We consider one stock \( S\left( \cdot \right) = {S}_{1}\left( \cdot \right) \) driven by a single Brownian motion, we assume (7.1)-(7.3), and we denote \( {\sigma }_{11} \) by \( \sigma \) . A European call option corresponds to \( \varphi \left( x\right) = {\left( x - q\right) }^{ ... | \[ \widehat{\varphi }\left( x\right) = \left\{ \begin{array}{ll} {\left( \frac{\beta - 1}{q}\right) }^{\beta - 1}{\left( \frac{x}{\beta }\right) }^{\beta }, & \text{ if }0 < x \leq \frac{\beta q}{\beta - 1}, \\ x - q, & \text{ if }x \geq \frac{\beta q}{\beta - 1}. \end{array}\right. \] (7.25) The function \( \widehat{\... | Yes |
Example 7.4 (European put option): We assume again (7.1)-(7.3) and consider one stock. A European put option corresponds to \( \varphi \left( x\right) = {\left( q - x\right) }^{ + } \) , where \( q \geq 0 \) . We consider again \( K = \left\lbrack {\alpha ,\beta }\right\rbrack \) with \( - \infty \leq \alpha \leq 0 \le... | \[ \widehat{\varphi }\left( x\right) = \left\{ \begin{array}{ll} q - x, & \text{ if }0 < x \leq \frac{\alpha q}{\alpha - 1}, \\ {\left( \frac{\left| \alpha - 1\right| }{q}\right) }^{\alpha - 1}{\left( \frac{x}{\left| \alpha \right| }\right) }^{\alpha }, & \text{ if }x \geq \frac{\alpha q}{\alpha - 1} \end{array}\right.... | Yes |
Example 8.8 (Incomplete market): Consider the case \( K = \{ p \in \) \( \left. {{\mathbb{R}}^{N};{p}_{M + 1} = \cdots = {p}_{N} = 0}\right\} \) of Example 4.1(iii), where there are only \( M \) stocks available for investment, but these are driven by the \( N \) -dimensional Brownian motion \( W\left( \cdot \right) \)... | \[ \zeta \left( \nu \right) + {p}^{\prime }\nu = 0,\;\forall p \in K,\;\nu \in \widetilde{K}. \] | Yes |
Example 10.2 (Prohibition of short-selling): We consider a market with constant coefficients and one stock, i.e., \( N = 1 \) . When short-selling is prohibited (Example 9.7(ii), \( K = \left\lbrack {0,\infty ),{K}_{ - } = ( - \infty ,0}\right\rbrack \) ), we have \( \widetilde{K} = \lbrack 0,\infty ) \) , \( \zeta \le... | \[ \check{\varphi }\left( x\right) = \mathop{\inf }\limits_{{\nu \geq 0}}\varphi \left( {x{e}^{-\nu }}\right) ,\;\forall x > 0. \] For a European call, we have \[ \check{\varphi }\left( x\right) = \mathop{\inf }\limits_{{\nu \geq 0}}{\left( x{e}^{-\nu } - q\right) }^{ + } = 0,\;\forall x > 0, \] and the lower hedging v... | Yes |
Example 10.3 (Prohibition of borrowing): We consider again a market with constant coefficients and one stock, i.e., \( N = 1 \) . When borrowing from the money market is prohibited (Example 9.7(vi), \( K = ( - \infty ,1\rbrack ,{K}_{ - } = \) \( \lbrack 1,\infty )) \), we have \( \widetilde{K} = ( - \infty ,0\rbrack ,\... | \[ \check{\varphi }\left( x\right) = \mathop{\inf }\limits_{{\nu \leq 0}}\left\lbrack {{e}^{\nu }\varphi \left( {x{e}^{-\nu }}\right) }\right\rbrack ,\;\forall x > 0. \] For a European call option, we have \[ \check{\varphi }\left( x\right) = \mathop{\inf }\limits_{{\nu \leq 0}}{\left( x - q{e}^{\nu }\right) }^{ + } = ... | Yes |
Proposition 5.1 (Weak Duality): Suppose (5.2) and (5.3) hold. Then, for any given \( y \in \left( {0,\infty }\right) \) and with \( x = {\mathcal{X}}_{{\nu }_{y}}\left( y\right) \) , (i) there exists an optimal consumption and portfolio-proportion process pair \( \left( {\widehat{c},\widehat{p}}\right) \in {\mathcal{A}... | Proof. First, let us note that (5.3), (3.10), and (3.27) imply \[ V\left( {x;K}\right) \leq \widetilde{V}\left( y\right) + {xy},\forall y > 0,\forall x > 0. \] (5.5) Now fix \( y \in \left( {0,\infty }\right) \), let \( x = {\mathcal{X}}_{{\nu }_{y}}\left( y\right) \), and note that the assumption \( {\widetilde{V}}_{{... | Yes |
Example 6.7 (Utility functions of power type): Fix \( \beta \in \left( {-\infty ,1}\right) \smallsetminus \{ 0\} \) and assume\n\n\[ \n{U}_{1}\left( {t, x}\right) = {U}_{2}\left( x\right) = \frac{1}{\beta }{x}^{\beta },\;0 \leq t \leq T, x > 0.\n\] | We know from Example 3.8.13 that\n\n\[ \n{\mathcal{X}}_{\widehat{\nu }}\left( {t, y}\right) = k\left( t\right) {y}^{\frac{1}{\beta - 1}},\;{G}_{\widehat{\nu }}\left( {t, y}\right) = \frac{1}{\beta }k\left( t\right) {y}^{\frac{\beta }{\beta - 1}},\n\]\n\n\[ \n\widetilde{V}\left( {t, y}\right) = \frac{1 - \beta }{\beta }... | Yes |
Example 7.2 (Logarithmic utility, incomplete market): \( {U}_{1}\left( {t, x}\right) = \) \( {U}_{2}\left( x\right) = \log x \) for every \( \left( {t, x}\right) \in \left\lbrack {0, T}\right\rbrack \times \left( {0,\infty }\right) \). | This is Example 4.2, specialized to the case of an incomplete market; i.e., \( K \) given by (7.1). The expression in (4.23) to be minimized over \( \xi \in {\mathbb{R}}^{L} \) is\n\n\[ \frac{1}{2}{\begin{Vmatrix}\widetilde{\theta }\left( t\right) + {\rho }^{\prime }\left( t\right) \left( a\left( t\right) + \xi - r\lef... | Yes |
Example 8.5 (Logarithmic utilities, general coefficients): In the special case \( {U}_{1}\left( {t, x}\right) = {U}_{2}\left( x\right) = \log x,0 \leq t \leq T, x > 0 \), we have \( {\widetilde{U}}_{1}\left( {t, y}\right) = {\widetilde{U}}_{2}\left( y\right) = \) \( - \left( {1 + \log y}\right), y > 0 \) (see Example 4... | \[ {\widetilde{V}}_{\nu }\left( y\right) = - \left( {T + 1}\right) \left( {1 + \log y}\right) - {\int }_{0}^{T}E\left\lbrack {\log {H}_{\nu }\left( t\right) }\right\rbrack {dt} - E\left\lbrack {\log {H}_{\nu }\left( T\right) }\right\rbrack . \] For \( \nu \left( \cdot \right) \in \mathcal{D} \), we have \[ - E\left\lbr... | Yes |
Assume that \( {U}_{1}\left( {t, x}\right) = {U}_{2}\left( x\right) = \frac{1}{\beta }{x}^{\beta },0 \leq t \leq T, x > 0 \), for some \( \beta < 0 \) , \( \beta \neq 0 \) . Assume further that \( r\left( \cdot \right), R\left( \cdot \right) ,\sigma \left( \cdot \right) \), and \( \theta \left( \cdot \right) \) are det... | We take \( {\widehat{\nu }}_{1}\left( t\right) \) to be the (nonrandom) minimizer of\n\n\[ - \left( {1 - \beta }\right) {\nu }_{1} + \frac{1}{2}{\begin{Vmatrix}\theta \left( t\right) + {\nu }_{1}{\sigma }^{-1}\left( t\right) {1}_{N}\end{Vmatrix}}^{2}\text{ over }{\nu }_{1} \in \left\lbrack {-\left( {R\left( t\right) - ... | Yes |
Theorem 1.1.1 (Cauchy-Schwarz inequality).\n\n\\[ \left| {\\langle \\mathbf{a},\\mathbf{b}\\rangle }\\right| \\leq \\parallel \\mathbf{a}\\parallel \\cdot \\parallel \\mathbf{b}\\parallel \\] | Proof. The inequality is trivial if either \\( \\mathbf{a} \\) or \\( \\mathbf{b} \\) is zero, so we assume that neither is. If we let \\( \\mathbf{x} = \\frac{\\mathbf{a}}{\\parallel \\mathbf{a}\\parallel } \\) and \\( \\mathbf{y} = \\frac{\\mathbf{b}}{\\parallel \\mathbf{b}\\parallel } \\), then \\( \\parallel \\math... | Yes |
Proposition 1.1.1 (Triangle inequalities). Let \( x, y \in {\mathbb{R}}^{n} \) . Then\n\n1. \( \parallel \mathbf{x} \pm \mathbf{y}\parallel \leq \parallel \mathbf{x}\parallel + \parallel \mathbf{y}\parallel \) ;\n\n2. \( \left| {\parallel \mathbf{x}\parallel - \parallel \mathbf{y}\parallel }\right| \leq \parallel \math... | Proof. 1. As above,\n\n\[ \parallel \mathbf{x} \pm \mathbf{y}{\parallel }^{2} = \langle \mathbf{x} \pm \mathbf{y},\mathbf{x} \pm \mathbf{y}\rangle \]\n\n\[ = \parallel \mathbf{x}{\parallel }^{2} \pm 2\langle \mathbf{x},\mathbf{y}\rangle + \parallel \mathbf{y}{\parallel }^{2} \]\n\n\[ \leq \parallel \mathbf{x}{\parallel... | Yes |
Let us consider the vector field (Fig. 1.4)\n\n\[ \mathbf{F} : {\mathbb{R}}^{2} \smallsetminus \{ 0\} \rightarrow {\mathbb{R}}^{2} \]\n\ndefined by\n\n\[ \mathbf{F}\left( {x, y}\right) = \left( {-\frac{y}{\sqrt{{x}^{2} + {y}^{2}}},\frac{x}{\sqrt{{x}^{2} + {y}^{2}}}}\right) . \] | Clearly, \( \mathbf{F}\left( {x, y}\right) \) is a unit vector, and if we place this vector at the point \( \left( {x, y}\right) \), we see that it is a tangent vector at \( \left( {x, y}\right) \) to the circle centered at the origin that passes through this point. | Yes |
Example 1.2.3 (Velocity field of a fluid). For every point \( \left( {x, y, z}\right) \) of an open set \( U \subset \) \( {\mathbb{R}}^{3} \) let \( \mathbf{F}\left( {x, y, z}\right) \) represent the velocity of a fluid at the position \( \left( {x, y, z}\right) \) at a given fixed time. Then | \[ \mathbf{F} : U \subset {\mathbb{R}}^{3} \rightarrow {\mathbb{R}}^{3} \] is a vector field. | Yes |
The vector field \( \mathbf{F}\left( {x, y, z}\right) = \left( {-y, x,0}\right) \) represents a rotation about \( \mathbf{N} = \left( {0,0,1}\right) \) . We observe that \( \operatorname{Curl}\mathbf{F} = \left( {0,0,1}\right) \) gives the direction of the rotation axis (Fig. 1.6). | We will deduce from Stokes's theorem that this is no coincidence (see Corollary 9.4.1). | No |
Theorem 1.2.2. Let \( f : U \subset {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) be a function of class \( {C}^{2} \) on the open set \( U \) . Then for all \( \mathbf{x} \in U \) and all \( i, j = 1,2,\ldots, n \) , | \[ \frac{{\partial }^{2}f}{\partial {x}_{i}\partial {x}_{j}}\left( \mathbf{x}\right) = \frac{{\partial }^{2}f}{\partial {x}_{j}\partial {x}_{i}}\left( \mathbf{x}\right) . \] | Yes |
Theorem 1.2.3. Let \( \mathbf{F} : U \subset {\mathbb{R}}^{3} \rightarrow {\mathbb{R}}^{3} \) be a vector field of class \( {C}^{2} \) on the open set U. Then\n\n\[ \text{Div}\left( {\operatorname{Curl}\mathbf{F}}\right) = 0\text{.} \] | Proof. \( {}^{1} \) Since \( \mathbf{F} = \left( {{f}_{1},{f}_{2},{f}_{3}}\right) \), Schwarz’s theorem concerning the symmetry of second derivatives gives\n\n---\n\n\( {}^{1} \) In Chap. 6 we will present an alternative argument (Corollary 6.3.1) based on properties of the exterior differential.\n\n---\n\n\n\n\[ \oper... | No |
The line segment joining two points \( \mathbf{x},\mathbf{y} \in {\mathbb{R}}^{n} \) is the arc \( \left\lbrack {\mathbf{x},\mathbf{y}}\right\rbrack \mathrel{\text{:=}} \) \( \mathbf{\gamma }\left( \left\lbrack {0,1}\right\rbrack \right) \), where \( \mathbf{\gamma } : \left\lbrack {0,1}\right\rbrack \rightarrow {\math... | \[ \mathbf{y}\left( t\right) = \mathbf{x} + t\left( {\mathbf{y} - \mathbf{x}}\right) = t\mathbf{y} + \left( {1 - t}\right) \mathbf{x}. \] | Yes |
Lemma 2.1.1. Let \( c \) be a real number: The function \( f : \mathbb{R} \rightarrow \mathbb{R} \) , \[ f\left( t\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} c{\mathrm{e}}^{-\frac{1}{{t}^{2}}}, & t < 0, \\ 0, & t = 0, \\ {\mathrm{e}}^{-\frac{1}{{t}^{2}}}, & t > 0, \end{array}\right. \] is of class \( {C}^{\in... | Proof. We show that for each \( n \in \mathbb{N} \cup \{ 0\} \) there is a polynomial \( {P}_{n} \) such that \[ {f}^{\left( n\right) }\left( t\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} c\frac{{P}_{n}\left( t\right) }{{t}^{3n}}{\mathrm{e}}^{-\frac{1}{{t}^{2}}}, & t < 0, \\ 0, & t = 0, \\ \frac{{P}_{n}\left( ... | Yes |
Lemma 2.1.2. Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow {\mathbb{R}}^{n} \) be a path and let \( P \mathrel{\text{:=}} \left\{ {a = {t}_{1} < \cdots < {t}_{k} = b}\right\} \) be a partition of \( \left\lbrack {a, b}\right\rbrack \) . If \( Q \) is another partition of \( \left\lbrack {a, b}\right\rbra... | Proof. We can assume without any loss of generality that \( Q \) is obtained from \( P \) by adding a single point. Thus, we assume \( Q = P \cup \{ s\} \), where \( {t}_{j} < s < {t}_{j + 1} \) . Then\n\n\[ L\left( {\mathbf{\gamma }, Q}\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{i \neq j}}\begin{Vmatrix}{\math... | Yes |
Corollary 2.1.1. Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow {\mathbb{R}}^{n} \) be a piecewise \( {C}^{1} \) path. Then \( \gamma \) is rectifiable and \( L\left( \mathbf{\gamma }\right) = {\int }_{a}^{b}\begin{Vmatrix}{{\mathbf{\gamma }}^{\prime }\left( t\right) }\end{Vmatrix}\mathrm{d}t \) . | Proof. Let \( \left\{ {a = {t}_{1} < {t}_{2} < \cdots < {t}_{k} = b}\right\} \) be a partition of \( \left\lbrack {a, b}\right\rbrack \) such that \( {\mathbf{\gamma }}_{j} \mathrel{\text{:=}} {\mathbf{\gamma }}_{\mid \left\lbrack {{t}_{j},{t}_{j + 1}}\right\rbrack } \) is of class \( {C}^{1} \) for every \( j = 1,\ldo... | Yes |
Lemma 2.2.1. Let \( P \mathrel{\text{:=}} \left\{ {a = {t}_{1} < {t}_{2} < \cdots < {t}_{k} = b}\right\} \) be a partition of the interval \( \left\lbrack {a, b}\right\rbrack \) . For any continuous function \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) and for each selection \( {u}_{j} \in \left\lb... | Proof. By the Heine-Cantor theorem, \( f \) is uniformly continuous, which means that for every \( \varepsilon > 0 \) one can find \( \delta > 0 \) such that\n\n\[ \left| {f\left( s\right) - f\left( t\right) }\right| \leq \varepsilon \]\n\nwhenever \( s, t \in \left\lbrack {a, b}\right\rbrack \) and \( \left| {s - t}\r... | Yes |
Example 2.3.1. Let \( g : U \subset {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) be a function of class \( {C}^{1} \) on the open set \( U \) . The differential of \( g \) at point \( \mathbf{x} \in U \) is the linear mapping \( \mathrm{d}g\left( \mathbf{x}\right) : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) given by | \[ \left( {\mathrm{d}g}\right) \left( \mathbf{x}\right) \left( \mathbf{h}\right) = \mathop{\sum }\limits_{{j = 1}}^{n}\frac{\partial g}{\partial {x}_{j}}\left( \mathbf{x}\right) {h}_{j} = \mathop{\sum }\limits_{{j = 1}}^{n}\frac{\partial g}{\partial {x}_{j}}\left( \mathbf{x}\right) \mathrm{d}{x}_{j}\left( \mathbf{h}\ri... | Yes |
Theorem 2.4.1. Let \( U \) and \( V \) be open subsets of \( {\mathbb{R}}^{n} \) and \( {\mathbb{R}}^{m} \) respectively. If the mappings \( \mathbf{F} : U \rightarrow V \) and \( \mathbf{G} : V \rightarrow {\mathbb{R}}^{p} \) are differentiable at \( \mathbf{a} \in U \) and \( \mathbf{F}\left( \mathbf{a}\right) \in V ... | \[ \mathrm{d}\mathbf{H}\left( \mathbf{a}\right) = \mathrm{d}\mathbf{G}\left( {\mathbf{F}\left( \mathbf{a}\right) }\right) \circ \mathrm{d}\mathbf{F}\left( \mathbf{a}\right) ,\] or, in terms of their associated matrices, \[ {\mathbf{H}}^{\prime }\left( \mathbf{a}\right) = {\mathbf{G}}^{\prime }\left( {\mathbf{F}\left( \... | Yes |
Corollary 2.4.1. Let \( V \) be an open subset of \( {\mathbb{R}}^{m},\gamma : \left\lbrack {a, b}\right\rbrack \rightarrow V \) a path that has a derivative at each point of \( \left\lbrack {a, b}\right\rbrack \), and suppose \( \mathbf{G} : V \rightarrow {\mathbb{R}}^{m} \) is differentiable at each point of \( \math... | Proof. We write \( \mathbf{\gamma }\left( t\right) = \left( {{\gamma }_{1}\left( t\right) ,\ldots ,{\gamma }_{m}\left( t\right) }\right) \), where \( {\gamma }_{j} : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) has a derivative on \( \left\lbrack {a, b}\right\rbrack \) for each \( j = 1,\ldots, n \) (or \... | Yes |
Corollary 2.4.2. Let \( \alpha : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be differentiable at each point of \( \left\lbrack {a, b}\right\rbrack \) and let \( \mathbf{\beta } : \left\lbrack {c, d}\right\rbrack \rightarrow {\mathbb{R}}^{m} \) be a path that has a derivative at each point of \( \left\lb... | \[ {\left( \mathbf{\beta } \circ \mathbf{\alpha }\right) }^{\prime }\left( t\right) = {\mathbf{\beta }}^{\prime }\left( {\mathbf{\alpha }\left( t\right) }\right) {\mathbf{\alpha }}^{\prime }\left( t\right) \] for every \( t \in \left\lbrack {a, b}\right\rbrack \) . Furthermore, if \( \alpha \) is \( {C}^{q} \) on \( \l... | Yes |
Proposition 2.4.1. If \( \alpha \) and \( \beta \) are two equivalent paths and one of them, say \( \beta \), is piecewise \( {C}^{1} \), then \( \mathbf{\alpha } \) is piecewise \( {C}^{1} \) and \( L\left( \mathbf{\alpha }\right) = L\left( \mathbf{\beta }\right) \) . | Proof. Write \( \alpha = \beta \circ \varphi \), where \( \varphi \) is as in Definition 2.4.1 and let \( Q = \) \( \left\{ {c = {u}_{1} < \cdots < {u}_{k} = d}\right\} \) be a partition of \( \left\lbrack {c, d}\right\rbrack \) such that \( {\mathbf{\beta }}_{\mid \left\lbrack {{u}_{j},{u}_{j + 1}}\right\rbrack } \) i... | Yes |
Proposition 2.4.2. Let \( \omega \) be a continuous 1-form on \( U \), and let \( \mathbf{\alpha } \) and \( \mathbf{\beta } \) be two piecewise \( {C}^{1} \) paths in \( U \) with \( \mathbf{\alpha } \sim \mathbf{\beta } \). Then\n\n\[ \n{\int }_{\alpha }\omega = {\int }_{\beta }\omega \n\] | Proof. Let us first assume that \( \mathbf{\alpha } \) and \( \mathbf{\beta } \) are paths of class \( {C}^{1} \). Then, employing the same notation as the previous proof, we have\n\n\[ \n{\int }_{\mathbf{\beta }}\mathbf{\omega } = {\int }_{\varphi \left( a\right) }^{\varphi \left( b\right) }\mathbf{\omega }\left( {\ma... | Yes |
Proposition 2.4.3. Let \( \omega \) be a continuous 1-form on an open subset \( U \) of \( {\mathbb{R}}^{n} \) and let \( \mathbf{\alpha },\mathbf{\beta },\mathbf{\gamma } \) be three piecewise \( {C}^{1} \) paths in \( U \) with the final point of \( \mathbf{\alpha } \) coinciding with the initial point of \( \mathbf{... | Proof. Since \( {\left( -\gamma \right) }^{\prime }\left( t\right) = - {\gamma }^{\prime }\left( {-t}\right) \), the substitution \( u = - t \) gives\n\n\[{\int }_{-\gamma }\omega = {\int }_{-b}^{-a}\omega \left( {\gamma \left( {-t}\right) }\right) \left( {-{\gamma }^{\prime }\left( {-t}\right) }\right) \mathrm{d}t\]\n... | Yes |
Let \( \mathbf{F} : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2} \) be the vector field defined by \( \mathbf{F}\left( {x, y}\right) = \left( {x, y}\right) \) and let us consider the paths (Fig. 2.4)\n\n\[{\mathbf{\gamma }}_{1} : \left\lbrack {0,1}\right\rbrack \rightarrow {\mathbb{R}}^{2}\text{ and }{\mathbf{\gamma }... | \n\[{\int }_{{\mathbf{y}}_{1}}\mathbf{F} = {\int }_{0}^{1}\langle \left( {t, t}\right) ,\left( {1,1}\right) \rangle \mathrm{d}t = {\int }_{0}^{1}\left( {t + t}\right) \mathrm{d}t = 1.\]\n\nAlso\n\n\[{\int }_{{\mathbf{\gamma }}_{2}}\mathbf{F} = {\int }_{0}^{1}\left\langle {\left( {t,{t}^{2}}\right) ,\left( {1,{2t}}\righ... | Yes |
Let \( {\mathbf{\gamma }}_{1} \) and \( {\mathbf{\gamma }}_{2} \) be the paths given in Example 2.5.1 but let us consider instead the vector field \[ \mathbf{F}\left( {x, y}\right) = \left( {-y + \frac{3}{8}, x - \frac{1}{2}}\right) . \] | Then \[ {\int }_{{\mathbf{\gamma }}_{1}}\mathbf{F} = {\int }_{0}^{1}\left\langle {\left( {-t + \frac{3}{8}, t - \frac{1}{2}}\right) ,\left( {1,1}\right) }\right\rangle \mathrm{d}t = {\int }_{0}^{1} - \frac{1}{8}\mathrm{\;d}t = - \frac{1}{8}, \] while \[ {\int }_{{\mathbf{\gamma }}_{2}}\mathbf{F} = {\int }_{0}^{1}\left\... | Yes |
Theorem 2.5.1. Let \( g : U \subset {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) be a function of class \( {C}^{1} \) on an open set \( U \) and \( \mathbf{\gamma } : \left\lbrack {a, b}\right\rbrack \rightarrow U \) a piecewise \( {C}^{1} \) path. Then\n\n\[{\int }_{\mathbf{y}}\mathbf{\nabla }g = {\int }_{\mathbf{y}}\ma... | Proof. Since \( \mathrm{d}g \) is the 1 -form associated to the vector field \( \mathbf{\nabla }g \), the first equation follows from our observation immediately preceding Sect. 2.4. For the second equation, we take the usual expansion of the 1 -form \( \mathrm{d}g \),\n\n\[ \mathrm{d}g = \mathop{\sum }\limits_{{j = 1}... | Yes |
Lemma 2.5.1. Let \( U \) be starlike with respect to the point \( a \in U \) . If \( y \in B\left( {x, R}\right) \subset U \) , then the triangle (Fig. 2.6) with vertex \( \{ \mathbf{a},\mathbf{x},\mathbf{y}\} \) is contained in \( U \) . | Proof. Let \( \mathbf{z} = \alpha \mathbf{a} + \beta \mathbf{x} + \gamma \mathbf{y} \) be given, where \( \alpha ,\beta ,\gamma \geq 0 \) and \( \alpha + \beta + \gamma = 1 \) . If \( \alpha = 1 \), then \( z = a \in U \) . In other case, we put\n\n\[ \mathbf{u} = \frac{\beta }{1 - \alpha }\mathbf{x} + \frac{\gamma }{1... | Yes |
Theorem 2.5.4. If the open set \( U \) is starlike with respect to the point \( \mathbf{a} \in U \), then the conditions of Theorem 2.5.3 are equivalent to the following:\n\n(4) If \( \mathbf{\gamma } \) is the boundary of a triangle contained in \( U \), then\n\n\[{\int }_{\gamma }\omega = 0\] | Proof. Since condition (2) clearly implies (4), it is enough to prove that (4) implies (1). We do this by arguing that the function \( f \) defined by\n\n\[f\left( \mathbf{x}\right) \mathrel{\text{:=}} {\int }_{\left\lbrack \mathbf{a},\mathbf{x}\right\rbrack }\mathbf{F}\]\nis, in fact, a potential for the vector field ... | Yes |
Example 2.5.3. Let\n\n\\[ \n\\omega = - \\frac{y}{{x}^{2} + {y}^{2}} \\cdot \\mathrm{d}x + \\frac{x}{{x}^{2} + {y}^{2}} \\cdot \\mathrm{d}y \n\\]\n\nwhich is a continuous 1 -form in \\( U \\mathrel{\\text{:=}} {\\mathbb{R}}^{2} \\smallsetminus \\{ \\left( {0,0}\\right) \\} \\) . Then \\( \\omega \\) is not exact, since | \\[\n\\mathbf{\\gamma } : \\left\\lbrack {0,{2\\pi }}\\right\\rbrack \\rightarrow {\\mathbb{R}}^{2},\\mathbf{\\gamma }\\left( t\\right) \\mathrel{\\text{:=}} \\left( {\\cos \\left( t\\right) ,\\sin \\left( t\\right) }\\right) ,\n\\]\n\ndefines a closed path contained in \\( U \\) for which \\( {\\int }_{\\gamma }\\omeg... | Yes |
Example 2.5.4. The gravitational field is conservative. | Let us consider a particle of mass \( M \) located at the origin. The force of attraction exerted on a particle of unit mass located at point \( \left( {x, y, z}\right) \in {\mathbb{R}}^{3} \smallsetminus \{ \left( {0,0,0}\right) \} \) is\n\n\[ \mathbf{F}\left( {x, y, z}\right) = - \frac{GM}{{\left( {x}^{2} + {y}^{2} +... | Yes |
Theorem 2.5.5. Let \( F : U \subset {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) be a conservative vector field of class \( {C}^{1} \) on an open set \( U \) and write \( \mathbf{F} \mathrel{\text{:=}} \left( {{f}_{1},{f}_{2},\ldots ,{f}_{n}}\right) \) . Then\n\n\[ \frac{\partial {f}_{j}}{\partial {x}_{k}}\left( \m... | Proof. By hypothesis, there is a function \( g : U \subset {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) such that \( \nabla g = \mathbf{F} \) . Therefore \( {f}_{j} = \frac{\partial g}{\partial {x}_{j}} \), and we can write\n\n\[ \frac{\partial {f}_{j}}{\partial {x}_{k}}\left( \mathbf{x}\right) = \frac{{\partial }^{2}g}{... | Yes |
Proposition 2.5.1. Suppose \( f : \left\lbrack {a, b}\right\rbrack \times \left\lbrack {c, d}\right\rbrack \rightarrow \mathbb{R} \) is a function of class \( {C}^{1} \) and \( g \) :\n\n\( \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) is defined by\n\[ g\left( x\right) = {\int }_{c}^{d}f\left( {x, t}\righ... | The proof of this statement follows the proof of \( \left( 3\right) \Rightarrow \left( 1\right) \) in Theorem 2.5.3, but the interested reader can find stronger results concerning the derivative of an integral, for example in [15, Theorem 9.42, p. 236]. | No |
Theorem 2.5.6 (Poincaré’s lemma). Let \( F : U \subset {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) be a vector field of class \( {C}^{1},\mathbf{F} \mathrel{\text{:=}} \left( {{f}_{1},{f}_{2},\ldots ,{f}_{n}}\right) \), on an open set \( U \) that is starlike with respect to the point \( \mathbf{a} \). If\n\n\[ \f... | Proof. We define \( g : U \subset {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) by\n\n\[ g\left( \mathbf{x}\right) \mathrel{\text{:=}} {\int }_{0}^{1}\mathop{\sum }\limits_{{j = 1}}^{n}{f}_{j}\left( {\mathbf{a} + t\left( {\mathbf{x} - \mathbf{a}}\right) }\right) \cdot \left( {{x}_{j} - {a}_{j}}\right) \mathrm{d}t. \]\n\nI... | Yes |
Lemma 2.6.1. Let \( K \) be a compact set that is a region of type \( I \) and let \( P \) be a continuous function on \( K \) admitting continuous partial derivative \( \frac{\partial P}{\partial v} \) on a neighborhood of \( K \) . Then\n\n\[ \n{\int }_{\gamma }P\mathrm{\;d}x = - {\iint }_{K}\frac{\partial P}{\partia... | Proof. Since \( {\mathbf{\gamma }}_{2}^{\prime }\left( t\right) = {\mathbf{\gamma }}_{4}^{\prime }\left( t\right) = \left( {0,1}\right) ,{\mathbf{\gamma }}_{1}^{\prime }\left( t\right) = \left( {1,{f}_{1}^{\prime }\left( t\right) }\right) \), and \( {\mathbf{\gamma }}_{3}^{\prime }\left( t\right) = \left( {1,{f}_{2}^{\... | Yes |
Theorem 2.6.3. Let \( K \) be a region of type I or a region of type II and let \( \omega = \) \( P\mathrm{\;d}x + Q\mathrm{\;d}y \) be a 1 -form of class \( {C}^{1} \) on some open rectangle containing \( K \) . Then \[ {\int }_{y}P\mathrm{\;d}x + Q\mathrm{\;d}y = {\iint }_{K}\left( {\frac{\partial Q}{\partial x} - \f... | Proof. We choose \( a, b, c \), and \( d \) such that \( K \) is contained in the rectangle \( R = \left\lbrack {a, b}\right\rbrack \times \) \( \left\lbrack {c, d}\right\rbrack \) and we assume that the 1 -form \( \mathbf{\omega } \) is defined and is of class \( {C}^{1} \) on the open rectangle \( T \) containing \( ... | Yes |
Theorem 2.6.4. Let \( K \subset {\mathbb{R}}^{2} \) be a simple region and let \( \omega = P\mathrm{\;d}x + Q\mathrm{\;d}y \) be a 1 -form of class \( {C}^{1} \) on an open neighborhood of \( K \) . Then\n\n\[ \n{\int }_{\partial K}\omega = {\iint }_{K}\left( {\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial ... | Proof. Since \( K \) is a region of type I, we have\n\n\[ \n{\int }_{\partial K}P\mathrm{\;d}x = - {\iint }_{K}\frac{\partial P}{\partial y}\mathrm{\;d}\left( {x, y}\right) \n\]\n\nand since it is also a region of type II, we obtain\n\n\[ \n{\int }_{\partial K}Q\mathrm{\;d}y = {\iint }_{K}\frac{\partial Q}{\partial x}\... | Yes |
Theorem 2.6.5 (Poincaré’s lemma). Let \( \mathbf{F} = \left( {P, Q}\right) \) be a vector field of class \( {C}^{1} \) on the open and starlike set \( U \in {\mathbb{R}}^{2} \) and suppose that\n\n\[ \frac{\partial Q}{\partial x}\left( {x, y}\right) = \frac{\partial P}{\partial y}\left( {x, y}\right) \]\n\nfor every \(... | Proof. Let \( \mathbf{\omega } = P\mathrm{\;d}x + Q\mathrm{\;d}y \) be the 1 -form associated with the vector field \( \mathbf{F} \) . It is enough to prove that\n\n\[ {\int }_{\gamma }\omega = 0 \]\n\nwhenever \( \mathbf{\gamma } \) is the boundary of some triangle \( K \) contained in \( U \) . Since \( K \) is a sim... | Yes |
Lemma 2.7.1. Let \( \varphi : \left\lbrack {c, d}\right\rbrack \rightarrow {\mathbb{R}}^{n} \) and \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow {\mathbb{R}}^{n} \) be two injective paths of class \( {C}^{1} \) such that\n\n\[ \mathbf{\gamma }\left( \left\lbrack {a, b}\right\rbrack \right) \subset \mathbf{\v... | Proof. Since \( \varphi \) is \( {C}^{1} \), the intermediate value theorem implies that \( {\varphi }_{k}^{\prime }\left( s\right) > 0 \) for all \( s \in \left\lbrack {c, d}\right\rbrack \) or \( {\varphi }_{k}^{\prime }\left( s\right) < 0 \) for all \( s \in \left\lbrack {c, d}\right\rbrack \) . Without loss of gene... | Yes |
Lemma 2.7.2. Let \( \varphi : \left\lbrack {c, d}\right\rbrack \rightarrow {\mathbb{R}}^{n} \) be a simple path. Then\n\n\[ \mathbf{\varphi } : \left( {c, d}\right) \rightarrow \mathbf{\varphi }\left( \left( {c, d}\right) \right) \]\n\nis a homeomorphism. | Proof. Let \( \left\{ {t}_{n}\right\} \) be a sequence in \( \left( {c, d}\right) \) such that\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathbf{\varphi }\left( {t}_{n}\right) = \mathbf{\varphi }\left( {t}_{0}\right) \]\n\nfor some \( {t}_{0} \in \left( {c, d}\right) \) . In order to conclude that \( \left\{... | Yes |
The regular \( n \) -surfaces of class \( {C}^{p} \) in \( {\mathbb{R}}^{n} \) are the open subsets of \( {\mathbb{R}}^{n} \) . | In fact, let us assume that \( M \subset {\mathbb{R}}^{n} \) is a regular \( n \) -surface of class \( {C}^{1} \) . According to Definition 3.1.1, for each point \( {\mathbf{x}}_{0} \in M \) there exist an open set \( A \subset {\mathbb{R}}^{n} \) and a mapping\n\n\[ \mathbf{\varphi } : A \subset {\mathbb{R}}^{n} \righ... | Yes |
Example 3.1.2. Let \( M \) be a regular \( k \) -surface of class \( {C}^{p} \) in \( {\mathbb{R}}^{n} \) and let \( U \) be an open set in \( {\mathbb{R}}^{n} \) such that \( M \cap U \neq \varnothing \) . Then \( M \cap U \) is a regular \( k \) -surface of class \( {C}^{p} \) in \( {\mathbb{R}}^{n} \) . | Let \( {\mathbf{x}}_{0} \in M \cap U \) and let \( \left( {A,\mathbf{\varphi }}\right) \) be a coordinate system of \( M \) such that \( {\mathbf{x}}_{0} \in \mathbf{\varphi }\left( A\right) \) . We define \( B \mathrel{\text{:=}} {\varphi }^{-1}\left( {M \cap U}\right) \), which is an open subset of \( A \) . If \( \p... | Yes |
The Bernoulli lemniscate is not a regular curve (Fig. 3.2). | The equation of the lemniscate in polar coordinates is\n\n\[ \n{\rho }^{2} = 2\cos \left( {2\theta }\right) \n\]\n\nand it is given by\n\n\[ \nM \mathrel{\text{:=}} \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} : {\left( {x}^{2} + {y}^{2}\right) }^{2} = 2\left( {{x}^{2} - {y}^{2}}\right) }\right\} .\n\]\n\nIf \( U... | Yes |
Let \( \mathbf{g} : A \subset {\mathbb{R}}^{k} \rightarrow {\mathbb{R}}^{n - k} \) be a mapping of class \( {C}^{p} \) on the open set \( A \) . Then the graph of \( \mathbf{g} \) ,\n\n\[ \n{G}_{g} \mathrel{\text{:=}} \{ \left( {\mathbf{t},\mathbf{g}\left( \mathbf{t}\right) }\right) : \mathbf{t} \in A\} \subset {\mathb... | In fact, we consider the mapping\n\n\[ \n\mathbf{\varphi } : A \subset {\mathbb{R}}^{k} \rightarrow {\mathbb{R}}^{n}\n\]\n\ngiven by\n\n\[ \n\mathbf{\varphi }\left( t\right) \mathrel{\text{:=}} \left( {t,\mathbf{g}\left( t\right) }\right)\n\]\n\n(1) Then \( \varphi \) is injective, and if \( B \subset A \) is an open s... | Yes |
Example 3.1.5. The unit sphere (Fig. 3.4) in \( {\mathbb{R}}^{3} \) ,\n\n\[ S \mathrel{\text{:=}} \left\{ {\left( {x, y, z}\right) \in {\mathbb{R}}^{3} : {x}^{2} + {y}^{2} + {z}^{2} = 1}\right\} ,\]\n\nis a regular surface of class \( {C}^{\infty } \) . | Consider \( A \mathrel{\text{:=}} \left\{ {\left( {s, t}\right) \in {\mathbb{R}}^{2} : {s}^{2} + {t}^{2} < 1}\right\} \) and let \( g : A \rightarrow \mathbb{R} \) be the mapping of class \( {C}^{\infty } \) given by\n\n\[ g\left( {s, t}\right) = \sqrt{1 - \left( {{s}^{2} + {t}^{2}}\right) }\]\n\nBy Example 3.1.4, \( \... | Yes |
Proposition 3.2.1. Let \( 1 \leq k < n \) and \( M \subset {\mathbb{R}}^{n}, M \neq \varnothing \), be given. Assume that for each \( {\mathbf{x}}_{0} \in M \) there exists a function of class \( {C}^{p} \) on the open set \( U \) ,\n\n\[ \mathbf{\Phi } : U \subset {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n - k} \]\n... | Proof. Let \( {\mathbf{x}}_{0} = \left( {{x}_{1}^{0},\ldots ,{x}_{n}^{0}}\right) \in M \) and \( \mathbf{\Phi } = \left( {{\Phi }_{1},\ldots ,{\Phi }_{n - k}}\right) \) be as in the hypothesis. After a permutation of the coordinates if necessary, we can assume that\n\n\[ \frac{\partial \left( {{\mathbf{\Phi }}_{1},\ldo... | Yes |
Let \( f : U \subset {\mathbb{R}}^{2} \rightarrow \mathbb{R} \) be a function of class \( {C}^{p} \) on the open set \( U \) and let \( c \) be a real number such that\n\n\[ \Gamma \mathrel{\text{:=}} \{ \left( {x, y}\right) \in U : f\left( {x, y}\right) = c\} \]\n\nis nonempty and \( \nabla f\left( {x, y}\right) \neq ... | To see this, it is enough to apply Proposition 3.2.1 with \( \Phi : U \rightarrow \mathbb{R} \) ,\n\n\[ \Phi \left( {x, y}\right) = f\left( {x, y}\right) - c. \]\n\nThe level curve is the projection onto the \( {xy} \) -plane of the intersection of the graph of \( f \) with the horizontal plane \( z = c \) . | Yes |
The level surface\n\n\[ S \mathrel{\text{:=}} \{ \left( {x, y, z}\right) \in U : f\left( {x, y, z}\right) = c\} \]\n\nof the function \( f : U \subset {\mathbb{R}}^{3} \rightarrow \mathbb{R} \) of class \( {C}^{p} \) on the open set \( U \) is a regular surface of class \( {C}^{p} \) provided that \( \nabla f\left( {x,... | Again, it suffices to apply Proposition 3.2.1 to \( \Phi \left( {x, y, z}\right) = f\left( {x, y, z}\right) - c \) . | No |
For every \( a > 0 \), the cone\n\n\[ \n{x}^{2} + {y}^{2} = a{z}^{2} \n\]\n\nwith the vertex \( \left( {0,0,0}\right) \) removed is a regular surface of class \( {C}^{\infty } \). | Let \( U \mathrel{\text{:=}} {\mathbb{R}}^{3} \smallsetminus \{ \left( {0,0,0}\right) \} \) and consider the function\n\n\[ \n\Phi : U \rightarrow \mathbb{R} \n\]\n\nof class \( {C}^{\infty } \), defined by\n\n\[ \n\Phi \left( {x, y, z}\right) \mathrel{\text{:=}} {x}^{2} + {y}^{2} - a{z}^{2}. \n\]\n\nThen the cone with... | Yes |
The sphere \[ {S}^{n - 1} \mathrel{\text{:=}} \left\{ {\left( {{x}_{1},\ldots ,{x}_{n}}\right) \in {\mathbb{R}}^{n} : {x}_{1}^{2} + \cdots + {x}_{n}^{2} = 1}\right\} \] is a regular \( \left( {n - 1}\right) \) -surface of class \( {C}^{\infty } \) in \( {\mathbb{R}}^{n}, n \geq 2 \). | Indeed, define \( \Phi : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) by \[ \Phi \left( \mathbf{x}\right) \mathrel{\text{:=}} \left( {\mathop{\sum }\limits_{{k = 1}}^{n}{x}_{k}^{2}}\right) - 1 \] Then \( {S}^{n - 1} = {\Phi }^{-1}\left( 0\right) \) and \( {\Phi }^{\prime }\left( \mathbf{x}\right) = \left( {2{x}_{1},\ldot... | Yes |
Lemma 3.3.1. Let \( M \) be a regular \( k \) -surface of class \( {C}^{p} \) in \( {\mathbb{R}}^{n},\left( {A,\varphi }\right) \) a coordinate system of \( M \), and \( \mathbf{\alpha } : V \subset {\mathbb{R}}^{m} \rightarrow M \) a mapping defined on the open set \( V \) . If \( \mathbf{\alpha }\left( {\mathbf{u}}_{... | Proof. We provide the proof only for the case that \( \mathbf{\alpha } \) is of class \( {C}^{q} \) on \( V \) . We take \( {t}_{0} \in A \) such that\n\n\[ \n\mathbf{\alpha }\left( {\mathbf{u}}_{0}\right) = \mathbf{\varphi }\left( {\mathbf{t}}_{0}\right) \n\]\n\nWe now set\n\n\[ \n\mathbf{\varphi } = \left( {{\varphi ... | Yes |
Theorem 3.3.1. Let \( \left( {{A}_{1},{\varphi }_{1}}\right) ,\left( {{A}_{2},{\varphi }_{2}}\right) \) be two coordinate systems of a regular \( k \) - surface \( M \) in \( {\mathbb{R}}^{n} \) of class \( {C}^{p} \) such that\n\n\[ D \mathrel{\text{:=}} {\varphi }_{1}\left( {A}_{1}\right) \cap {\varphi }_{2}\left( {A... | Proof. It follows from Lemma 3.3.1, with \( \varphi = {\varphi }_{2} \) and \( \alpha = {\varphi }_{1} \), that \( {\varphi }_{2}^{-1} \circ {\varphi }_{1} \) is a function of class \( {C}^{p} \) on its domain of definition, which is precisely \( {\varphi }_{1}^{-1}\left( D\right) \) . | Yes |
Corollary 3.3.1. Let \( M \) be a regular \( k \) -surface of class \( {C}^{p} \) in \( {\mathbb{R}}^{n}, A \subset {\mathbb{R}}^{k} \) an open set, and \[ \mathbf{\varphi } : A \subset {\mathbb{R}}^{k} \rightarrow M \subset {\mathbb{R}}^{n} \] an injective mapping of class \( {C}^{p} \) such that \( {\mathbf{\varphi }... | Proof. We fix \( {t}_{0} \in A \) and put \( {\mathbf{x}}_{0} \mathrel{\text{:=}} \mathbf{\varphi }\left( {t}_{0}\right) \) . We may choose a coordinate system \( \left( {B,\mathbf{\psi }}\right) \) of \( M \) such that \( {\mathbf{x}}_{0} \in \mathbf{\psi }\left( B\right) \) . By Theorem 3.3.1, the mapping \[ \mathbf{... | Yes |
Let us consider the mapping\n\n\[ \varphi : \left( {-1,\frac{\pi }{2}}\right) \rightarrow {\mathbb{R}}^{2} \]\n\ndefined by\n\n\[ \mathbf{\varphi }\left( t\right) = \left( {t,0}\right) \]\n\nfor \( - 1 < t \leq 0 \) and\n\n\[ \mathbf{\varphi }\left( t\right) = \left( {t\cos t, t\frac{\sin \left( {2t}\right) }{2}}\right... | To see this, suppose there exists a coordinate system \( \left( {A,\mathbf{\gamma }}\right) \) with \( \left( {0,0}\right) \in \mathbf{\gamma }\left( A\right) \) . Let \( U \) be an open ball centered at \( \left( {0,0}\right) \) such that \( U \cap M \subset \mathbf{\gamma }\left( A\right) \) . Since \( \mathbf{\gamma... | Yes |
Example 3.3.2 (Spherical coordinates). Let \( S \) be the sphere with radius \( R \) centered at the origin in \( {\mathbb{R}}^{3} \) (Fig. 3.12), \[ A \mathrel{\text{:=}} \left\{ {\left( {s, t}\right) \in {\mathbb{R}}^{2} : 0 < s < \pi ,0 < t < {2\pi }}\right\} , \] and let us consider \[ \mathbf{\varphi }\left( {s, t... | We show that \( {\varphi }^{\prime }\left( {s, t}\right) \) has rank 2 for every \( \left( {s, t}\right) \in A \) . To this end, we determine that \[ \frac{\partial \left( {x, y}\right) }{\partial \left( {s, t}\right) } = {R}^{2}\cos s\sin s \] \[ \frac{\partial \left( {y, z}\right) }{\partial \left( {s, t}\right) } = ... | Yes |
Example 3.4.1 (Tangent plane to the cone \( {x}^{2} + {y}^{2} = 2{z}^{2} \) at the point \( \left( {1,1,1}\right) \) ). | We already proved in Example 3.2.3 that\n\n\[ S \mathrel{\text{:=}} \left\{ {\left( {x, y, z}\right) \neq \left( {0,0,0}\right) : {x}^{2} + {y}^{2} = 2{z}^{2}}\right\} \]\n\nis a regular surface of class \( {C}^{\infty } \). We now choose a coordinate system for this cone. We take\n\n\[ \mathbf{\varphi } : A \subset {\... | Yes |
Proposition 3.4.1. Let \( 1 \leq k < n \) be given and let \( M = {\Phi }^{-1}\left( \mathbf{0}\right) \) be the regular \( k \) - surface where\n\n\[ \mathbf{\Phi } : U \subset {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n - k},\mathbf{\Phi } = \left( {{\Phi }_{1},{\Phi }_{2},\ldots ,{\Phi }_{n - k}}\right) ,\]\n\nis a... | Proof. It follows from Proposition 3.2.1 that \( M \) is a regular \( k \) -surface of class \( {C}^{1} \) in \( {\mathbb{R}}^{n} \) . If \( \mathbf{\alpha } : \left( {-\delta ,\delta }\right) \rightarrow M \) is differentiable at \( t = 0 \) and \( \mathbf{\alpha }\left( 0\right) = {\mathbf{x}}_{0} \), then\n\n\[ {\Ph... | Yes |
Example 3.4.2 (Tangent hyperplane to a graph). Let \( g : A \subset {\mathbb{R}}^{n - 1} \rightarrow \mathbb{R} \) be a function of class \( {C}^{1} \) on an open set \( A \) and\n\n\[ \mathbf{\varphi } : A \rightarrow {\mathbb{R}}^{n};\mathbf{\varphi }\left( \mathbf{t}\right) = \left( {\mathbf{t}, g\left( \mathbf{t}\r... | Finally, the equation of the tangent hyperplane to the graph of \( g \) at the point \( {\mathbf{x}}_{0} \) is\n\n\[ \left( {{x}_{1} - {x}_{1}^{0}}\right) \frac{\partial g}{\partial {t}_{1}}\left( {t}_{0}\right) + \cdots + \left( {{x}_{n - 1} - {x}_{n - 1}^{0}}\right) \frac{\partial g}{\partial {t}_{n - 1}}\left( {t}_{... | Yes |
Example 3.4.3 (Normal vector to a level surface). Let \( f : U \subset {\mathbb{R}}^{3} \rightarrow \mathbb{R} \) be a function of class \( {C}^{1} \) on an open set \( U \) such that for all \( \left( {x, y, z}\right) \in S \) ,\n\n\[ \nabla f\left( {x, y, z}\right) \neq \left( {0,0,0}\right) \]\n\nwhere \( S \) repre... | From Example 3.2.2, \( S \) is a regular surface of class \( {C}^{1} \) . Moreover, the vector\n\n\[ \nabla f\left( {{x}_{0},{y}_{0},{y}_{0}}\right) \]\n\nis normal to the surface at the point \( \left( {{x}_{0},{y}_{0},{z}_{0}}\right) \) . In fact, it suffices to apply Proposition 3.4.1 with \( \Phi \mathrel{\text{:=}... | No |
Theorem 3.4.2. Let \( M \subset {\mathbb{R}}^{n} \) be a nonempty set and \( 1 \leq k < n \) . The following conditions are equivalent:\n\n1. \( M \) is a \( k \) -dimensional regular surface in \( {\mathbb{R}}^{n} \) of class \( {C}^{p} \) .\n\n2. For every \( {\mathbf{x}}_{0} \in M \) there exist an open neighborhood... | Proof. Some of the necessary implications have already been proved. For example, (1) implies (2) is proven in Proposition 3.1.1, while (2) implies (1) holds by Example 3.1.4. (3) implies (2) is contained in the proof of Proposition 3.2.1. We need only show that (2) implies (3). To do this, we represent the points \( \m... | Yes |
Theorem 4.1.1. Let \( P \) be the parallelepiped spanned by the linearly independent vectors \( \left\{ {{\mathbf{a}}_{1},\ldots ,{\mathbf{a}}_{k}}\right\} \) . Then\n\n\[ k\text{-area}\left( P\right) = \sqrt{\det \left( \left\langle {{\mathbf{a}}_{i},{\mathbf{a}}_{j}}\right\rangle \right) } = \sqrt{\det \left( {{B}^{\... | Proof. Let \( V \) be the \( k \) -dimensional vector subspace spanned by \( \left\{ {{\mathbf{a}}_{1},\ldots ,{\mathbf{a}}_{k}}\right\} \) and let \( \Psi : V \rightarrow {\mathbb{R}}^{k} \) be a linear isomorphism preserving the inner product. Consider the linear mapping\n\n\[ \mathbf{L} : {\mathbb{R}}^{k} \rightarro... | Yes |
Lemma 4.2.1. Let \( K \mathrel{\text{:=}} \left\lbrack {a, b}\right\rbrack \times \left\lbrack {c, d}\right\rbrack \) ,\n\n\[ a = {s}_{0} < {s}_{1} < \cdots < {t}_{m} = b \]\n\na partition of \( \left\lbrack {a, b}\right\rbrack \) into subintervals of length \( h \), and\n\n\[ c = {t}_{0} < {t}_{1} < \cdots < {t}_{l} =... | Proof. Define \( {I}_{i, j} \mathrel{\text{:=}} \left\lbrack {{s}_{i},{s}_{i + 1}}\right\rbrack \times \left\lbrack {{t}_{j},{t}_{j + 1}}\right\rbrack \) . Then\n\n\[ {\iint }_{K}F\left( {s, t}\right) \mathrm{d}\left( {s, t}\right) = \mathop{\sum }\limits_{{i = 0}}^{{m - 1}}\mathop{\sum }\limits_{{j = 0}}^{{l - 1}}{\ii... | Yes |
Example 4.2.1. Let \( \varphi : {\mathbb{R}}^{k} \rightarrow {\mathbb{R}}^{n} \) be a linear mapping preserving the dot product. Then \( V = \mathbf{\varphi }\left( {\mathbb{R}}^{k}\right) \) is a \( k \) -dimensional vector subspace of \( {\mathbb{R}}^{n} \) and \( \left( {{\mathbb{R}}^{k},\mathbf{\varphi }}\right) \)... | \n\[\n\operatorname{area}\left( {K,\mathbf{\varphi }}\right) = {\int }_{{\mathbb{R}}^{k}}{\chi }_{K}\left( \mathbf{t}\right) \mathrm{d}\mathbf{t}\n\]\n\nIn fact, since \( \varphi \) is a linear mapping, we have that \( {\varphi }^{\prime }\left( t\right) \) is the matrix of \( \varphi \) with respect to the canonical b... | Yes |
Lemma 4.2.2. Let \( M \) be a regular \( k \) -surface in \( {\mathbb{R}}^{n} \) with a finite atlas \[ {\left\{ \left( {A}_{j},{\mathbf{\varphi }}_{j}\right) \right\} }_{j = 1}^{m}\text{.} \] Then there is a partition of \( M \) into subsets \( \left\{ {B}_{j}\right\} \) with the property that \( {B}_{j} = \) \( {\mat... | Proof. According to Definition 3.1.1, for every \( j = 1,\ldots, m \) there is an open set \( {U}_{j} \) in \( {\mathbb{R}}^{n} \) such that \( {\varphi }_{j}\left( {A}_{j}\right) = {U}_{j} \cap M \) . Define \( {B}_{1} \mathrel{\text{:=}} {\varphi }_{1}\left( {A}_{1}\right) \), and for each \( j = 2,\ldots m \) , \[ {... | Yes |
Example 4.2.3. Let \( g : A \subset {\mathbb{R}}^{2} \rightarrow \mathbb{R} \) be a mapping of class \( {C}^{1} \) on the open set \( A \) . Then \( S \mathrel{\text{:=}} {G}_{g} \) is a regular surface in \( {\mathbb{R}}^{3} \) . We plan to express the area of \( S \) in terms of the function \( g \) . | We already know that\n\n\[ \mathbf{\varphi } : A \subset {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{3},\;\left( {s, t}\right) \mapsto \left( {s, t, g\left( {s, t}\right) }\right) ,\]\n\nis a parameterization of \( S \) and that\n\n\[ \frac{\partial \varphi }{\partial s}\left( {s, t}\right) \times \frac{\partial \varphi... | Yes |
The bases \( \left\{ {{\mathbf{v}}_{1},{\mathbf{v}}_{2},{\mathbf{v}}_{3}}\right\} \) and \( \left\{ {{\mathbf{v}}_{3},{\mathbf{v}}_{1},{\mathbf{v}}_{2}}\right\} \) of \( {\mathbb{R}}^{3} \) define the same orientation. However, \( \left\{ {{\mathbf{v}}_{1},{\mathbf{v}}_{2},{\mathbf{v}}_{3}}\right\} \) and \( \left\{ {{... | In the first case, the change-of-basis matrix is\n\n\[ \left( \begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right) \]\n\nwhich has determinant equal to 1 . In the second case, the change-of-basis matrix is\n\n\[ \left( \begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right) \]\n\n... | Yes |
Proposition 5.1.1. Let\n\n\[ \mathbf{\varphi } : \left\lbrack {0,1}\right\rbrack \rightarrow {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \times \cdots \times {\mathbb{R}}^{n},\;\mathbf{\varphi } = \left( {{\mathbf{\varphi }}_{1},\ldots ,{\mathbf{\varphi }}_{n}}\right) \]\n\nbe a continuous mapping such that \( {\left\{ {\... | Proof. Let \( f\left( t\right) \) denote the determinant of the matrix with vector columns \( \left\lbrack {{\mathbf{\varphi }}_{1}\left( t\right) }\right. \) , \( \left. {{\mathbf{\varphi }}_{2}\left( t\right) ,\ldots ,{\mathbf{\varphi }}_{n}\left( t\right) }\right\rbrack \) . Then \( f \) is a continuous function on ... | Yes |
Proposition 5.1.2. Let \( \left\{ {{v}_{1},{v}_{2}}\right\} \) be two linearly independent vectors in \( {\mathbb{R}}^{3} \) . Then the ordered basis \( \left\{ {{\mathbf{v}}_{1} \times {\mathbf{v}}_{2},{\mathbf{v}}_{1},{\mathbf{v}}_{2}}\right\} \) of \( {\mathbb{R}}^{3} \) is positively oriented. | Proof. With respect to the canonical basis, the matrix of the linear transformation \( f : {\mathbb{R}}^{3} \rightarrow {\mathbb{R}}^{3} \) defined by \( f\left( {\mathbf{e}}_{1}\right) = {\mathbf{v}}_{1} \times {\mathbf{v}}_{2}, f\left( {\mathbf{e}}_{2}\right) = {\mathbf{v}}_{1} \), and \( f\left( {\mathbf{e}}_{3}\rig... | Yes |
Proposition 5.1.3. Let \( V \) be a two-dimensional vector subspace of \( {\mathbb{R}}^{3} \) and let \( {\mathcal{B}}_{1} \mathrel{\text{:=}} \left\{ {{\mathbf{v}}_{1},{\mathbf{v}}_{2}}\right\} \) and \( {\mathcal{B}}_{2} \mathrel{\text{:=}} \left\{ {{\mathbf{w}}_{1},{\mathbf{w}}_{2}}\right\} \) be two ordered bases o... | Proof. Both \( {\mathbf{v}}_{1} \times {\mathbf{v}}_{2} \) and \( {\mathbf{w}}_{1} \times {\mathbf{w}}_{2} \) are orthogonal vectors to \( V \), and so (since \( V \) has a one-dimensional orthogonal complement) there exists \( \lambda \neq 0 \) such that \( {\mathbf{v}}_{1} \times {\mathbf{v}}_{2} = \) \( \lambda \lef... | Yes |
Lemma 5.2.1. Let \( V \) and \( W \) be \( n \) -dimensional real vector spaces, \[ L : V \rightarrow W \] a linear isomorphism, and \( {\left\{ {\mathbf{u}}_{j}\right\} }_{j = 1}^{n},{\left\{ {\mathbf{v}}_{j}\right\} }_{j = 1}^{n} \) two bases of \( V \) . Then \[ {\left\{ {\mathbf{u}}_{j}\right\} }_{j = 1}^{n} \sim {... | Proof. Denote by \( f : V \rightarrow V \) and \( g : W \rightarrow W \) the linear isomorphisms defined by \( \mathbf{f}\left( {\mathbf{u}}_{i}\right) = {\mathbf{v}}_{i},\mathbf{g}\left( {\mathbf{L}{\mathbf{u}}_{i}}\right) = \mathbf{L}{\mathbf{v}}_{i} \) . Then \( \left( {{\mathbf{L}}^{-1} \circ \mathbf{g} \circ \math... | Yes |
Lemma 5.2.2. Let \( \left( {A,\varphi }\right) \) be a coordinate system of a regular \( k \) -surface \( M \) in \( {\mathbb{R}}^{n} \) and define \( \mathbf{L} : {\mathbb{R}}^{k} \rightarrow {\mathbb{R}}^{k} \) by \( \mathbf{L}\left( \mathbf{t}\right) = \left( {{t}_{1},\ldots ,{t}_{k - 2},{t}_{k},{t}_{k - 1}}\right) ... | Proof. It is clear that \( \left( {B,\psi }\right) \) is a coordinate system of \( M \) . Set \( S \mathrel{\text{:=}} \varphi \left( A\right) = \psi \left( B\right) \) . For every\n\n\[ \mathbf{x} = \mathbf{\varphi }\left( \mathbf{t}\right) \in S \]\n\nthe orientation of \( {T}_{x}M \) given by the coordinate system \... | Yes |
Lemma 5.2.4. Let \( f, g : A \subset {\mathbb{R}}^{p} \rightarrow {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) be continuous on the open set \( A \) and let \( h : A \subset {\mathbb{R}}^{p} \rightarrow \mathbb{R} \) be such that \( f\left( \mathbf{x}\right) = h\left( \mathbf{x}\right) \mathbf{g}\left( \mathbf{x}\right) \... | Proof. For any \( {\mathbf{x}}_{0} \in A \) there exists \( j \in \{ 1,\ldots, n\} \) such that \( {g}_{j}\left( {\mathbf{x}}_{0}\right) \neq 0 \), and by continuity, there then exists \( r > 0 \) with \( {g}_{j}\left( \mathbf{x}\right) \neq 0 \) for all \( \mathbf{x} \in B\left( {{\mathbf{x}}_{0}, r}\right) \) . It fo... | Yes |
The sphere \( M \mathrel{\text{:=}} \left\{ {\left( {x, y, z}\right) \in {\mathbb{R}}^{3} : {x}^{2} + {y}^{2} + {z}^{2} = {R}^{2}}\right\} \) is orientable. | Proof. It suffices to consider the continuous vector field \( \mathbf{F} : M \rightarrow {\mathbb{R}}^{3} \) defined by \( \mathbf{F}\left( {x, y, z}\right) \mathrel{\text{:=}} \frac{1}{R}\left( {x, y, z}\right) \) . It is obvious that \( \mathbf{F}\left( {x, y, z}\right) \) is normal to the sphere at the point \( \lef... | Yes |
Proposition 6.1.1. (1) If \( f, g \in {\Lambda }^{k}\left( {\mathbb{R}}^{n}\right) \) and \( f\left( {e}_{\mathbf{I}}\right) = g\left( {e}_{\mathbf{I}}\right) \) for any strictly increasing \( k \) -tuple \( \mathbf{I} \), then \( f = g \) . | Proof. (1) After applying property (c) of Definition 6.1.1 as many times as necessary, we deduce that \( f\left( {{\mathbf{e}}_{{i}_{1}},\ldots ,{\mathbf{e}}_{{i}_{k}}}\right) = g\left( {{\mathbf{e}}_{{i}_{1}},\ldots ,{\mathbf{e}}_{{i}_{k}}}\right) \) for all \( {i}_{1},\ldots ,{i}_{k} \) . It then follows from propert... | Yes |
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