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Proposition 6.1.2. For any \( n \in \mathbb{N} \) and \( 1 \leq k \leq n \), the set\n\n\[ \n\left\{ {\mathrm{d}{x}_{\mathbf{I}} : \mathbf{I} \in \mathcal{J}}\right\} \n\] \n\nis a basis of \( {\Lambda }^{k}\left( {\mathbb{R}}^{n}\right) \) . | Proof. Let us assume\n\n\[ \n\mathop{\sum }\limits_{{\mathbf{I} \in \mathcal{J}}}{a}_{\mathbf{I}}\mathrm{d}{x}_{\mathbf{I}} = 0 \n\] \n\nThen for each strictly increasing \( k \) -tuple \( \mathbf{J} \) we have\n\n\[ \n0 = \left( {\mathop{\sum }\limits_{{\mathbf{I} \in \mathcal{J}}}{a}_{\mathbf{I}}\mathrm{d}{x}_{\mathb... | Yes |
Proposition 6.1.3. (1) There is a linear isomorphism from \( {\mathbb{R}}^{n} \) to \( {\Lambda }^{1}\left( {\mathbb{R}}^{n}\right) \), given by | \[ {\mathbb{R}}^{n} \rightarrow {\Lambda }^{1}\left( {\mathbb{R}}^{n}\right) ,\mathbf{a} = \left( {{a}_{1},\ldots ,{a}_{n}}\right) \mapsto \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{j} \cdot \mathrm{d}{x}_{j}. \] | Yes |
Proposition 6.2.1. Let \( \omega \) be a \( k \) -form, \( \varphi \) an \( s \) -form, and \( \theta \) an \( r \) -form on \( U \subset {\mathbb{R}}^{n} \) . Then:\n\n(1) \( \left( {\omega \land \varphi }\right) \land \mathbf{\theta } = \omega \land \left( {\varphi \land \mathbf{\theta }}\right) \),\n\n(2) \( \omega ... | Proof. Leaving the other parts to the reader, we will prove (3) by way of an informal induction argument. In shortened form, we can write\n\n\[ \mathbf{\omega } \land \mathbf{\varphi } = \mathop{\sum }\limits_{{\mathbf{I} \in \mathcal{J}}}\mathop{\sum }\limits_{{\mathbf{J} \in \mathcal{J}}}{f}_{\mathbf{I}}{g}_{\mathbf{... | No |
Corollary 6.2.1. Let \( {\varphi }_{1},\ldots ,{\varphi }_{k} \in {\Lambda }^{1}\left( {\mathbb{R}}^{n}\right) \) be given. Then for each \( \left( {{v}_{1},\ldots ,{v}_{k}}\right) \in \) \( {\left( {\mathbb{R}}^{n}\right) }^{k} \), and each \( x \in {\mathbb{R}}^{n} \)\n\n\[ \left( {{\varphi }_{1} \land \cdots \land {... | Proof. We fix \( {\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{k} \in {\mathbb{R}}^{n} \) and consider the mappings\n\n\[ S, T : {\left( {\mathbb{R}}^{n}\right) }^{ * } \times \cdots \times {\left( {\mathbb{R}}^{n}\right) }^{ * } \rightarrow \mathbb{R} \]\n\ndefined by\n\n\[ S\left( {{\varphi }_{1},\ldots ,{\varphi }_{k}}\rig... | Yes |
The exterior differential of a differential form of degree 1 on \( {\mathbb{R}}^{2} \) . | Let\n\n\[ \mathbf{F} = \left( {P, Q}\right) \]\n\nbe a vector field of class \( {C}^{1} \) on an open set \( U \) in \( {\mathbb{R}}^{2} \) and let us consider the associated differential form of degree 1 ,\n\n\[ \omega \left( {x, y}\right) = P\left( {x, y}\right) \mathrm{d}x + Q\left( {x, y}\right) \mathrm{d}y. \]\n\n... | Yes |
The exterior differential of a differential form of degree 1 on \( {\mathbb{R}}^{3} \). | Let\n\n\[ \mathbf{F} = \left( {{f}_{1},{f}_{2},{f}_{3}}\right) \]\n\nbe a vector field of class \( {C}^{1} \) in an open set \( U \) of \( {\mathbb{R}}^{3} \). We intend to calculate the exterior differential of the associated differential form of degree 1 ,\n\n\[ \omega = {f}_{1}\mathrm{\;d}x + {f}_{2}\mathrm{\;d}y + ... | Yes |
Example 6.3.3 (The exterior differential of a differential form of degree 2 on \( {\mathbb{R}}^{3} \) ).\n\nLet\n\n\[ \mathbf{F} = \left( {{f}_{1},{f}_{2},{f}_{3}}\right) \]\n\nbe a vector field of class \( {C}^{1} \) in an open set \( U \) of \( {\mathbb{R}}^{3} \) . We intend to calculate the exterior differential of... | We get\n\n\[ \mathrm{d}\omega = \frac{\partial {f}_{1}}{\partial x}\mathrm{\;d}x \land \mathrm{d}y \land \mathrm{d}z + \frac{\partial {f}_{2}}{\partial y}\mathrm{\;d}y \land \mathrm{d}z \land \mathrm{d}x + \frac{\partial {f}_{3}}{\partial z}\mathrm{\;d}z \land \mathrm{d}x \land \mathrm{d}y. \]\n\nUsing\n\n\[ \mathrm{d}... | Yes |
Proposition 6.3.1. Let \( \omega \) and \( \eta \) be two k-forms, \( \varphi \) an \( s \) -form, and \( f \) a function, each of class \( {C}^{1} \) on an open set \( U \) of \( {\mathbb{R}}^{n} \) . Then\n\n(1) \( \mathrm{d}\left( {\omega + \eta }\right) = \mathrm{d}\omega + \mathrm{d}\eta \) ,\n\n(2) \( \mathrm{d}\... | Proof. (1) This is a simple verification.\n\n(2) It suffices to consider the case\n\n\[ \mathbf{\omega }\left( \mathbf{x}\right) = g\left( \mathbf{x}\right) \mathrm{d}{x}_{{i}_{1}} \land \cdots \land \mathrm{d}{x}_{{i}_{k}},1 \leq {i}_{1} < \cdots < {i}_{k} \leq n, \]\n\nwhere \( g : U \rightarrow \mathbb{R} \) is a fu... | Yes |
Corollary 6.3.1. Let \( F \mathrel{\text{:=}} \left( {{f}_{1},{f}_{2},{f}_{3}}\right) \) be a vector field of class \( {C}^{2} \) in an open set \( U \) of \( {\mathbb{R}}^{3} \) . Then\n\n\[ \operatorname{Div}\left( {\mathbf{\nabla } \times \mathbf{F}}\right) = 0\text{in}U\text{.} \] | Proof. If\n\n\[ \omega = {f}_{1}\mathrm{\;d}x + {f}_{2}\mathrm{\;d}y + {f}_{3}\mathrm{\;d}z \]\n\nis the differential form of degree 1 associated to the vector field \( \mathbf{F} \), then \( \mathbf{\nabla } \times \mathbf{F} \) is the vector field associated to the differential form (of degree 2) \( \mathrm{d}\mathbf... | Yes |
Let \( \mathbf{T} : U \subset {\mathbb{R}}^{m} \rightarrow G \subset {\mathbb{R}}^{n} \) be a mapping of class \( {C}^{1},\mathbf{T} = \) \( \left( {{T}_{1},\ldots ,{T}_{n}}\right) \) . For each \( 1 \leq i \leq n \) we have \( {\mathbf{T}}^{ * }\mathrm{\;d}{x}_{i} = \mathrm{d}{T}_{i} \) . | For \( \mathbf{u} \in U \subset {\mathbb{R}}^{m} \) we write \( \mathbf{x} = \mathbf{T}\left( \mathbf{u}\right) \), so \( {x}_{i} = {T}_{i}\left( \mathbf{u}\right) \) . By definition,\n\n\[ \left( {{\mathbf{T}}^{ * }\mathrm{\;d}{x}_{i}}\right) \left( \mathbf{u}\right) \left( \mathbf{v}\right) = \mathrm{d}{x}_{i}\left( ... | Yes |
Corollary 6.4.1. Let \( \mathbf{T} : U \subset {\mathbb{R}}^{m} \rightarrow G \subset {\mathbb{R}}^{n} \) be a mapping of class \( {C}^{1} \) on the open set \( U \) and let \( \omega : G \subset {\mathbb{R}}^{n} \rightarrow {\Lambda }^{k}\left( {\mathbb{R}}^{n}\right) \) be a differential form of degree \( k \) on the... | Proof. Consider the differential 1-form \( {\varphi }_{j} = \mathrm{d}{x}_{{i}_{j}}, j = 1,\ldots, k \) . Clearly \( {\varphi }_{j}\left( \mathbf{x}\right) = \mathrm{d}{x}_{{i}_{j}} \) for every \( \mathbf{x} \in G \) . Now,\n\n\[ {\mathbf{T}}^{ * }\left( {{\varphi }_{1} \land \cdots \land {\varphi }_{k}}\right) \left(... | Yes |
Proposition 6.4.1. Let \( T : U \subset {\mathbb{R}}^{m} \rightarrow G \subset {\mathbb{R}}^{n} \) be a mapping of class \( {C}^{1} \) on the open set \( U \) and let \( \mathbf{\omega } \) and \( {\mathbf{\omega }}_{j} \) be differential forms of degree \( k \) on the open set \( G \) in \( {\mathbb{R}}^{n} \) . Furth... | Proof. (1) is a consequence of Definition 6.4.1 and the chain rule, while (2), (3), and (4) are immediate from the definition. | No |
In this example we will show that the definition provided in Chap. 2 of the integral of a differential 1-form \( \omega \) along a path \( \gamma \) (or equivalently the integral of its associated vector field \( \mathbf{F} \) along \( \mathbf{\gamma } \) ) can be written in terms of the pullback. | Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow {\mathbb{R}}^{n} \) be a path of class \( {C}^{1} \) whose arc is contained in an open set \( U \) and let \( \mathbf{F} = \left( {{f}_{1},\ldots ,{f}_{n}}\right) \) be a continuous vector field on \( U \) with associated differential form of degree 1 :\n\n\[... | Yes |
Example 6.4.4. Let \( S \) be a plane in \( {\mathbb{R}}^{3} \) that contains the point \( \mathbf{P} \) and let \( M \) denote the underlying vector subspace, that is, \( M = S - \mathbf{P} \) . We consider a linear isometry \( \mathbf{L} \) : \( {\mathbb{R}}^{2} \rightarrow M \) and assume that the surface \( S \) is... | Since \( {\mathbf{T}}^{ * }\left( \mathbf{\omega }\right) \) is a differential form of degree 2 on \( {\mathbb{R}}^{2} \), then, denoting by \( \left( {s, t}\right) \) the coordinates of an arbitrary point,\n\n\[ {\mathbf{T}}^{ * }\left( \mathbf{\omega }\right) = H \cdot \mathrm{d}s \land \mathrm{d}t \]\n\nThe function... | Yes |
Theorem 6.4.1. Let \( \mathbf{T} : U \subset {\mathbb{R}}^{k} \rightarrow G \subset {\mathbb{R}}^{n} \) be a mapping of class \( {C}^{1}\left( {\mathbf{x} = \mathbf{T}\left( \mathbf{u}\right) }\right) \) in an open set \( U \) and let \( \mathbf{\omega } : G \subset {\mathbb{R}}^{n} \rightarrow {\Lambda }^{k}\left( {\m... | Proof. It suffices to prove (1). Recall that \( {\Lambda }^{k}\left( {\mathbb{R}}^{k}\right) \) is a 1-dimensional vector space and a basis is given by\n\n\[ \left\{ {\mathrm{d}{u}_{1} \land \cdots \land \mathrm{d}{u}_{k}}\right\} \]\n\nBy Corollary 6.4.1,\n\n\[ \left( {{\mathbf{T}}^{ * }\mathbf{\omega }}\right) \left(... | Yes |
Theorem 6.5.1. Let \( \mathbf{T} : U \subset {\mathbb{R}}^{n} \rightarrow \mathbf{T}\left( U\right) \subset {\mathbb{R}}^{n} \) be a bijective mapping between two open sets \( U \) and \( \mathbf{T}\left( U\right) \) such that \( \mathbf{T} \) and \( {\mathbf{T}}^{-1} \) are both of class \( {C}^{1} \) . Let \( f : \ma... | \[ {\int }_{\mathbf{T}\left( U\right) }f = {\int }_{U}\left( {f \circ \mathbf{T}}\right) \left| {J\mathbf{T}}\right| \] where \( J\mathbf{T}\left( \mathbf{x}\right) \) is the determinant of the matrix of \( \mathrm{d}\mathbf{T}\left( \mathbf{x}\right) ,\mathbf{x} \in U \) . | Yes |
Theorem 6.5.2. Let \( \mathbf{T} : U \subset {\mathbb{R}}^{n} \rightarrow \mathbf{T}\left( U\right) \subset {\mathbb{R}}^{n} \) be a bijective mapping between two open sets \( U \) and \( \mathbf{T}\left( U\right) \) such that \( \mathbf{T} \) and \( {\mathbf{T}}^{-1} \) are both of class \( {C}^{1} \) . Let \( K \) be... | \[ {\int }_{\mathbf{T}\left( K\right) }f = {\int }_{K}\left( {f \circ \mathbf{T}}\right) \left| {J\mathbf{T}}\right| \] | Yes |
Let \( U \) and \( V \) be open sets in \( {\mathbb{R}}^{2} \) and \( \mathbf{T} : U \rightarrow V \) a transformation of class \( {C}^{2} \) on \( U \) with strictly positive Jacobian. Let \( \left( {x, y}\right) = \mathbf{T}\left( {s, t}\right) \). Suppose that \( K \) is a compact subset of \( U \) whose (topologica... | In fact, we consider \( \eta \mathrel{\text{:=}} {\mathbf{T}}^{ * }\mathbf{\omega } = {f}_{1}\mathrm{\;d}s + {f}_{2}\mathrm{\;d}t \), which is a differential form of class \( {C}^{1} \) in \( U \). According to Example 6.4.3, \( {\mathbf{\gamma }}^{ * }\left( \mathbf{\eta }\right) = h\left( t\right) \cdot \mathrm{d}t \... | Yes |
Let \( S \) be a plane in \( {\mathbb{R}}^{3} \) that contains the point \( \mathbf{P} \), and let \( M \) denote the corresponding vector subspace. Let \( \mathbf{L} : {\mathbb{R}}^{2} \rightarrow M \) be a linear isometry and assume that the surface \( S \) is oriented according to the parameterization\n\n\[ \mathbf{... | To this end, let\n\n\[ \omega = {f}_{1}\mathrm{\;d}x + {f}_{2}\mathrm{\;d}y + {f}_{3}\mathrm{\;d}z \]\n\nbe the differential form of degree 1 associated to \( \mathbf{F} \) . Put \( \mathbf{v} \mathrel{\text{:=}} {\mathbf{T}}^{ * }\left( \mathbf{\omega }\right) \), which is a differential form of degree \( 1, v = P \cd... | Yes |
Proposition 6.6.1. Let \( \alpha ,\beta ,\gamma : \left\lbrack {a, b}\right\rbrack \rightarrow U \subset {\mathbb{R}}^{n} \) be three continuous paths.\n\n(1) If \( \mathbf{\alpha } \) is homotopic to \( \mathbf{\beta } \), then \( \mathbf{\beta } \) is homotopic to \( \mathbf{\alpha } \) . | Proof. (1) If \( \mathbf{H} : \left\lbrack {a, b}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \rightarrow U \) is a homotopy from \( \mathbf{\alpha } \) to \( \mathbf{\beta } \) (Definition 6.6.2), then \( \widetilde{\mathbf{H}} : \left\lbrack {a, b}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \rightarr... | Yes |
Theorem 6.6.2. Let \( \alpha ,\beta : \left\lbrack {a, b}\right\rbrack \rightarrow U \subset {\mathbb{R}}^{n} \) be two paths that are piecewise of class \( {C}^{1} \) and satisfy (a) or (b):\n\n(a) \( \mathbf{\alpha }\left( a\right) = \mathbf{\beta }\left( a\right) ,\mathbf{\alpha }\left( b\right) = \mathbf{\beta }\le... | Proof. Since\n\n\[ \n\mathrm{d}\omega = 0, \n\]\n\nit follows from Theorem 6.6.1 that \( \mathbf{\omega } \) is exact on each ball contained in \( U \) . Let\n\n\[ \n\mathbf{H} : \left\lbrack {a, b}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \rightarrow U \n\]\n\nbe a homotopy according to Definition 6.6.2. Fo... | Yes |
Theorem 6.6.3. Let \( U \subset {\mathbb{R}}^{n} \) be a simply connected open set and suppose that \( \mathbf{\omega } \) is a closed differential form of degree 1 and class \( {C}^{1} \) . Then \( \omega \) is exact in \( U \) . | Proof. If \( \mathbf{\alpha } \) is a closed path that is piecewise of class \( {C}^{1} \) and contained in \( U \), then\n\n\[ \n{\int }_{\alpha }\omega = 0 \n\]\n\nby Theorem 6.6.2, since \( \alpha \) is strictly homotopic to a constant path. Without loss of generality we can assume that \( U \) is connected (for if ... | Yes |
Example 6.6.2. (a) Every open convex set is simply connected. | (a) Let \( U \) be an open convex set and let \( \mathbf{\alpha } : \left\lbrack {a, b}\right\rbrack \rightarrow U \) be a closed path. We let \( {\mathbf{x}}_{0} \mathrel{\text{:=}} \mathbf{\alpha }\left( a\right) = \mathbf{\alpha }\left( b\right) \) and define\n\n\[ \mathbf{H} : \left\lbrack {a, b}\right\rbrack \time... | Yes |
Corollary 6.6.1. Let \( U \subset {\mathbb{R}}^{2} \) be a simply connected open set. A vector field \( \mathbf{F} = \) \( \left( {{f}_{1},{f}_{2}}\right) \) of class \( {C}^{1} \) in \( U \) is conservative if and only if | \[ \frac{\partial {f}_{1}}{\partial y}\left( {x, y}\right) = \frac{\partial {f}_{2}}{\partial x}\left( {x, y}\right) \] for every \( \left( {x, y}\right) \in U \) . | Yes |
Theorem 7.1.1. Under the conditions of Definition 7.1.1, let\n\n\[ \left( {{\mathbf{\varphi }}^{ * }\mathbf{\omega }}\right) \left( \mathbf{t}\right) = g\left( \mathbf{t}\right) \cdot \mathrm{d}{t}_{1} \land \cdots \land \mathrm{d}{t}_{k} \]\n\nfor \( t \in A \) . Then\n\n\[ {\int }_{\mathbf{\varphi }\left( \Delta \rig... | Proof. Let\n\n\[ \mathbf{\omega }\left( \mathbf{x}\right) = \mathop{\sum }\limits_{{1 \leq {i}_{1} < \cdots < {i}_{k} \leq n}}{f}_{{i}_{1},\ldots ,{i}_{k}}\left( \mathbf{x}\right) \cdot \mathrm{d}{x}_{{i}_{1}} \land \cdots \land \mathrm{d}{x}_{{i}_{k}}. \]\n\nFrom Definitions 6.1.2 and 7.1.1,\n\n\[ {\int }_{\varphi \le... | Yes |
Let \( \mathbf{\omega } = f\mathrm{\;d}{t}_{1} \land \cdots \land \mathrm{d}{t}_{k} \) be a continuous differential \( k \) -form with compact support contained in the open set \( W \subset {\mathbb{R}}^{k} \). Then \[ {\int }_{\left( W,{\mathrm{{id}}}_{{\mathbb{R}}^{k}}\right) }\mathbf{\omega } = {\int }_{W}f\left( \m... | Indeed, every continuous function with compact support is Lebesgue integrable; hence the integrals exist, and \[ {\int }_{\left( W,{\operatorname{Id}}_{{\mathbb{R}}^{k}}\right) }\mathbf{\omega } = {\int }_{W}\mathbf{\omega }\left( {\operatorname{Id}\left( \mathbf{t}\right) }\right) \left( {\frac{\partial \operatorname{... | Yes |
Proposition 7.2.1. The vector \( {v}_{1} \times \cdots \times {v}_{n - 1} \) is orthogonal to each of the vectors \( {\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{n - 1} \) | Proof. It suffices to observe that\n\n\[ \left\langle {{\mathbf{v}}_{1} \times \cdots \times {\mathbf{v}}_{n - 1},{\mathbf{v}}_{i}}\right\rangle \]\n\n is the determinant of a matrix with two equal rows: both row 1 and row \( i + 1 \) consist of the coordinates of the vector \( {\mathbf{v}}_{i} \). | Yes |
Proposition 7.2.2. Let \( {\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{n - 1} \) be linearly independent vectors in \( {\mathbb{R}}^{n} \) . Then \( \left\{ {{\mathbf{v}}_{1} \times \cdots \times {\mathbf{v}}_{n - 1},{\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{n - 1}}\right\} \) is a positively oriented basis of \( {\mathbb{R}}^{... | Proof. If \( {\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{n - 1} \) are linearly independent vectors in \( {\mathbb{R}}^{n} \), then by Proposition 7.2.1, \( \left\{ {{\mathbf{v}}_{1} \times \cdots \times {\mathbf{v}}_{n - 1},{\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{n - 1}}\right\} \) is a basis of \( {\mathbb{R}}^{n} \) . The... | Yes |
Proposition 7.2.3. (1) Let \( {v}_{1},\ldots ,{v}_{n - 1} \) be vectors in \( {\mathbb{R}}^{n} \) . Then the jth coordinate of\n\n\( {\mathbf{v}}_{1} \times \cdots \times {\mathbf{v}}_{n - 1} \) is given by\n\n\[ \n{\left( -1\right) }^{j - 1}\mathrm{\;d}{x}_{1} \land \cdots \land \widehat{\mathrm{d}{x}_{j}} \land \cdot... | Proof. From Definition 6.1.2, we have that\n\n\[ \n\mathrm{d}{x}_{1} \land \cdots \land \widehat{\mathrm{d}{x}_{j}} \land \cdots \land \mathrm{d}{x}_{n}\left( {{\mathbf{v}}_{1},\ldots ,{\mathbf{v}}_{n - 1}}\right) \n\]\n\nis the determinant of the matrix obtained by deleting the first row and the \( j \) th column in t... | No |
Proposition 7.3.1. Definition 7.3.1 is independent of the chosen atlas. | Proof. Let \( {\left\{ \left( {A}_{j},{\mathbf{\varphi }}_{j}\right) \right\} }_{j = 1}^{m} \) and \( {\left\{ \left( {U}_{i},{\mathbf{\psi }}_{i}\right) \right\} }_{i = 1}^{s} \) be two atlases of \( M \), each one compatible with its orientation, and let \( \left\{ {Y}_{j}\right\} \) and \( \left\{ {X}_{i}\right\} \)... | Yes |
Theorem 7.3.1. Suppose that \( M \) is a compact and oriented regular \( k \) -surface and \( \omega : M \rightarrow {\Lambda }^{k}\left( {\mathbb{R}}^{n}\right) \) is a continuous differential \( k \) -form. Then the integral \( {\int }_{M}\omega \) exists. | Proof. For every \( \mathbf{x} \in M \), one can find a coordinate system \( \left( {{A}_{x},{\mathbf{\varphi }}_{\mathbf{x}}}\right) \) of \( M \), compatible with the orientation, and an open set \( {V}_{x} \) in \( {\mathbb{R}}^{k} \), contained in a compact subset of \( {A}_{x} \), \[ {V}_{x} \subset \overline{{V}_... | Yes |
Let \( M \) be an oriented regular surface contained in the open set \( U \subset {\mathbb{R}}^{3},\mathbf{F} = \left( {{f}_{1},{f}_{2},{f}_{3}}\right) \) a continuous vector field in \( U \), and\n\n\[ \omega \mathrel{\text{:=}} {f}_{1} \cdot \mathrm{d}y \land \mathrm{d}z + {f}_{2} \cdot \mathrm{d}z \land \mathrm{d}x ... | We also use the notation\n\n\[ {\int }_{M}\omega = {\int }_{M}\langle \mathbf{F},\mathbf{N}\rangle \mathrm{d}S = {\int }_{M}\mathbf{F} \cdot \mathrm{d}\mathbf{S} \]\n\nHence, if \( M = \varphi \left( A\right) \), where \( \left( {A,\varphi }\right) \) is a coordinate system of \( M \) compatible with the orientation\n\... | Yes |
Proposition 7.3.2. Let \( M \) be an oriented regular surface contained in the open set \( U \subset {\mathbb{R}}^{3} \) and let \( \mathbf{F} = \mathbf{N} \) be the continuous vector field of unit normal vectors defined by the orientation of \( M \) . If \( M \) admits a finite atlas, then\n\n\[ \operatorname{area}M =... | Proof. It suffices to show that if \( \left( {A,\varphi }\right) \) is a coordinate system of \( M \) compatible with the orientation and \( \Delta \subset A \) is a measurable subset, then\n\n\[ \operatorname{area}\left( {\Delta ,\mathbf{\varphi }}\right) = {\int }_{\mathbf{\varphi }\left( \Delta \right) }\langle \mat... | Yes |
Lemma 7.3.1. Let \( M \) be an (orientable) regular curve in \( {\mathbb{R}}^{n} \) . Then there is a continuous function\n\n\[ \mathbf{T} : M \rightarrow {\mathbb{R}}^{n} \]\n\nsuch that for each \( \mathbf{x} \in M,\mathbf{T}\left( \mathbf{x}\right) \) is a unit tangent vector to \( M \) at \( \mathbf{x} \) . Moreove... | Proof. For every \( \mathbf{x} \in M \) let \( \left( {\left( {a, b}\right) ,\mathbf{\gamma }}\right) \) be a coordinate system of \( M \), of class \( {C}^{1} \) , compatible with the orientation,\n\n\[ \mathbf{\gamma } : \left( {a, b}\right) \rightarrow {\mathbb{R}}^{n} \]\n\nsuch that \( \mathbf{x} = \mathbf{\gamma ... | Yes |
Example 7.3.3. Let \( M \) be a regular curve in \( {\mathbb{R}}^{n} \) of class \( {C}^{1} \) and \( \mathbf{F} = \left( {{f}_{1},\ldots ,{f}_{n}}\right) \) a continuous vector field in \( M \) . We consider its associated differential form of degree 1\n\n\[ \omega = \mathop{\sum }\limits_{{j = 1}}^{n}{f}_{j}\mathrm{\... | From now on, we may use any of the following expressions to represent the integral of the vector field \( \mathbf{F} \) over the regular curve \( M \) :\n\n\[ {\int }_{M}\mathbf{F} \cdot \mathrm{d}\mathbf{s} \mathrel{\text{:=}} {\int }_{M}\langle \mathbf{F},\mathbf{T}\rangle \mathrm{d}s \mathrel{\text{:=}} {\int }_{M}\... | Yes |
Lemma 8.1.1. Let \( a > 0 \) and suppose the function \( f : ( - a,0\rbrack \rightarrow \mathbb{R} \) has derivative of order \( p \) at each \( s \in \left( {-a,0}\right) \) and left derivative of order \( p,1 \leq p < \infty \), at \( s = 0 \) . Then there exist numbers \( {\alpha }_{0},{\alpha }_{1},\ldots ,{\alpha ... | Proof. We can select scalars \( {\alpha }_{0},\ldots ,{\alpha }_{p} \) such that\n\n\[ {\alpha }_{0} + {\alpha }_{1} + \cdots + {\alpha }_{p} = 1 \]\n\nand\n\n\[ \mathop{\sum }\limits_{{l = 1}}^{p}{l}^{m}{\alpha }_{l} = {\left( -1\right) }^{m} \]\n\nfor every \( m = 1,2,\ldots, p \) . Indeed, this linear system has a u... | Yes |
Lemma 8.2.1. Let \( M \) be a regular \( k \) -surface with boundary of class \( {C}^{p} \). Let \( {\mathbf{x}}_{0} \in M \) and \( \left( {\mathbb{B},\mathbf{\psi }}\right) \) a coordinate system of \( M \) such that\n\n\[ \n{\mathbf{\psi }}^{-1}\left( {\mathbf{x}}_{0}\right) = \left( {0,{s}_{2}^{0},\ldots ,{s}_{k}^{... | Proof. Let\n\n\[ \nD \mathrel{\text{:=}} \varphi \left( \mathbb{A}\right) \cap \psi \left( \mathbb{B}\right) \n\]\n\nand put\n\n\[ \n{\mathbf{s}}_{0} \mathrel{\text{:=}} {\mathbf{\psi }}^{-1}\left( {\mathbf{x}}_{0}\right) ,\;{\mathbf{t}}_{0} \mathrel{\text{:=}} {\mathbf{\varphi }}^{-1}\left( {\mathbf{x}}_{0}\right) = \... | Yes |
Let \( M \) be a compact regular \( n \) -surface with boundary in \( {\mathbb{R}}^{n} \). Then \( M \smallsetminus \partial M \) is an open set in \( {\mathbb{R}}^{n} \) and \( \partial M \) coincides with the topological boundary of \( M \). | We first show that \( M \smallsetminus \partial M \) is an open set in \( {\mathbb{R}}^{n} \). For every \( {\mathbf{x}}_{0} \in M \smallsetminus \partial M \) let \( \left( {\mathbb{A},\mathbf{\varphi }}\right) \) be a coordinate system of \( M \), \[ \mathbf{\varphi } : \mathbb{A} \subset {\mathbb{H}}^{n} \rightarrow... | Yes |
Proposition 8.2.2. Let \( M \) be a regular \( k \) -surface of class \( {C}^{p} \) in \( {\mathbb{R}}^{n} \) . Then \( M \) is a regular \( k \) -surface with boundary of class \( {C}^{p} \) and \( \partial M = \varnothing \) . | Proof. According to Definition 3.1.1, for each \( {\mathbf{x}}_{0} \in M \) there is coordinate system \( \left( {A,\varphi }\right) \) of \( M \) such that \( {\mathbf{x}}_{0} \in \varphi \left( A\right) \) . Take \( {\mathbf{t}}_{0} \in A \) and \( r > 0 \) with \( B\left( {{\mathbf{t}}_{0}, r}\right) \subset A \) an... | Yes |
Lemma 8.2.2. For each half-space \( \mathbb{S} \) in \( {\mathbb{R}}^{k} \) there exists an affine mapping\n\n\[ \mathbf{R} : {\mathbb{R}}^{k} \rightarrow {\mathbb{R}}^{k} \]\n\nsuch that \( \mathbf{R}\left( {\mathbb{H}}^{k}\right) = \mathbb{S},\mathbf{R}\left( {\partial {\mathbb{H}}^{k}}\right) = \partial \mathbb{S} \... | Proof. Given the half-space \( \mathbb{S} = \left\{ {t \in {\mathbb{R}}^{k} : \pi \left( t\right) \leq a}\right\} \), we consider an orthonormal basis \( \left\{ {{\mathbf{v}}_{1},{\mathbf{v}}_{2},\ldots ,{\mathbf{v}}_{k}}\right\} \) of \( {\mathbb{R}}^{k} \) with the properties that \( \pi \left( {\mathbf{v}}_{1}\righ... | Yes |
Proposition 8.2.3. Let \( M = {M}_{1} \cup {M}_{2} \) satisfy \( {M}_{1} \cap {M}_{2} = \varnothing \), where \( {M}_{1} \) is a regular \( k \) -surface of class \( {C}^{p} \) and for each point \( {\mathbf{x}}_{0} \in {M}_{2} \) there exist a half-space \( \mathbb{S} \) in \( {\mathbb{R}}^{k} \) and a mapping\n\n\[ \... | Proof. For every \( {\mathbf{x}}_{0} \in {M}_{1} \), we can argue as in Proposition 8.2.2 in order to find a coordinate system \( \left( {\mathbb{B},\mathbf{\psi }}\right) \) such that in accordance with Definition 8.2.1, \( {\mathbf{x}}_{0} \in \mathbf{\psi }\left( \mathbb{B}\right) \) and the first coordinate of \( {... | Yes |
Corollary 8.3.1. Let \( S \) be a regular \( n \) -surface contained in an open set \( U \) in \( {\mathbb{R}}^{n} \) and let \( f : U \subset {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) be a function of class \( {C}^{p} \) such that \( \nabla f\left( x\right) \neq 0 \) for all \( x \in S \cap {f}^{-1}\left( 0\right) \)... | Proof. The hypothesis implies that the interior of \( M \) is nonempty and hence is a regular \( n \) -surface. According to Proposition 8.2.3, we need to check the conditions in Definition 8.2.1 of regular surface with boundary at each point of the topological boundary of \( M \) . Fix \( {\mathbf{x}}_{0} \in {\partia... | Yes |
Proposition 8.3.1. Let \( M \) be a regular \( n \) -surface with boundary of class \( {C}^{p} \) and \( {\mathbf{x}}_{0} \) a point in \( \partial M \) . We assume that there exist \( r > 0 \) and a mapping\n\n\[ \n\Phi : B\left( {{\mathbf{x}}_{0}, r}\right) \subset {\mathbb{R}}^{n} \rightarrow \mathbb{R}\n\]\n\nof cl... | Proof. From Proposition 3.4.1 we already know that \( \mathbf{N} \) is a normal vector to \( {T}_{{x}_{0}}\left( {\partial M}\right) \) . Moreover,\n\n\[ \n0 < \langle \mathbf{N},\mathbf{N}\rangle = {D}_{\mathbf{N}}\Phi \left( {\mathbf{x}}_{0}\right) = \mathop{\lim }\limits_{{t \rightarrow 0}}\frac{\Phi \left( {{\mathb... | Yes |
Let\n\n\[ M \mathrel{\text{:=}} \left\{ {\left( {x, y, z}\right) \in {\mathbb{R}}^{3} : {x}^{2} + {y}^{2} + {z}^{2} = 1, z \geq 0}\right\} \]\n\nbe the upper hemisphere of the unit sphere. Then \( M \) is a regular 2-surface with boundary in \( {\mathbb{R}}^{3} \), and the boundary is the unit circle in the \( {xy} \) ... | To check this, consider the mappings given by\n\n\[ f\left( {x, y, z}\right) = - z,\;\Phi \left( {x, y, z}\right) = {x}^{2} + {y}^{2} + {z}^{2} - 1. \]\nThen\n\n\[ \mathbf{\nabla }\Phi \left( {x, y, z}\right) = \left( {{2x},{2y},{2z}}\right) \neq \left( {0,0,0}\right) \]\n\nin \( {\mathbb{R}}^{3} \smallsetminus \{ \lef... | Yes |
The truncated paraboloid\n\n\\[ \nM \\mathrel{\\text{:=}} \\left\\{ {\\left( {x, y, z}\\right) \\in {\\mathbb{R}}^{3} : z = {x}^{2} + {y}^{2}, z \\leq 1}\\right\\} \n\\]\n\nis a regular 2-surface with boundary whose boundary is the unit circle in the plane \\( z = 1 \\)\n\n\\[ \n\\partial M = \\left\\{ {\\left( {x, y, ... | Again we apply Criterion 8.3.1, this time with the mappings\n\n\\[ \n\\Phi \\left( {x, y, z}\\right) = {x}^{2} + {y}^{2} - z, f\\left( {x, y, z}\\right) = z - 1.\n\\]\n\nEasily we see that\n\n\\[ \n\\mathbf{\\nabla }\\Phi \\left( {x, y, z}\\right) = \\left( {{2x},{2y}, - 1}\\right) \\neq \\left( {0,0,0}\\right) ,\n\\]\... | Yes |
Example 8.3.3. Already we have seen in Example 8.2.4 that the closed unit ball\n\n\\[ \nM \\mathrel{\\text{:=}} \\left\\{ {\\left( {x, y, z}\\right) \\in {\\mathbb{R}}^{3} : {x}^{2} + {y}^{2} + {z}^{2} \\leq 1}\\right\\} \n\\]\n\nis a regular 3-surface with boundary in \\( {\\mathbb{R}}^{3} \\) and that the boundary is... | This conclusion can alternatively be arrived by application of Theorem 8.3.2 in the limiting case of \\( k = n = 3 \\) and \\( S = {\\mathbb{R}}^{3} \\) upon considering the function\n\n\\[ \nf : {\\mathbb{R}}^{3} \\rightarrow \\mathbb{R};f\\left( {x, y, z}\\right) = {x}^{2} + {y}^{2} + {z}^{2} - 1. \n\\]\n\nThe normal... | Yes |
The annulus \( M \mathrel{\text{:=}} \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} : 1 < {x}^{2} + {y}^{2} \leq 4}\right\} \) is a regular 2- surface with boundary. The boundary is the circle with radius 2 centered at the origin. | Again we apply Theorem 8.3.2 in a limiting case. Take\n\n\[ S \mathrel{\text{:=}} \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} : 1 < {x}^{2} + {y}^{2}}\right\} \]\n\nwhich is a regular 2-surface in \( {\mathbb{R}}^{2} \) because it is an open set. Now define\n\n\[ f : {\mathbb{R}}^{2} \rightarrow \mathbb{R},\;f\l... | Yes |
We consider the half-space\n\n\[ \mathbb{S} \mathrel{\text{:=}} \left\{ {\mathbf{t} \in {\mathbb{R}}^{n} : \pi \left( \mathbf{t}\right) \leq a}\right\} \]\n\n(see Definition 8.2.3). Suppose this is oriented so that \( \left( {\mathbb{S} \smallsetminus \partial \mathbb{S},\mathrm{{id}}}\right) \) is a coordinate system ... | We observe that the tangent space to \( \partial \mathbb{S} \) at a point is exactly the \( \left( {n - 1}\right) \) - dimensional vector subspace \( \operatorname{Ker}\pi \) (see, for instance, Proposition 3.4.1). Put \( {v}_{1} \mathrel{\text{:=}} N \) and consider an orthonormal basis \( \left\{ {{\mathbf{v}}_{2},{\... | Yes |
Let \( 0 < \varepsilon < 1 \) and\n\n\[ M \mathrel{\text{:=}} \left\{ {\left( {x, y, z}\right) \in {\mathbb{R}}^{3} : {x}^{2} + {y}^{2} + {z}^{2} = 1, z \geq \varepsilon }\right\} \]\n\nbe given. We wish to analyze the orientation induced on \( M \smallsetminus \partial M \) and on \( \partial M \) by the coordinate sy... | Recall that \( \mathbb{B} \mathrel{\text{:=}} ( - r,0\rbrack \times \left( {{\theta }_{0} - \pi ,{\theta }_{0} + \pi }\right) \), where \( {r}^{2} = 1 - {\varepsilon }^{2} \), and\n\n\[ \psi : \mathbb{B} \subset {\mathbb{H}}^{2} \rightarrow M \]\n\nis given by\n\n\[ \mathbf{\psi }\left( {s,\theta }\right) \mathrel{\tex... | Yes |
Proposition 9.2.1. Let \( {x}_{0} \in {\mathbb{R}}^{n} \) and \( r > 0 \) be given. There is a function \( h : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) of class \( {C}^{\infty } \) such that \( h\left( \mathbf{x}\right) > 0 \) if \( \mathbf{x} \in B\left( {{\mathbf{x}}_{0}, r}\right) \) and \( h\left( \mathbf{x}\righ... | Proof. Put \( h\left( \mathbf{x}\right) \mathrel{\text{:=}} {g}_{0}\left( \frac{{\begin{Vmatrix}\mathbf{x} - {\mathbf{x}}_{0}\end{Vmatrix}}^{2}}{{r}^{2}}\right) \) . Since\n\n\[ \varphi \left( \mathbf{x}\right) \mathrel{\text{:=}} {\begin{Vmatrix}\mathbf{x} - {\mathbf{x}}_{0}\end{Vmatrix}}^{2} = \mathop{\sum }\limits_{... | Yes |
Lemma 9.2.1. (a) Let \( V \subset {\mathbb{R}}^{n} \) be an open set, \( K \subset V \) a closed subset, and \( h \in {C}^{p}\left( V\right) \) a function of class \( {C}^{p} \) on \( V,1 \leq p \leq \infty \), such that \( h\left( \mathbf{x}\right) = 0 \) if \( x \in V \smallsetminus K \) . Then\n\n\[ f\left( \mathbf{... | Proof. (a) Since \( h\left( \mathbf{x}\right) = 0 \) on the open set \( V \cap \left( {{\mathbb{R}}^{n} \smallsetminus K}\right) \) and \( V \cup \left( {{\mathbb{R}}^{n} \smallsetminus K}\right) = {\mathbb{R}}^{n} \), we see that \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) is well defined. The function is of cla... | Yes |
Corollary 9.2.1. Let \( M \) be a compact regular \( k \) -surface with boundary and\n\n\[ \omega : U \rightarrow {\Lambda }^{r}\left( {\mathbb{R}}^{n}\right) \]\n\na differential form of degree \( r \) and class \( {C}^{1} \) on an open neighborhood \( U \) of \( M \) . Then there is an open neighborhood \( V \) of \(... | Proof. By compactness, one can find a finite family \( {\left\{ \left( {\mathbb{A}}_{j},{\mathbf{\varphi }}_{j}\right) \right\} }_{j = 1}^{m} \) of coordinate systems and bounded open sets \( {\mathbb{B}}_{j} \) (for the relative topology of \( {\mathbb{H}}^{k} \) ) such that\n\n\[ {\mathbb{B}}_{j} \subset \overline{{\... | Yes |
Lemma 9.3.1. Let \( \omega \) be a differential form of degree \( k - 1 \) and class \( {C}^{1} \) on an open neighborhood of a regular \( k \) -surface with boundary \( M \) in \( {\mathbb{R}}^{n} \) . Let \( \left( {\mathbb{A},\mathbf{\varphi }}\right) \) be a coordinate system of \( M \) . Writing the differential f... | Proof. Note that \( \widehat{\varphi } = \varphi \circ \mathbf{J} \), where \( \mathbf{J} \) is the inclusion\n\n\[ \n\mathbf{J} : \widehat{\mathbb{A}} \rightarrow \mathbb{A},\;\mathbf{J}\left( \mathbf{s}\right) \mathrel{\text{:=}} \left( {0,\mathbf{s}}\right) . \n\]\n\nMaking the change of variables\n\n\[ \n{t}_{1} = ... | Yes |
Lemma 9.3.2. The exterior differential of the differential form \( \mathbf{v} \) of degree \( \left( {k - 1}\right) \) and class \( {C}^{1} \) defined on an open subset of \( {\mathbb{R}}^{k} \) by\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{k}{f}_{j}\left( {{t}_{1},\ldots ,{t}_{k}}\right) \cdot \mathrm{d}{t}_{1} \land \cdo... | Proof. According to Definition 6.3.1,\n\n\[ \mathrm{d}\mathbf{v}\left( {{t}_{1},\ldots ,{t}_{k}}\right) = \mathop{\sum }\limits_{{j = 1}}^{k}\mathrm{\;d}{f}_{j}\left( {{t}_{1},\ldots ,{t}_{k}}\right) \land \mathrm{d}{t}_{1} \land \cdots \land \widehat{\mathrm{d}{t}_{j}} \land \cdots \land \mathrm{d}{t}_{k}. \]\n\nMoreo... | Yes |
Theorem 9.4.4 (Classical Stokes’s theorem). Let \( M \) be a compact and orientable regular 2-surface with boundary in \( {\mathbb{R}}^{3} \) of class \( {C}^{2} \) and let \( \mathbf{F} = \left( {{f}_{1},{f}_{2},{f}_{3}}\right) \) be a vector field of class \( {C}^{1} \) on an open neighborhood of \( M \) . Then\n\n\[... | Proof. We consider the degree-1 differential form\n\n\[ \omega = {f}_{1}\mathrm{\;d}x + {f}_{2}\mathrm{\;d}y + {f}_{3}\mathrm{\;d}z. \]\n\nAs we saw in Example 6.3.2,\n\n\[ \mathrm{d}\omega = \left( {\frac{\partial {f}_{3}}{\partial y} - \frac{\partial {f}_{2}}{\partial z}}\right) \mathrm{d}y \land \mathrm{d}z + \left(... | Yes |
Corollary 9.4.1. Let \( \\mathbf{F} = \\left( {{f}_{1},{f}_{2},{f}_{3}}\\right) \) be a vector field of class \( {C}^{1} \) on an open set \( U \\subset \) \( {\\mathbb{R}}^{3} \) . Given a point \( \\mathbf{a} \\in U \) and a unit vector \( \\mathbf{N} \\in {\\mathbb{R}}^{3} \), we denote by \( {C}_{\\varepsilon } \) ... | Proof. For every \( \\varepsilon > 0 \) we have\n\n\[ \n\\langle \\left( {\\mathbf{\\nabla } \\times \\mathbf{F}}\\right) \\left( \\mathbf{a}\\right) ,\\mathbf{N}\\rangle = \\frac{1}{\\pi {\\varepsilon }^{2}}{\\int }_{{D}_{\\varepsilon }}\\langle \\left( {\\mathbf{\\nabla } \\times \\mathbf{F}}\\right) \\left( \\mathbf... | Yes |
Theorem 9.4.5 (Green's theorem). Let \( M \) be a compact and orientable regular 2- surface with boundary of class \( {C}^{2} \) in \( {\mathbb{R}}^{2} \) and let \( \omega = P\mathrm{\;d}x + Q\mathrm{\;d}y \) be a differential form of degree 1 and class \( {C}^{1} \) on an open neighborhood of \( M \) . Then\n\n\[ \n{... | The proof is a simple consequence of the fact that\n\n\[ \n\mathrm{d}\omega = \left( {\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}}\right) \mathrm{d}x \land \mathrm{d}y.\n\] | No |
Proposition 9.5.2. Let \( M \) be an oriented regular \( k \) -surface with boundary. Suppose there is a mapping \( f \) of class \( {C}^{2} \) on an open neighborhood \( W \) of the closed unit cube\n\n\[ f : W \rightarrow {\mathbb{R}}^{n} \]\n\nsuch that \( f : U \rightarrow M \) is a bijection and \( {f}^{\prime } \... | Proof. Again (and throughout this section) we denote by \( I = {\left( 0,1\right) }^{k} \) the open unit cube. According to Criterion 8.6.1, \( \mathbf{f}\left( I\right) = M \smallsetminus \partial M \) and \( \left( {I,\mathbf{f}}\right) \) is a coordinate system of \( M \smallsetminus \partial M \) . First suppose th... | Yes |
Lemma 9.5.1. \( \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}{\iiint }_{{K}_{\varepsilon }}\operatorname{Div}\mathbf{F}\left( {x, y, z}\right) \mathrm{d}\left( {x, y, z}\right) = {\iiint }_{K}\operatorname{Div}\mathbf{F}\left( {x, y, z}\right) \mathrm{d}\left( {x, y, z}\right) \) . | Proof. Since \( F \) is continuous on the compact set \( K \), there exists \( C > 0 \) a positive constant such that \( \left| {\operatorname{Div}\mathbf{F}\left( {x, y, z}\right) }\right| \leq C \) for every \( \left( {x, y, z}\right) \in K \) . Since\n\n\[ K \smallsetminus {K}_{\varepsilon } = \mathbf{f}\left( \bar{... | Yes |
Lemma 9.5.2. Let \( C \) be a face of the cube without edges, \( U \), such that \( f\left( C\right) \) is a regular surface and assume that \( f\left( C\right) \) is oriented so that if \( \left( {V,\mathbf{\psi }}\right) \) is a coordinate system of \( C \) compatible with the orientation of the cube, then \( \left( ... | Proof. Suppose that \( C \) is the face of the cube defined by \( {x}_{1} = 1 \) . Then \( {J}_{\varepsilon } \) is the face of \( {I}_{\varepsilon } \) defined by \( {x}_{1} = 1 - \varepsilon \) . Since the mapping\n\n\[ \left( {u, v}\right) \rightarrow \left( {1, u, v}\right) \]\n\ndefines a positively oriented coord... | Yes |
Lemma 9.5.3. Suppose that \( \frac{\partial f}{\partial y} \times \frac{\partial f}{\partial z} = 0 \) on the face of the cube defined by \( {x}_{1} = 0 \) . Denote by \( {J}_{\varepsilon } \) the face of \( {I}_{\varepsilon } \) defined by \( {x}_{1} = \varepsilon \) . Then\n\n\[ \mathop{\lim }\limits_{{\varepsilon \r... | Proof. There is a constant \( C > 0 \) such that\n\n\[ \parallel \mathbf{F}\left( {x, y, z}\right) \parallel \leq C \]\n\nwhenever \( \left( {x, y, z}\right) \in K \) . Moreover, if we let\n\n\[ {M}_{\varepsilon } \mathrel{\text{:=}} \sup \left\{ {\begin{Vmatrix}{\left( {\frac{\partial f}{\partial y} \times \frac{\part... | Yes |
Lemma 9.5.4. Let \( f \) be periodic with period 1 with respect to a coordinate and let \( {J}_{\varepsilon ,0},{J}_{\varepsilon ,1} \) be the two parallel faces of \( {I}_{\varepsilon } \) in which that coordinate is constant. Then\n\n\[ \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}\left( {{\int }_{f\left( {J}_{... | Proof. Suppose that\n\n\[ \mathbf{f}\left( {x,0, z}\right) = \mathbf{f}\left( {x,1, z}\right) \]\n\nwhenever \( 0 \leq x, z \leq 1 \) . Then \( {J}_{\varepsilon ,0} \) is the face of \( {I}_{\varepsilon } \) defined by \( y = \varepsilon \) and \( {J}_{\varepsilon ,1} \) is the face of \( {I}_{\varepsilon } \) defined ... | Yes |
Consider the full cylindrical pipe (Fig. 9.4)\n\n\\[ \nK \\mathrel{\\text{:=}} \\left\\{ {\\left( {x, y, z}\\right) \\in {\\mathbb{R}}^{3} : {x}^{2} + {y}^{2} \\leq 1;0 \\leq z \\leq 1}\\right\\} .\n\\]\n\nTake\n\n\\[ \n\\mathbf{f} : {\\mathbb{R}}^{3} \\rightarrow {\\mathbb{R}}^{3},\\;\\mathbf{f}\\left( {r, s, t}\\righ... | The boundary of \\( {K}_{\\varepsilon } \\) consists of six connected components, namely, the images under \\( f \\) of the faces (without edges) of \\( {I}_{\\varepsilon } \\) . We analyze each one of these components.\n\nImage of the faces parallel to the plane \\( \\{ 0\\} \\times {\\mathbb{R}}^{2} \\) . First we fi... | Yes |
Consider the half-ball (Fig. 9.5)\n\n\\[ \nK \\mathrel{\\text{:=}} \\left\\{ {\\left( {x, y, z}\\right) \\in {\\mathbb{R}}^{3} : {x}^{2} + {y}^{2} + {z}^{2} \\leq 1;z \\geq 0}\\right\\} \n\\]\n\nand take\n\n\\[ \n\\mathbf{f}\\left( {r, s, t}\\right) \\mathrel{\\text{:=}} \\left( {r\\sin \\left( {\\frac{\\pi }{2}s}\\rig... | Proceeding as in Example 9.5.1, we see that whenever\n\n\\[ \n\\omega \\mathrel{\\text{:=}} {F}_{1} \\cdot \\mathrm{d}y \\land \\mathrm{d}z + {F}_{2} \\cdot \\mathrm{d}z \\land \\mathrm{d}x + {F}_{3} \\cdot \\mathrm{d}x \\land \\mathrm{d}y \n\\]\n\nis a differential form of degree 2 and class \\( {C}^{1} \\) on an open... | Yes |
Consider the truncated cone\n\n\[ K \mathrel{\text{:=}} \left\{ {\left( {x, y, z}\right) \in {\mathbb{R}}^{3} : {x}^{2} + {y}^{2} = {z}^{2};0 \leq z \leq 1}\right\} \]\nand the mapping \( \mathbf{f} : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2} \) defined by\n\n\[ \mathbf{f}\left( {t, r}\right) \mathrel{\text{:=}} \l... | To do this, denote by \( {I}_{\varepsilon } \) the square \( {\left\lbrack \varepsilon ,1 - \varepsilon \right\rbrack }^{2} \) less its vertices and take\n\n\[ {M}_{\varepsilon } = f\left( {I}_{\varepsilon }\right) \]\nwhich is a regular surface with boundary in \( {\mathbb{R}}^{3} \) for which the general Stokes’s the... | Yes |
Proposition 9.6.1. The divergence of the vector field \( \mathbf{F} \) vanishes. | Proof. We first evaluate\n\n\[ \frac{\partial {F}_{1}}{\partial x} = - {GMm}\left( {\frac{1}{\parallel \mathbf{r}{\parallel }^{3}} - \frac{3{\left( x - {x}_{0}\right) }^{2}}{\parallel \mathbf{r}{\parallel }^{5}}}\right) ,\] \n\n\[ \frac{\partial {F}_{2}}{\partial y} = - {GMm}\left( {\frac{1}{\parallel \mathbf{r}{\paral... | Yes |
Lemma 9.6.1. Let \( B \) be the closed ball centered at a point \( \mathbf{P} \) and with radius \( R > 0 \) . The flux of the vector field \( \mathbf{F} \) across \( \partial B \) is\n\n\[ - {4\pi GMm}\text{.} \] | Proof. The unit normal vector to \( \partial B \) at a point \( \left( {x, y, z}\right) \in \partial B \), pointing to the exterior of \( B \), is given by\n\n\[ N = \frac{r}{\parallel r\parallel }.\]\n\nHence, for\n\n\[ \mathbf{F} = - \frac{GMm}{\parallel \mathbf{r}{\parallel }^{3}}\mathbf{r} \]\n\nit turns out that\n... | Yes |
Theorem 9.6.1 (Gauss’s law). Let \( U \) be a regular 3-surface with boundary of class \( {C}^{2} \) in \( {\mathbb{R}}^{3} \) and assume that\n\n\[ S \mathrel{\text{:=}} \partial U \]\n\nis a compact regular surface that is oriented according to the normal vector pointing to the exterior of \( U \) .\n\n(1) If \( \mat... | Proof. We first suppose that \( \mathbf{P} \notin U \) . In this case, \( \mathbf{F} \) is a vector field of class \( {C}^{1} \) on a neighborhood of \( U \), and by the divergence theorem, the flux of \( \mathbf{F} \) across \( S \) coincides with\n\n\[ {\iiint }_{U}\operatorname{Div}\mathbf{F}\left( {x, y, z}\right) ... | Yes |
Let’s graph the curve given by the polar equation \( r = 6\cos \theta \) (Figure 1.88). We can begin to get a feeling for the graph by compiling values, as in the adjacent tabulation. | The polar equation \( r = 6\cos \theta \) that defines the curve requires a little ingenuity to convert to the corresponding Cartesian equation. The trick is to multiply both sides of the equation by \( r \). Doing so, we obtain\n\n\[ {r}^{2} = {6r}\cos \theta \]\n\nNow (1) and (2) immediately give\n\n\[ {x}^{2} + {y}^... | Yes |
Lemma 2 With \( h \) and \( {S}_{b} \) as in Lemma 1,\n\n\[ \mathop{\lim }\limits_{{b \rightarrow 0}}{\oiint }_{{S}_{b}}h\left( \mathbf{x}\right) {\nabla }_{\mathbf{x}}\left( \frac{1}{\parallel \mathbf{r} - \mathbf{x}\parallel }\right) \cdot d\mathbf{S} = - {4\pi h}\left( \mathbf{r}\right) . \] | PROOF Let \( \mathbf{n} = \left( {\mathbf{x} - \mathbf{r}}\right) /\parallel \mathbf{r} - \mathbf{x}\parallel \), the normalization of \( \mathbf{x} - \mathbf{r} \) . Straightforward calculations yield\n\n\[ {\nabla }_{\mathbf{x}}\left( \frac{1}{\parallel \mathbf{r} - \mathbf{x}\parallel }\right) = - \frac{\mathbf{x} -... | Yes |
\[ \left( {\forall n\text{ integer,}n \geq 2}\right) \left( {\exists p\text{ prime }}\right) n/p\text{ is an integer }\;\text{ is true,}\] | \[ \left( {\exists p\text{ prime}}\right) \left( {\forall n\text{ integer,}n \geq 2}\right) n/p\text{ is an integer is false. }\] | No |
Example 0.2.2 (Order of quantifiers in defining continuity). In the definitions of continuity and uniform continuity, the order of quantifiers really counts. A function \( f \) is continuous if for all \( x \), and for all \( \epsilon > 0 \), there exists \( \delta > 0 \) such that for all \( y \), if \( \left| {x - y}... | \[ \left( {\forall x}\right) \left( {\forall \epsilon > 0}\right) \left( {\exists \delta > 0}\right) \left( {\forall y}\right) \left| {x - y}\right| < \delta \text{ implies }\left| {f\left( x\right) - f\left( y\right) }\right| < \epsilon ,\] which can also be written \[ \left( {\forall x}\right) \left( {\forall \epsilo... | Yes |
What is the natural domain of the formula\n\n\[ f\left( x\right) = \sqrt{{x}^{2} - {3x} + 2}? \] | This can be evaluated only if \( {x}^{2} - {3x} + 2 \geq 0 \), which happens if \( x \leq 1 \) or \( x \geq 2 \) . So the natural domain is \( \left( {-\infty ,1\rbrack \cup \lbrack 2,\infty }\right) \) . | Yes |
Example 0.4.10 (Inverse image). Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be the (noninvert-ible) mapping \( f\left( x\right) = {x}^{2} \) . The inverse image of \( \{ - 1,4,9,{16}\} \) under \( f \) is \( \{ - 4, - 3, - 2,2,3,4\} \) : | \[ {f}^{-1}\left( {\{ - 1,4,9,{16}\} }\right) = \{ - 4, - 3, - 2,2,3,4\} .\;\bigtriangleup \] | Yes |
Example 0.4.14 (Composition of two functions). If \( f\left( x\right) = x - 1 \) , and \( g\left( x\right) = {x}^{2} \), then | \n\[
\left( {f \circ g}\right) \left( x\right) = f\left( {g\left( x\right) }\right) = {x}^{2} - 1.\;\bigtriangleup
\] | Yes |
Proposition 0.4.15 (Composition is associative).\n\n\[ \left( {f \circ g}\right) \circ h = f \circ \left( {g \circ h}\right) . \] | Proof. This is simply the computation\n\n\[ \left( {\left( {f \circ g}\right) \circ h}\right) \left( x\right) = \left( {f \circ g}\right) \left( {h\left( x\right) }\right) = f\left( {g\left( {h\left( x\right) }\right) }\right) \;\text{whereas} \]\n\n---\n\n\[ \left( {f \circ \left( {g \circ h}\right) }\right) \left( x\... | Yes |
Theorem 0.5.3 (The real numbers are complete). Every nonempty subset \( X \subset \mathbb{R} \) that has an upper bound has a least upper bound \( \sup X \) . Every nonempty subset \( X \subset \mathbb{R} \) that has a lower bound has a greatest lower bound \( \inf X \) . | Proof. We will construct successive decimals of \( \sup X \) . Suppose that \( x \in X \) is an element (which we know exists, since \( X \neq \phi \) ) and that \( a \) is an upper bound. We will assume that \( x > 0 \) (the case \( x \leq 0 \) is slightly different). If \( x = a \), we are done: the least upper bound... | Yes |
If \( \left| r\right| < 1 \), then \[ \mathop{\sum }\limits_{{n = 0}}^{\infty }a{r}^{n} = \frac{a}{1 - r} \] | Indeed, the following subtraction shows that \( {S}_{n}\left( {1 - r}\right) = a - a{r}^{n + 1} \) : \[ {S}_{n}\overset{\text{ def }}{ = }a + {ar} + a{r}^{2} + a{r}^{3} + \cdots + a{r}^{n} \] \[ {S}_{n}r = \;{ar} + a{r}^{2} + a{r}^{3} + \cdots + a{r}^{n} + a{r}^{n + 1} \] \[ {S}_{n}\left( {1 - r}\right) = a - a{r}^{n +... | Yes |
Theorem 0.5.7. A nondecreasing sequence \( n \mapsto {a}_{n} \) of real numbers converges if and only if it is bounded. | Proof. If a sequence \( n \mapsto {a}_{n} \) of real numbers converges, it is clearly bounded. If it is bounded, then (by Theorem 0.5.3) it has a least upper bound \( A \) . We claim that \( A \) is the limit. This means that for any \( \epsilon > 0 \) , there exists \( N \) such that if \( n > N \), then \( \left| {{a... | Yes |
Theorem 0.5.8 (Absolute convergence implies convergence). If the series of absolute values \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }\left| {a}_{n}\right| \;\text{ converges, then so does the series }\mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}. \] | Proof. The series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }\left( {{a}_{n} + \left| {a}_{n}\right| }\right) \) is a series of nonnegative numbers, so the partial sums \( {b}_{m} = \mathop{\sum }\limits_{{n = 1}}^{m}\left( {{a}_{n} + \left| {a}_{n}\right| }\right) \) are nondecreasing. They are also bounded: \[ {b}_{... | Yes |
Theorem 0.5.9 (Intermediate value theorem). If \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) is a continuous function and \( c \) is a number such that \( f\left( a\right) \leq c \) and \( f\left( b\right) \geq c \) , then there exists \( {x}_{0} \in \left\lbrack {a, b}\right\rbrack \) such that \( ... | Proof. Let \( X \) be the set of \( x \in \left\lbrack {a, b}\right\rbrack \) such that \( f\left( x\right) \leq c \) . Note that \( X \) is nonempty ( \( a \) is in it) and it has an upper bound, namely \( b \), so that it has a least upper bound, which we call \( {x}_{0} \) . We claim \( f\left( {x}_{0}\right) = c \)... | Yes |
Proposition 0.6.1. A mapping \( f : E \rightarrow \mathcal{P}\left( E\right) \) is never onto. | Proof. Let \( A \subset E \) be the set of \( x \) such that \( x \notin f\left( x\right) \) . We will show that \( A \) is not in the image of \( f \), and thus that \( f \) is not onto. Suppose \( A \) is in the image of \( f \) . Then \( A \) must be of the form \( f\left( y\right) \) for an appropriate \( y \in E \... | Yes |
Example 0.7.1 (Multiplying complex numbers). | \\left( {2 + i}\\right) \\left( {1 - {3i}}\\right) = \\left( {2 + 3}\\right) + i\\left( {1 - 6}\\right) = 5 - {5i};\\;{\\left( 1 + i\\right) }^{2} = {2i}.\\;\\bigtriangleup \\;{0.7.4} | Yes |
Proposition 0.7.7 (Roots of a complex number). Every complex number \( \;z = r\left( {\cos \theta + i\sin \theta }\right) \; \) with \( \;r \neq 0\; \) has \( \;n\; \) distinct complex \( \;n \) th roots, the numbers \[ {r}^{1/n}\left( {\cos \frac{\theta + {2k\pi }}{n} + i\sin \frac{\theta + {2k\pi }}{n}}\right) ,\;k =... | Proof. All that needs to be checked is that 1. \( {\left( {r}^{1/n}\right) }^{n} = r \), which is true by definition; 2. \[ \cos n\frac{\theta + {2k\pi }}{n} = \cos \theta \;\text{ and }\;\sin n\frac{\theta + {2k\pi }}{n} = \sin \theta \] which is true since \( n\frac{\theta + {2k\pi }}{n} = \theta + {2k\pi } \), and s... | Yes |
Example 1.1.6 (Subspaces of \( {\mathbb{R}}^{2} \) ). Let us first give some examples of subsets that are not subspaces. The unit circle \( {S}^{1} \subset {\mathbb{R}}^{2} \) of equation \( {x}^{2} + {y}^{2} = 1 \) is not a subspace of \( {\mathbb{R}}^{2} \). | It is not closed under addition: for instance, \( \left\lbrack \begin{array}{l} 1 \\ 0 \end{array}\right\rbrack \in {S}^{1} \) and \( \left\lbrack \begin{array}{l} 0 \\ 1 \end{array}\right\rbrack \in {S}^{1} \), but \( \left\lbrack \begin{array}{l} 1 \\ 0 \end{array}\right\rbrack + \left\lbrack \begin{array}{l} 0 \\ 1 ... | Yes |
Example 1.2.2 (Addition of matrices; multiplication by scalars). | \[ \left\lbrack \begin{array}{rr} 1 & 0 \\ 2 & - 1 \\ 4 & 2 \end{array}\right\rbrack + \left\lbrack \begin{array}{rr} 0 & - 3 \\ 1 & - 2 \\ 3 & 1 \end{array}\right\rbrack = \left\lbrack \begin{array}{rr} 1 & - 3 \\ 3 & - 3 \\ 7 & 3 \end{array}\right\rbrack \text{ and }2\left\lbrack \begin{array}{rr} 1 & 4 \\ - 2 & 3 \e... | Yes |
Example 1.2.3 (Matrix multiplication). The first entry of the first row of \( {AB} \) is obtained by multiplying, one by one, the entries of the first row of \( A \) by those of the first column of \( B \), and adding these products together: in equation 1.2.1, \( \left( {2 \times 1}\right) + \left( {-1 \times 3}\right... | \[ \begin{array}{r} \left\lbrack \begin{array}{l} B \end{array}\right\rbrack \\ \left\lbrack \begin{array}{l} A \end{array}\right\rbrack \left\lbrack \begin{array}{l} {AB} \end{array}\right\rbrack \\ \underset{A}{\underbrace{\left\lbrack \begin{array}{ll} 2 & - 1 \\ 3 & 2 \end{array}\right\rbrack }} \end{array}\;\under... | Yes |
Example 1.2.5 (Matrix multiplication is not commutative). | \[ \begin{matrix} & \left\lbrack \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\right\rbrack & & & & & \left\lbrack \begin{matrix} 0 & 1 \\ 1 & 1 \end{matrix}\right\rbrack & \\ \left\lbrack \begin{matrix} 0 & 1 \\ 1 & 1 \end{matrix}\right\rbrack & \left\lbrack \begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}\right\rbrack & & \te... | Yes |
Example 1.2.7 (The \( i \) th column of \( {AB} \) is \( A{\overrightarrow{\mathbf{b}}}_{i} \) ). The second column of the product \( {AB} \) is the product of \( A \) and the second column of \( B \) : |  | Yes |
Proposition 1.2.9 (Matrix multiplication is associative). If \( A \) is an \( n \times m \) matrix, \( B \) an \( m \times p \) matrix, and \( C \) a \( p \times q \) matrix, so that \( \left( {AB}\right) C \) and \( A\left( {BC}\right) \) are both defined, then they are equal: | Proof. Figure 1.2.5 shows that the \( \left( {i, j}\right) \) th entry of both \( A\left( {BC}\right) \) and \( \left( {AB}\right) C \) depend only on the \( i \) th row of \( A \) and the \( j \) th column of \( C \) (but on all the entries of \( B \) ). Without loss of generality we can assume that \( A \) is a line ... | Yes |
Example 1.2.12 (A matrix with neither right nor left inverse). The matrix \( \left\lbrack \begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right\rbrack \) does not have a right or a left inverse. | To see this, assume it has a right inverse. Then there exists a matrix \( \left\lbrack \begin{array}{ll} a & b \\ c & d \end{array}\right\rbrack \) such that\n\n\[ \left\lbrack \begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right\rbrack \left\lbrack \begin{array}{ll} a & b \\ c & d \end{array}\right\rbrack = \left\lbrack... | Yes |
Proposition 1.2.15 (Inverse of product of matrices). If \( A \) and \( B \) are invertible matrices, then \( {AB} \) is invertible, and the inverse is given by\n\n\[{\left( AB\right) }^{-1} = {B}^{-1}{A}^{-1}.\] | Proof. The computation\n\n\[ \left( {AB}\right) \left( {{B}^{-1}{A}^{-1}}\right) = A\left( {B{B}^{-1}}\right) {A}^{-1} = A{A}^{-1} = I \]\n\nand a similar one for \( \left( {{B}^{-1}{A}^{-1}}\right) \left( {AB}\right) \) prove the result. | Yes |
Example 1.2.21 (Matrices and probabilities). Suppose you have three reference books on a shelf: a thesaurus, a French dictionary, and an English dictionary. Each time you consult one of these books, you put it back on the shelf at the far left. When you need a reference, we denote by \( {P}_{1} \) the probability that ... | <table><thead><tr><th></th><th>(123)</th><th>(132)</th><th>\( \left( \begin{array}{lll} 2 & 1 & 3 \end{array}\right) \)</th><th>\( \left( \begin{matrix} 2 & 3 & 1 \end{matrix}\right) \)</th><th>\( \left( \begin{matrix} 3 & 1 & 2 \end{matrix}\right) \)</th><th>\( \left( \begin{matrix} 3 & 2 & 1 \end{matrix}\right) \)</t... | Yes |
How many walks of length \( n \) are there that go from a vertex to itself, or, more generally, from a given vertex to a different vertex? | As we will see in Proposition 1.2.23, we answer that question by raising to the \( n \) th power the adjacency matrix of the graph. The adjacency matrix for our cube is the \( 8 \times 8 \) matrix \( A \) whose rows and columns are labeled by the vertices \( {V}_{1},\ldots ,{V}_{8} \), and such that the \( \left( {i, j... | No |
Proposition 1.2.23. For any graph formed of vertices connected by edges, the number of possible walks of length \( n \) from vertex \( {V}_{i} \) to vertex \( {V}_{j} \) is given by the \( \left( {i, j}\right) \) th entry of the matrix \( {A}^{n} \) formed by taking the \( n \) th power of the graph’s adjacency matrix ... | Proof. The proof is by induction, in the context of the graph above; the general case is the same. Let \( {B}_{n} \) be the \( 8 \times 8 \) matrix whose \( \left( {i, j}\right) \) th entry is the number of walks from \( {V}_{i} \) to \( {V}_{j} \) of length \( n \), for a graph with eight vertices; we must prove \( {B... | Yes |
In a food processing plant making three types of frozen dinners, one might associate the number of different dinners produced to the total ingredients needed (beef, chicken, noodles, cream, salt, ... ). As shown in Figure 1.3.1, this mapping is given by multiplication (on the left) by the matrix \( A \), giving the amo... | \[ \left( {{.25} \times {60}}\right) + \left( {{.20} \times {30}}\right) + \left( {0 \times {40}}\right) = {21}. \] | Yes |
Consider the transformation \( T \) that reflects with respect to a line through the origin; one such transformation is shown in Figure 1.3.4. | The | No |
Example 1.3.6 (Identity transformation). The identity transformation id : \( {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) is linear and is given by the matrix \( {I}_{n} \) . Applying this transformation to a subset of \( {\mathbb{R}}^{n} \) leaves it unchanged. | \( \;\bigtriangleup \) | No |
Example 1.3.9 (Rotation by an angle \( \theta \) ). Figure 1.3.10 shows that the transformation \( R \) giving rotation by \( \theta \) counterclockwise around the origin is linear, and that its matrix is | \[ \left\lbrack {R\left( {\overrightarrow{\mathbf{e}}}_{1}\right), R\left( {\overrightarrow{\mathbf{e}}}_{2}\right) }\right\rbrack = \left\lbrack \begin{array}{rr} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array}\right\rbrack . \] | Yes |
Theorem 1.3.10 (Composition corresponds to matrix multiplication). Suppose \( S : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{m} \) and \( T : {\mathbb{R}}^{m} \rightarrow {\mathbb{R}}^{l} \) are linear transformations given by the matrices \( \left\lbrack S\right\rbrack \) and \( \left\lbrack T\right\rbrack \) respecti... | Proof of Theorem 1.3.10. The following computation shows that \( T \circ S \) is linear:\n\n\[ \left( {T \circ S}\right) \left( {a\overrightarrow{\mathbf{v}} + b\overrightarrow{\mathbf{w}}}\right) = T\left( {S\left( {a\overrightarrow{\mathbf{v}} + b\overrightarrow{\mathbf{w}}}\right) }\right) = T\left( {{aS}\left( \ove... | Yes |
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