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Proposition 1 (Limit of a composition of mappings) Let \( Y \) be a set with base \( {\mathcal{B}}_{Y} \) and \( g : Y \rightarrow Z \) a mapping of \( Y \) into a topological space \( Z \) having a limit over the base \( {\mathcal{B}}_{Y} \). Let \( X \) be a set with base \( {\mathcal{B}}_{X} \) and \( f : X \rightar... | For the proof see Theorem 5 of Sect. 3.2. | No |
Proposition 2 (Cauchy criterion for existence of the limit of a mapping) Let \( X \) be a set with a base \( \mathcal{B} \), and let \( f : X \rightarrow Y \) be a mapping of \( X \) into a complete metric space \( \left( {Y, d}\right) \). A necessary and sufficient condition for the mapping \( f \) to have a limit ove... | For the proof see Theorem 4 of Sect. 3.2. | No |
Theorem 1 (Criterion for continuity) A mapping \( f : X \rightarrow Y \) of a topological space \( \left( {X,{\tau }_{X}}\right) \) into a topological space \( \left( {Y,{\tau }_{Y}}\right) \) is continuous if and only if the pre-image of every open (resp. closed) subset of \( y \) is open (resp. closed) in \( X \) . | Proof Since the pre-image of a complement is the complement of the pre-image, it suffices to prove the assertions for open sets.\n\nWe first show that if \( f \in C\left( {X, Y}\right) \) and \( {G}_{Y} \in {\tau }_{Y} \), then \( {G}_{X} = {f}^{-1}\left( {G}_{Y}\right) \) belongs to \( {\tau }_{X} \) . If \( {G}_{X} =... | Yes |
Proposition 3 (Continuity of a composition of continuous mappings) Let \( \left( {X,{\tau }_{X}}\right) \) , \( \left( {Y,{\tau }_{Y}}\right) \) and \( \left( {Z,{\tau }_{Z}}\right) \) be topological spaces. If the mapping \( g : Y \rightarrow Z \) is continuous at a point \( b \in Y \) and the mapping \( f : X \righta... | This follows from the definition of continuity of a mapping and Proposition 1. | No |
Proposition 4 (Boundedness of a mapping in a neighborhood of a point of continuity) If a mapping \( f : X \rightarrow Y \) of a topological space \( \left( {X,\tau }\right) \) into a metric space \( \left( {Y, d}\right) \) is continuous at a point \( a \in X \), then it is bounded in some neighborhood of that point. | This proposition follows from the ultimate boundedness (over a base) of a mapping that has a limit. | No |
Proposition 5 A mapping \( f : X \rightarrow Y \) of a metric space \( \left( {X,{d}_{X}}\right) \) into a metric space \( \left( {Y,{d}_{Y}}\right) \) is continuous at the point \( a \in X \) if and only if \( \omega \left( {f, a}\right) = 0 \) . | This proposition follows immediately from the definition of continuity of a mapping at a point. | No |
Theorem 2 The image of a compact set under a continuous mapping is compact. | Proof Let \( f : K \rightarrow Y \) be a continuous mapping of the compact space \( \left( {K,{\tau }_{K}}\right) \) into a topological space \( \left( {Y,{\tau }_{Y}}\right) \), and let \( \left\{ {{G}_{Y}^{\alpha },\alpha \in A}\right\} \) be a covering of \( f\left( K\right) \) by sets that are open in \( Y \) . By ... | Yes |
Theorem 3 (Uniform continuity) A continuous mapping \( f : K \rightarrow Y \) of a compact metric space \( K \) into a metric space \( \left( {Y,{d}_{Y}}\right) \) is uniformly continuous. | In particular, if \( K \) is a closed interval in \( \mathbb{R} \) and \( Y = \mathbb{R} \), we again have the classical theorem of Cantor, the proof of which given in Sect. 4.2.2 carries over with almost no changes to this general case. | No |
Theorem 4 The image of a connected topological space under a continuous mapping is connected. | Proof Let \( f : X \rightarrow Y \) be a continuous mapping of a connected topological space \( \left( {X,{\tau }_{X}}\right) \) onto a topological space \( \left( {Y,{\tau }_{Y}}\right) \) . Let \( {E}_{Y} \) be an open-closed subset of \( Y \) . By Theorem 1, the pre-image \( {E}_{X} = {f}^{-1}\left( {E}_{Y}\right) \... | Yes |
As an important example of the application of the contraction mapping principle we shall prove, following Picard, an existence theorem for the solution of the differential equation \( {y}^{\prime }\left( x\right) = f\left( {x, y\left( x\right) }\right) \) satisfying an initial condition \( y\left( {x}_{0}\right) = {y}_... | Proof Equation (9.23) and the condition (9.22) can be jointly written as a single relation\n\n\[ y\left( x\right) = {y}_{0} + {\int }_{{x}_{0}}^{x}f\left( {t, y\left( t\right) }\right) \mathrm{d}t. \]\n\n(9.24)\n\nDenoting the right-hand side of this equality by \( A\left( y\right) \), we find that \( A \) : \( C\left(... | Yes |
As an illustration of what was just said, we shall seek a solution of the familiar equation\n\n\\[ \n{y}^{\prime } = y \n\\]\n\nwith the initial condition (9.22) on the basis of the contraction mapping principle. | In this case\n\n\\[ \n{Ay} = {y}_{0} + {\\int }_{{x}_{0}}^{x}y\\left( t\\right) \\mathrm{d}t \n\\]\n\nand the principle is applicable at least for \\( \\left| {x - {x}_{0}}\\right| \\leq q < 1 \\) .\n\nStarting from the initial approximation \\( y\\left( x\\right) \\equiv 0 \\), we construct successively the sequence \... | Yes |
Example 3 (Newton’s method of finding a root of the equation \( f\left( x\right) = 0 \) ) Suppose a real-valued function that is convex and has a positive derivative on a closed interval \( \left\lbrack {\alpha ,\beta }\right\rbrack \) assumes values of opposite signs at the endpoints of the interval. Then there is a u... | Take an arbitrary point \( {x}_{0} \in \left\lbrack {\alpha ,\beta }\right\rbrack \) and write the equation \( y = f\left( {x}_{0}\right) + {f}^{\prime }\left( {x}_{0}\right) \left( {x - {x}_{0}}\right) \) of the tangent to the graph of the function at the point \( \left( {{x}_{0}, f\left( {x}_{0}\right) }\right) \) . ... | Yes |
In analysis, besides the spaces \( {\mathbb{R}}^{n} \) and \( {\mathbb{C}}^{n} \) exhibited in Example 1, we encounter the space closest to them, which is the space \( \ell \) of sequences \( x = \) \( \left( {{x}^{1},\ldots ,{x}^{n},\ldots }\right) \) of real or complex numbers. The vector-space operations in \( \ell ... | The set of finite sequences (all of whose terms are zero from some point on) is a vector subspace \( \ell \) of the space \( \ell \), also infinite-dimensional. | No |
If for \( p \geq 1 \) we set\n\n\[ \parallel x{\parallel }_{p} \mathrel{\text{:=}} {\left( \mathop{\sum }\limits_{{i = 1}}^{n}{\left| {x}^{i}\right| }^{p}\right) }^{\frac{1}{p}} \]\n\nfor \( x = \left( {{x}^{1},\ldots ,{x}^{n}}\right) \in {\mathbb{R}}^{n} \), it follows from Minkowski’s inequality that we obtain a norm... | One can verify that\n\n\[ \parallel x{\parallel }_{{p}_{2}} \leq \parallel x{\parallel }_{{p}_{1}},\;\text{ if }1 \leq {p}_{1} \leq {p}_{2}, \]\n\nand that\n\n\[ \parallel x{\parallel }_{p} \rightarrow \max \left\{ {\left| {x}^{1}\right| ,\ldots ,\left| {x}^{n}\right| }\right\} \]\n\nas \( p \rightarrow + \infty \) . T... | Yes |
Example 6 The preceding example can be usefully generalized as follows. If \( X = \) \( {X}_{1} \times \cdots \times {X}_{n} \) is the direct product of normed vector spaces, one can introduce the norm of a vector \( x = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) in the direct product by setting\n\n\[ \parallel x{\para... | Naturally, inequalities (10.8) remain valid in this case as well.\n\nFrom now on, when the direct product of normed spaces is considered, unless the contrary is explicitly stated, it is assumed that the norm is defined in accordance with formula (10.9) (including the case \( p = + \infty \) ). | No |
Example 7 Let \( p \geq 1 \) . We denote by \( {\ell }_{p} \) the set of sequences \( x = \left( {{x}^{1},\ldots ,{x}^{n},\ldots }\right) \) of real or complex numbers such that the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left| {x}^{n}\right| }^{p} \) converges, and for \( x \in {\ell }_{p} \) we set\n\n\[... | Using Minkowski’s inequality, one can easily see that \( {\ell }_{p} \) is a normed vector space with respect to the standard vector-space operations and the norm (10.10). This is an infinite-dimensional space with respect to which \( {\mathbb{R}}_{p}^{n} \) is a vector subspace of finite dimension.\n\nAll the inequali... | Yes |
In the vector space \( C\left\lbrack {a, b}\right\rbrack \) of numerical-valued functions that are continuous on the closed interval \( \left\lbrack {a, b}\right\rbrack \), one usually considers the following norm:\n\n\[ \parallel f\parallel \mathrel{\text{:=}} \mathop{\max }\limits_{{x \in \left\lbrack {a, b}\right\rb... | We leave the verification of the norm axioms to the reader. We remark that this norm generates a metric on \( C\left\lbrack {a, b}\right\rbrack \) that is already familiar to us (see Sect. 9.5), and we know that the metric space that thereby arises is complete. Thus the vector space \( C\left\lbrack {a, b}\right\rbrack... | No |
One can also introduce another norm in \( C\left\lbrack {a, b}\right\rbrack \)\n\n\[ \parallel f{\parallel }_{p} \mathrel{\text{:=}} {\left( {\int }_{a}^{b}{\left| f\right| }^{p}\left( x\right) \mathrm{d}x\right) }^{\frac{1}{p}},\;p \geq 1, \] | It is easy to see (for example, Sect. 9.5) that the space \( C\left\lbrack {a, b}\right\rbrack \) with the norm (10.12) is not complete for \( 1 \leq p < + \infty \) . | No |
In \( {\ell }_{2} \) the inner product of the vectors \( x \) and \( y \) can be defined as\n\n\[\n\langle x, y\rangle \mathrel{\text{:=}} \mathop{\sum }\limits_{{i = 1}}^{\infty }{x}^{i}\overline{{y}^{i}}\n\] | The series in this expression converges absolutely since\n\n\[\n2\mathop{\sum }\limits_{{i = 1}}^{\infty }\left| {{x}^{i}\overline{{y}^{i}}}\right| \leq \mathop{\sum }\limits_{{i = 1}}^{\infty }{\left| {x}^{i}\right| }^{2} + \mathop{\sum }\limits_{{i = 1}}^{\infty }{\left| {y}^{i}\right| }^{2}.\n\] | Yes |
The following important inequality, known as the Cauchy-Bunyakovskii inequality, holds for the inner product:\n\n\[ \n{\left| \langle x, y\rangle \right| }^{2} \leq \langle x, x\rangle \cdot \langle y, y\rangle \n\]\n\nwhere equality holds if and only if the vectors \( x \) and \( y \) are collinear. | Proof Indeed, let \( a = \langle x, x\rangle, b = \langle x, y\rangle \), and \( c = \langle y, y\rangle \) . By hypothesis \( a \geq 0 \) and \( c \geq 0 \) . If \( c > 0 \), the inequalities\n\n\[ \n0 \leq \langle x + {\lambda y}, x + {\lambda y}\rangle = a + \bar{b}\lambda + b\bar{\lambda } + {c\lambda }\bar{\lambda... | Yes |
Example 4 We define the transformation \( A : C\left( {\left\lbrack {a, b}\right\rbrack ,\mathbb{R}}\right) \rightarrow C\left( {\left\lbrack {a, b}\right\rbrack ,\mathbb{R}}\right) \) by the formula\n\n\[ A\left( f\right) \mathrel{\text{:=}} {\int }_{a}^{x}f\left( t\right) \mathrm{d}t \]\n\nwhere \( x \) is a point ra... | All of these transformations are obviously linear. | No |
Example 9 Let \( X = {X}_{1} \times \cdots \times {X}_{m} \) be the vector space that is the direct product of the spaces \( {X}_{1},\ldots ,{X}_{m} \), and let \( A : X \rightarrow Y \) be a linear mapping of \( X \) into a vector space \( Y \) . Representing every vector \( x = \left( {{x}_{1},\ldots ,{x}_{m}}\right)... | Since the mapping \( A : X = {X}_{1} \times \cdots \times {X}_{m} \rightarrow Y \) is obviously linear for any linear mappings \( {A}_{i} : {X}_{i} \rightarrow Y \), we have shown that formula (10.22) gives the general form of any linear mapping \( A \in \mathcal{L}\left( {X = {X}_{1} \times \cdots \times {X}_{m};Y}\ri... | Yes |
Example 11 Combining Examples 9 and 10, we conclude that any linear mapping\n\n\\[ \nA : {X}_{1} \\times \\cdots \\times {X}_{m} = X \\rightarrow Y = {Y}_{1} \\times \\cdots \\times {Y}_{n} \n\\]\n\nof the direct product \\( X = {X}_{1} \\times \\cdots \\times {X}_{m} \\) of vector spaces into another direct product \\... | In particular, if \\( {X}_{1} = {X}_{2} = \\cdots = {X}_{m} = \\mathbb{R} \\) and \\( {Y}_{1} = {Y}_{2} = \\cdots = {Y}_{n} = \\mathbb{R} \\), then \\( {A}_{ij} : {X}_{j} \\rightarrow {Y}_{i} \\) are the linear mappings \\( \\mathbb{R} \\ni x \\mapsto {a}_{ij}x \\in \\mathbb{R} \\), each of which is given by a single n... | Yes |
Proposition 1 For a multilinear transformation \( A : {X}_{1} \times \cdots \times {X}_{n} \rightarrow Y \) mapping a product of normed spaces \( {X}_{1},\ldots ,{X}_{n} \) into a normed space \( Y \) the following conditions are equivalent:\n\na) A has a finite norm,\n\nb) \( A \) is a bounded transformation,\n\nc) \(... | Proof We prove a closed chain of implications a) \( \Rightarrow \) b) \( \Rightarrow \) c) \( \Rightarrow \) d) \( \Rightarrow \) a).\n\nIt is obvious from relation (10.27) that a) \( \Rightarrow \) b).\n\nLet us verify that b) \( \Rightarrow \) c), that is, that (10.29) implies that the operator \( A \) is continuous.... | Yes |
Proposition 2 The norm of a multilinear transformation is a norm in the vector space \( \mathcal{L}\left( {{X}_{1},\ldots ,{X}_{n};Y}\right) \) of continuous multilinear transformations. | Proof We observe first of all that by Proposition 1 the nonnegative number \( \parallel A\parallel < \) \( \infty \) is defined for every transformation \( A \in \mathcal{L}\left( {{X}_{1},\ldots ,{X}_{n};Y}\right) \) . Inequality (10.27) shows that \[ \parallel A\parallel = 0 \Leftrightarrow A = 0. \] Next, by definit... | Yes |
Proposition 3 If \( Y \) is a complete normed space, then \( \mathcal{L}\left( {{X}_{1},\ldots ,{X}_{n};Y}\right) \) is also a complete normed space. | Proof We shall carry out the proof for the space \( \mathcal{L}\left( {X;Y}\right) \) of continuous linear transformations. The general case, as will be clear from the reasoning below, differs only in requiring a more cumbersome notation.\n\nLet \( {A}_{1},{A}_{2},\ldots ,{A}_{n},\ldots \) be a Cauchy sequence in \( \m... | Yes |
Proposition 4 For each \( m \in \{ 1,\ldots, n\} \) there is a bijection between the spaces\n\n\[ \mathcal{L}\left( {{X}_{1},\ldots ,{X}_{m};\mathcal{L}\left( {{X}_{m + 1},\ldots ,{X}_{n};Y}\right) }\right) \;\text{ and }\;\mathcal{L}\left( {{X}_{1},\ldots ,{X}_{n};Y}\right) \]\n\nthat preserves the vector-space struct... | Proof We shall exhibit this isomorphism.\n\nLet \( \mathfrak{B} \in \mathcal{L}\left( {{X}_{1},\ldots ,{X}_{m};\mathcal{L}\left( {{X}_{m + 1},\ldots ,{X}_{n};Y}\right) }\right) \), that is, \( \mathfrak{B}\left( {{x}_{1},\ldots ,{x}_{m}}\right) \in \) \( \mathcal{L}\left( {{X}_{m + 1},\ldots ,{X}_{n};Y}\right) \) .\n\n... | No |
Proposition 1 If a mapping \( f : E \rightarrow Y \) is differentiable at an interior point \( x \) of a set \( E \subset X \), its differential \( L\left( x\right) \) at that point is uniquely determined. | Proof Thus we are verifying the uniqueness of the differential.\n\nLet \( {L}_{1}\left( x\right) \) and \( {L}_{2}\left( x\right) \) be linear mappings satisfying relation (10.31), that is\n\n\[ f\left( {x + h}\right) - f\left( x\right) - {L}_{1}\left( x\right) h = {\alpha }_{1}\left( {x;h}\right) ,\]\n\n(10.32)\n\n\[ ... | Yes |
Example 1 If \( f : U \rightarrow Y \) is a constant mapping of a neighborhood \( U = U\left( x\right) \subset X \) of the point \( x \), that is, \( f\left( U\right) = {y}_{0} \in Y \), then \( {f}^{\prime }\left( x\right) = 0 \in \mathcal{L}\left( {X;Y}\right) \) . | Proof Indeed, in this case it is obvious that\n\n\[ f\left( {x + h}\right) - f\left( x\right) - {0h} = {y}_{0} - {y}_{0} - 0 = 0 = o\left( h\right) . \] | Yes |
If the mapping \( f : X \rightarrow Y \) is a continuous linear mapping of a normed vector space \( X \) into a normed vector space \( Y \), then \( {f}^{\prime }\left( x\right) = f \in \mathcal{L}\left( {X;Y}\right) \) at any point \( x \in A \) . | Indeed,\n\n\[ f\left( {x + h}\right) - f\left( x\right) - {fh} = {fx} + {fh} - {fx} - {fh} = 0. \]\n\nWe remark that strictly speaking \( {f}^{\prime }\left( x\right) \in \mathcal{L}\left( {T{X}_{x};T{Y}_{f\left( x\right) }}\right) \) here and \( h \) is a vector of the tangent space \( T{X}_{x} \) . But parallel trans... | Yes |
Example 3 From the chain rule for differentiating a composition of mappings and the result of Example 2 one can conclude that if \( f : U \rightarrow Y \) is a mapping of a neighborhood \( U = U\left( x\right) \subset X \) of the point \( x \in X \) and is differentiable at \( x \), while \( A \in \mathcal{L}\left( {Y;... | \[ {\left( A \circ f\right) }^{\prime }\left( x\right) = A \circ {f}^{\prime }\left( x\right) . \] | Yes |
Now let \( A \in \mathcal{L}\left( {{X}_{1},\ldots ,{X}_{n};Y}\right) \), that is, \( A \) is a continuous \( n \) -linear transformation from the product \( {X}_{1} \times \cdots \times {X}_{n} \) of the normed vector spaces \( {X}_{1},\ldots ,{X}_{n} \) into the normed vector space \( Y \) . We shall prove that the m... | Proof Using the multilinearity of \( A \), we find that \[ A\left( {x + h}\right) - A\left( x\right) = A\left( {{x}_{1} + {h}_{1},\ldots ,{x}_{n} + {h}_{n}}\right) - A\left( {{x}_{1},\ldots ,{x}_{n}}\right) = \] \[ = A\left( {{x}_{1},\ldots ,{x}_{n}}\right) + A\left( {{h}_{1},{x}_{2},\ldots ,{x}_{n}}\right) + \] \[ + \... | Yes |
Example 6 Let \( U \) be the subset of \( \mathcal{L}\left( {X;Y}\right) \) consisting of the continuous linear transformations \( A : X \rightarrow Y \) having continuous inverse transformations \( {A}^{-1} : Y \rightarrow X \) (belonging to \( \mathcal{L}\left( {Y;X}\right) \) ). Consider the mapping\n\n\[ \nU \ni A ... | Proposition 2 proved below makes it possible to determine whether this mapping is differentiable.\n\nProposition 2 If \( X \) | No |
Example 7 Let \( X \) be a complete normed vector space. The important mapping\n\n\[ \exp : \mathcal{L}\left( {X;X}\right) \rightarrow \mathcal{L}\left( {X;X}\right) \]\n\nis defined as follows:\n\n\[ \exp A \mathrel{\text{:=}} E + \frac{1}{1!}A + \frac{1}{2!}{A}^{2} + \cdots + \frac{1}{n!}{A}^{n} + \cdots , \]\n\n(10.... | The series in (10.37) converges, since \( \mathcal{L}\left( {X;X}\right) \) is a complete space and \( \begin{Vmatrix}{\frac{1}{n!}{A}^{n}}\end{Vmatrix} \leq \frac{\parallel A{\parallel }^{n}}{n!} \), while the numerical series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{\parallel A{\parallel }^{n}}{n!} \) conver... | Yes |
We shall attempt to give a mathematical description of the instantaneous angular velocity of a rigid body with a fixed point \( o \) (a top). Consider an orthonormal frame \( \left\{ {{\mathbf{e}}_{1},{\mathbf{e}}_{2},{\mathbf{e}}_{3}}\right\} \) at the point \( o \) rigidly attached to the body. It is clear that the p... | Since \( O\left( t\right) \) is an orthogonal matrix, the relation\n\n\[ O\left( t\right) {O}^{ * }\left( t\right) = E \] \n\nholds at any time \( t \), where \( {O}^{ * }\left( t\right) \) is the transpose of \( O\left( t\right) \) and \( E \) is the identity matrix.\n\nWe remark that the product \( A \cdot B \) of ma... | Yes |
Consider the familiar situation when \( X = Y = \mathbb{R} \), and hence \( f : U \rightarrow \mathbb{R} \) is a real-valued function of a real argument. Since any linear mapping \( A \in \mathcal{L}\left( {\mathbb{R};\mathbb{R}}\right) \) reduces to multiplication by some number \( a \in \mathbb{R} \), that is, \( {Ah... | Next, since\n\n\[ \left( {{f}^{\prime }\left( {x + \delta }\right) - {f}^{\prime }\left( x\right) }\right) h = {f}^{\prime }\left( {x + \delta }\right) h - {f}^{\prime }\left( x\right) h = \]\n\n\[ = a\left( {x + \delta }\right) h - a\left( x\right) h = \left( {a\left( {x + \delta }\right) - a\left( x\right) }\right) h... | Yes |
This time suppose that \( X \) is the direct product \( {X}_{1} \times \cdots \times {X}_{n} \) of normed spaces. In this case the mapping (10.50) is a function \( f\left( x\right) = f\left( {{x}_{1},\ldots ,{x}_{m}}\right) \) of \( m \) variables \( {x}_{i} \in {X}_{i}, i = 1,\ldots, m \), with values in \( Y \) . If ... | The action of \( \mathrm{d}f\left( x\right) \) on a vector \( h = \left( {{h}_{1},\ldots ,{h}_{m}}\right) \), by formula (10.45), can be represented as \[ \mathrm{d}f\left( x\right) h = {\partial }_{1}f\left( x\right) {h}_{1} + \cdots + {\partial }_{m}f\left( x\right) {h}_{m}, \] where \( {\partial }_{i}f\left( x\right... | Yes |
Proposition 1 If \( K \) is a convex compact set in a normed space \( X \) and \( f \in \) \( {C}^{\left( 1\right) }\left( {K, Y}\right) \), where \( Y \) is also a normed space, then the mapping \( f : K \rightarrow Y \) satisfies a Lipschitz condition on \( K \), that is, there exists a constant \( M > 0 \) such that... | Proof By hypothesis \( {f}^{\prime } : K \rightarrow \mathcal{L}\left( {X;Y}\right) \) is a continuous mapping of the compact set \( K \) into the metric space \( \mathcal{L}\left( {X;Y}\right) \) . Since the norm is a continuous function on a normed space with its natural metric, the mapping \( x \mapsto \begin{Vmatri... | Yes |
Proposition 2 Under the hypotheses of Proposition 1 there exists a non-negative function \( \omega \left( \delta \right) \) tending to 0 as \( \delta \rightarrow + 0 \) such that\n\n\[ \left| {f\left( {x + h}\right) - f\left( x\right) - {f}^{\prime }\left( x\right) h}\right| \leq \omega \left( \delta \right) \left| h\r... | Proof By the corollary to the finite-increment theorem we can write\n\n\[ \left| {f\left( {x + h}\right) - f\left( x\right) - {f}^{\prime }\left( x\right) h}\right| \leq \mathop{\sup }\limits_{{0 < \theta < 1}}\begin{Vmatrix}{{f}^{\prime }\left( {x + {\theta h}}\right) - {f}^{\prime }\left( x\right) }\end{Vmatrix}\left... | Yes |
Theorem 2 Let \( U \) be a neighborhood of the point \( x \) in a normed space \( X = {X}_{1} \times \) \( \cdots \times {X}_{m} \), which is the direct product of the normed spaces \( {X}_{1} \times \cdots \times {X}_{m} \), and let \( f : U \rightarrow Y \) be a mapping of \( U \) into a normed space \( Y \) . If the... | Proof To simplify the writing we carry out the proof for the case \( m = 2 \) . We verify immediately that the mapping\n\n\[ \n{Lh} = {\partial }_{1}f\left( x\right) {h}_{1} + {\partial }_{2}f\left( x\right) {h}_{2}, \n\] \n\nwhich is linear in \( h = \left( {{h}_{1},{h}_{2}}\right) \), is the total differential of \( ... | Yes |
Theorem 1 If a mapping \( f : U \rightarrow Y \) from a neighborhood \( U = U\\left( x\\right) \) of a point \( x \) in a normed space \( X \) into a normed space \( Y \) has derivatives up to order \( n - 1 \) inclusive in \( U \) and has an nth order derivative \( {f}^{\\left( n\\right) }\\left( x\\right) \) at the p... | Proof We prove Taylor's formula by induction.\n\nFor \( n = 1 \) it is true by definition of \( {f}^{\\prime }\\left( x\\right) \) .\n\nAssume formula (10.70) is true for some \( \\left( {n - 1}\\right) \\in \\mathbb{N} \).\n\nThen by the mean-value theorem, formula (10.69) of Sect. 10.5, and the induction hypothesis, ... | Yes |
We shall show that the functional (10.72) is a differentiable mapping and find its differential. | We remark that the function (10.72) can be regarded as the composition of the mappings\n\n\[ {F}_{1} : {C}^{\left( 1\right) }\left( {\left\lbrack {a, b}\right\rbrack ,\mathbb{R}}\right) \rightarrow C\left( {\left\lbrack {a, b}\right\rbrack ,\mathbb{R}}\right) \]\n\n(10.73)\n\ndefined by the formula\n\n\[ {F}_{1}\left( ... | Yes |
Among all the curves in a plane joining two fixed points, find the curve that has minimal length. | We shall assume that a fixed Cartesian coordinate system has been chosen in the plane, in which the two points are, for example, \( \\left( {0,0}\\right) \) and \( \\left( {1,0}\\right) \) . We confine ourselves to just the curves that are the graphs of functions \( f \\in {C}^{\\left( 1\\right) }\\left( {\\left\\lbrac... | Yes |
Lemma 1 The measure of an interval in \( {\mathbb{R}}^{n} \) has the following properties.\n\na) It is homogeneous, that is, if \( \lambda {I}_{a, b} \mathrel{\text{:=}} {I}_{{\lambda a},{\lambda b}} \), where \( \lambda \geq 0 \), then\n\n\[ \left| {\lambda {I}_{a, b}}\right| = {\lambda }^{n}\left| {I}_{a, b}\right| \... | All these assertions follow easily from Definitions 1 and 2. | No |
Let \( f : I \rightarrow \mathbb{R} \) be a continuous real-valued function defined on an \( \left( {n - 1}\right) \) -dimensional interval \( I \subset {\mathbb{R}}^{n - 1} \) . We shall show that its graph in \( {\mathbb{R}}^{n} \) is a set of \( n \) -dimensional measure zero. | Proof Since the function \( f \) is uniformly continuous on \( I \), for \( \varepsilon > 0 \) we find \( \delta > 0 \) such that \( \left| {f\left( {x}_{1}\right) - f\left( {x}_{2}\right) }\right| < \varepsilon \) for any two points \( {x}_{1},{x}_{2} \in I \) such that \( \left| {{x}_{1} - {x}_{2}}\right| < \delta \)... | Yes |
Lemma 5 The following relations hold between the Darboux sums of a function \( f : I \rightarrow \mathbb{R} : \n\na) \( s\left( {f, P}\right) = \mathop{\inf }\limits_{\xi }\sigma \left( {f, P,\xi }\right) \leq \sigma \left( {f, P,\xi }\right) \leq \mathop{\sup }\limits_{\xi }\sigma \left( {f, P,\xi }\right) = S\left( {... | Proof Relations a) and b) follow immediately from Definitions 6 and 10, taking account, of course, of the definition of the greatest lower bound and least upper bound of a set of numbers.\n\nTo prove c) it suffices to consider the auxiliary partition \( P \) obtained by intersecting the intervals of the partitions \( {... | Yes |
Theorem 2 (Darboux) For any bounded function \( f : I \rightarrow \mathbb{R} \) , \n\n\[ \n\left( {\exists \mathop{\lim }\limits_{{\lambda \left( P\right) \rightarrow 0}}s\left( {f, P}\right) }\right) \land \left( {\mathop{\lim }\limits_{{\lambda \left( P\right) \rightarrow 0}}s\left( {f, P}\right) = \underline{\mathca... | Proof If we compare these assertions with Definition 11, it becomes clear that in essence all we have to prove is that the limits exist. We shall verify this for the lower Darboux sums.\n\nFix \( \varepsilon > 0 \) and a partition \( {P}_{\varepsilon } \) of the interval \( I \) for which \( s\left( {f;{P}_{\varepsilon... | Yes |
Theorem 3 (The Darboux criterion) A real-valued function \( f : I \rightarrow \mathbb{R} \) defined on an interval \( I \subset {\mathbb{R}}^{n} \) is integrable over that interval if and only if it is bounded on \( I \) and its upper and lower Darboux integrals are equal. | Thus,\n\n\[ f \in \mathcal{R}\left( I\right) \Leftrightarrow \left( {f\text{ is bounded on }I}\right) \land \left( {\underline{\mathcal{J}} = \overline{\mathcal{J}}}\right) . \]\n\nProof Necessity. If \( f \in \mathcal{R}\left( I\right) \), then by Proposition 1 the function \( f \) is bounded on \( I \) . It follows f... | Yes |
Lemma 3 If \( {I}_{1} \) and \( {I}_{2} \) are two intervals, both containing the set \( E \), then the integrals \[ {\int }_{{I}_{1}}f{\chi }_{E}\left( x\right) \mathrm{d}x\;\text{ and }\;{\int }_{{I}_{2}}f{\chi }_{E}\left( x\right) \mathrm{d}x \] either both exist or both fail to exist, and in the first case their va... | Proof Consider the interval \( I = {I}_{1} \cap {I}_{2} \) . By hypothesis \( I \supset E \) . The points of discontinuity of \( f{\chi }_{E} \) are either points of discontinuity of \( f \) on \( E \), or the result of discontinuities of \( {\chi }_{E} \), in which case they lie on \( \partial E \) . In any case, all ... | Yes |
Theorem 1 A function \( f : E \rightarrow \mathbb{R} \) is integrable over an admissible set if and only if it is bounded and continuous at almost all points of \( E \) . | Proof Compared with \( f \), the function \( f{\chi }_{E} \) may have additional points of discontinuity only on the boundary \( \partial E \) of \( E \), which by hypothesis is a set of measure zero. | No |
Proposition 2 Let \( {E}_{1} \) and \( {E}_{2} \) be admissible sets in \( {\mathbb{R}}^{n} \) and \( f \) a function defined on \( {E}_{1} \cup {E}_{2} \) . a) The following relations hold: \[ \left( {\exists {\int }_{{E}_{1} \cup {E}_{2}}f\left( x\right) \mathrm{d}x}\right) \Leftrightarrow \left( {\exists {\int }_{{E... | Proof Assertion a) follows from Lebesgue's criterion for existence of the Riemann integral over an admissible set (Theorem 1 of Sect. 11.2). Here it is only necessary to recall that the union and intersection of admissible sets are also admissible sets (Lemma 2 of Sect. 11.2). | No |
Proposition 3 If \( f \in \mathcal{R}\left( E\right) \), then \( \left| f\right| \in \mathcal{R}\left( E\right) \), and the inequality\n\n\[ \left| {{\int }_{E}f\left( x\right) \mathrm{d}x}\right| \leq {\int }_{E}\left| f\right| \left( x\right) \mathrm{d}x \] | Proof The relation \( \left| f\right| \in \mathcal{R}\left( E\right) \) follows from the definition of the integral over a set and the Lebesgue criterion for integrability of a function over an interval.\n\nThe inequality now follows from the corresponding inequality for Riemann sums and passage to the limit. | No |
Proposition 4 The following implication holds for a function \( f : E \rightarrow \mathbb{R} \) :\n\n\[ \left( {f \in \mathcal{R}\left( E\right) }\right) \land \left( {\forall x \in E\left( {f\left( x\right) \geq 0}\right) }\right) \Rightarrow {\int }_{E}f\left( x\right) \mathrm{d}x \geq 0. \] | Proof Indeed, if \( f\left( x\right) \geq 0 \) on \( E \), then \( f{\chi }_{E}\left( x\right) \geq 0 \) in \( {\mathbb{R}}^{n} \) . Then, by definition,\n\n\[ {\int }_{E}f\left( x\right) \mathrm{d}x = {\int }_{I \supset E}f{\chi }_{E}\left( x\right) \mathrm{d}x. \]\n\nThis last integral exists by hypothesis. But it is... | Yes |
Corollary 2 If \( f \in \mathcal{R}\left( E\right) \) and the inequalities \( m \leq f\left( x\right) \leq M \) hold at every point of the admissible set \( E \), then | \[ {m\mu }\left( E\right) \leq {\int }_{E}f\left( x\right) \mathrm{d}x \leq {M\mu }\left( E\right) . \] | Yes |
Corollary 5 If in addition to the hypotheses of Corollary 2 the function \( g \in \mathcal{R}\left( E\right) \) is nonnegative on \( E \), then\n\n\[ m{\int }_{E}g\left( x\right) \mathrm{d}x \leq {\int }_{E}{fg}\left( x\right) \mathrm{d}x \leq M{\int }_{E}g\left( x\right) \mathrm{d}x. \]\n\n | Proof Corollary 5 follows from the inequalities \( {mg}\left( x\right) \leq f\left( x\right) g\left( x\right) \leq {Mg}\left( x\right) \) taking account of the linearity of the integral and Corollary 1. It can also be proved directly by passing from integrals over \( E \) to the corresponding integrals over an interval... | No |
Corollary 1 If \( f \in \mathcal{R}\left( {X \times Y}\right) \), then for almost all \( x \in X \) (in the sense of Lebesgue) the integral \( {\int }_{X}f\left( {x, y}\right) \mathrm{d}y \) exists, and for almost all \( y \in Y \) the integral \( {\int }_{X}f\left( {x, y}\right) \mathrm{d}x \) exists. | Proof By the theorem just proved,\n\n\[{\int }_{X}\left( {{\int }_{Y}f\left( {x, y}\right) \mathrm{d}y - {\int }_{\bar{Y}}f\left( {x, y}\right) \mathrm{d}y}\right) \mathrm{d}x = 0.\]\n\nBut the difference of the upper and lower integrals in parentheses is nonnegative. We can therefore conclude by the lemma of Sect. 11.... | Yes |
Corollary 2 If the interval \( I \subset {\mathbb{R}}^{n} \) is the direct product of the closed intervals \( {I}_{i} = \) \( \left\lbrack {{a}^{i},{b}^{i}}\right\rbrack, i = 1,\ldots, n \), then\n\n\[{\int }_{I}f\left( x\right) \mathrm{d}x = {\int }_{{a}^{n}}^{{b}^{n}}\mathrm{\;d}{x}^{n}{\int }_{{a}^{n - 1}}^{{b}^{n -... | Proof This formula obviously results from repeated application of the theorem just proved. All the inner integrals on the right-hand side are to be understood as in the theorem. For example, one can insert the upper or lower integral sign throughout. | No |
Let \( f\left( {x, y, z}\right) = z\sin \left( {x + y}\right) \) . We shall find the integral of the restriction of this function to the interval \( I \subset {\mathbb{R}}^{3} \) defined by the relations \( 0 \leq x \leq \pi ,\left| y\right| \leq \pi /2 \) , \( 0 \leq z \leq 1 \) . | \[ {\iiint }_{I}f\left( {x, y, z}\right) \mathrm{d}x\mathrm{\;d}y\mathrm{\;d}z = {\int }_{0}^{1}\mathrm{\;d}z{\int }_{-\pi /2}^{\pi /2}\mathrm{\;d}y{\int }_{0}^{\pi }z\sin \left( {x + y}\right) \mathrm{d}x = \] \[ = {\int }_{0}^{1}\mathrm{\;d}z{\int }_{-\pi /2}^{\pi /2}\left( {-{\left. z\cos \left( x + y\right) \right|... | Yes |
Corollary 3 Let \( D \) be a bounded set in \( {\mathbb{R}}^{n - 1} \) and \( E = \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{n} \mid \left( {x \in D}\right) \land }\right. \) \( \left. \left( {{\varphi }_{1}\left( x\right) \leq y \leq {\varphi }_{2}\left( x\right) }\right) \right\} \) . If \( f \in \mathcal{R}\lef... | Proof Let \( {E}_{x} = \left\{ {y \in \mathbb{R} \mid {\varphi }_{1}\left( x\right) \leq y \leq {\varphi }_{2}\left( x\right) }\right\} \) if \( x \in D \) and \( {E}_{x} = \varnothing \) if \( x \notin D \) . We remark that \( {\chi }_{E}\left( {x, y}\right) = {\chi }_{D}\left( x\right) \cdot {\chi }_{{E}_{x}}\left( y... | Yes |
For the disk \( E = \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} \mid {x}^{2} + {y}^{2} \leq {r}^{2}}\right\} \) we obtain by this formula | \n\[
\mu \left( E\right) = {\int }_{-r}^{r}\left( {\sqrt{{r}^{2} - {y}^{2}} - \left( {-\sqrt{{r}^{2} - {y}^{2}}}\right) }\right) \mathrm{d}y = 2{\int }_{-r}^{r}\sqrt{{r}^{2} - {y}^{2}}\mathrm{\;d}y =
\]
\[
= 4{\int }_{0}^{r}\sqrt{{r}^{2} - {y}^{2}}\mathrm{\;d}y = 4{\int }_{0}^{\pi /2}r\cos \varphi \mathrm{d}\left( {r\... | Yes |
Corollary 5 Let \( E \) be a measurable set contained in the interval \( I \subset {\mathbb{R}}^{n} \) . Represent \( I \) as the direct product \( I = {I}_{x} \times {I}_{y} \) of the \( \left( {n - 1}\right) \) -dimensional interval \( {I}_{x} \) and the closed interval \( {I}_{y} \) . Then for almost all values \( {... | Proof Corollary 5 follows immediately from the theorem and Corollary 1, if we set \( f = {\chi }_{E} \) in both of them and take account of the relation \( {\chi }_{E}\left( {x, y}\right) = {\chi }_{{E}_{y}}\left( x\right) \) . | Yes |
Corollary 6 (Cavalieri’s \( {}^{6} \) principle) Let \( A \) and \( B \) be two solids in \( {\mathbb{R}}^{3} \) having volume (that is, Jordan-measurable). Let \( {A}_{c} = \{ \left( {x, y, z}\right) \in A \mid z = c\} \) and \( {B}_{c} = \{ \left( {x, y, z}\right) \in \) \( B \mid z = c\} \) be the sections of the so... | It is clear that Cavalieri’s principle can be stated for spaces \( {\mathbb{R}}^{n} \) of any dimension. | No |
Using formula (11.3), let us compute the volume \( {V}_{n} \) of the ball \( B = \left\{ {x \in {\mathbb{R}}^{n}\left| \right| x \mid \leq r}\right\} \) of radius \( r \) in the Euclidean space \( {\mathbb{R}}^{n} \). | It is obvious that \( {V}_{1} = 2 \) . In Example 2 we found that \( {V}_{2} = \pi {r}^{2} \) . We shall show that \( {V}_{n} = {c}_{n}{r}^{n} \), where \( {c}_{n} \) is a constant (which we shall compute below). Let us choose some diameter \( \left\lbrack {-r, r}\right\rbrack \) of the ball and for each point \( x \in... | Yes |
Theorem 1 If \( \varphi : {D}_{t} \rightarrow {D}_{x} \) is a diffeomorphism of a bounded open set \( {D}_{t} \subset {\mathbb{R}}^{n} \) onto a set \( {D}_{x} = \varphi \left( {D}_{t}\right) \subset {\mathbb{R}}^{n} \) of the same type, \( f \in \mathcal{R}\left( {D}_{x}\right) \), and \( \operatorname{supp}f \) is a ... | \[ {\int }_{{D}_{x} = \varphi \left( {D}_{t}\right) }f\left( x\right) \mathrm{d}x = {\int }_{{D}_{t}}f \circ \varphi \left( t\right) \left| {\det {\varphi }^{\prime }\left( t\right) }\right| \mathrm{d}t. \] | Yes |
Lemma 1 Let \( \varphi : {D}_{t} \rightarrow {D}_{x} \) be a diffeomorphism of an open set \( {D}_{t} \subset {\mathbb{R}}^{n} \) onto a set \( {D}_{x} \subset {\mathbb{R}}^{n} \) of the same type. Then the following assertions hold.\n\na) If \( {E}_{t} \subset {D}_{t} \) is a set of (Lebesgue) measure zero, its image ... | Proof We begin by remarking that every open subset \( D \) in \( {\mathbb{R}}^{n} \) can be represented as the union of a countable number of closed intervals (no two of which have any interior points in common). To do this, for example, one can partition the coordinate axes into closed intervals of length \( \Delta \)... | Yes |
Lemma 4 If \( {D}_{\tau }\overset{\psi }{ \rightarrow }{D}_{t}\overset{\varphi }{ \rightarrow }{D}_{x} \) are two diffeomorphisms for each of which formula (11.10) for change of variable in the integral holds, then it holds also for the composition \( \varphi \circ \psi : {D}_{\tau } \rightarrow {D}_{x} \) of these map... | Proof It suffices to recall that \( {\left( \varphi \circ \psi \right) }^{\prime } = {\varphi }^{\prime } \circ {\psi }^{\prime } \) and that \( \det {\left( \varphi \circ \psi \right) }^{\prime }\left( \tau \right) = \) \( \det {\varphi }^{\prime }\left( t\right) \det {\psi }^{\prime }\left( \tau \right) \), where \( ... | Yes |
Proposition 1 Let \( \varphi : {D}_{t} \rightarrow {D}_{x} \) be a diffeomorphism of a bounded open set \( {D}_{t} \subset \) \( {\mathbb{R}}^{n} \) onto a set \( {D}_{x} \subset {\mathbb{R}}^{n} \) of the same type; let \( {E}_{t} \) and \( {E}_{x} \) be subsets of \( {D}_{t} \) and \( {D}_{x} \) respectively and such... | Proof Indeed,\n\n\[{\int }_{{E}_{x}}f\left( x\right) \mathrm{d}x = {\int }_{{D}_{x}}\left( {f{\chi }_{{E}_{x}}}\right) \left( x\right) \mathrm{d}x = {\int }_{{D}_{t}}\left( {\left( {\left( {f{\chi }_{{E}_{x}}}\right) \circ \varphi }\right) \left| {\det {\varphi }^{\prime }}\right| }\right) \left( t\right) \mathrm{d}t =... | Yes |
Proposition 2 The value of the integral of a function \( f \) over a set \( E \subset {\mathbb{R}}^{n} \) is independent of the choice of Cartesian coordinate system in \( {\mathbb{R}}^{n} \). | Proof In fact the transition from one Cartesian coordinate system in \( {\mathbb{R}}^{n} \) to another Cartesian system has a Jacobian constantly equal to 1 in absolute value. By Proposition 1 this implies the equality\n\n\[ \n{\int }_{{E}_{x}}f\left( x\right) \mathrm{d}x = {\int }_{{E}_{t}}\left( {f \circ \varphi }\ri... | Yes |
Theorem 2 Let \( \varphi : {D}_{t} \rightarrow {D}_{x} \) be a mapping of a (Jordan) measurable set \( {D}_{t} \subset {\mathbb{R}}_{t}^{n} \) onto a set \( {D}_{x} \subset {\mathbb{R}}_{x}^{n} \) of the same type. Suppose that there are subsets \( {S}_{t} \) and \( {S}_{x} \) of \( {D}_{t} \) and \( {D}_{x} \) respect... | Proof By Lebesgue’s criterion the function \( f \) can have discontinuities in \( {D}_{x} \) and hence also in \( {D}_{x} \smallsetminus {S}_{x} \) only on a set of measure zero. By Lemma 1, the image of this set of discontinuities under the mapping \( {\varphi }^{-1} : {D}_{x} \smallsetminus {S}_{x} \rightarrow {D}_{t... | Yes |
Proposition 1 If a function \( f : E \rightarrow \mathbb{R} \) is nonnegative and the limit in Definition 2 exists for even one exhaustion \( \left\{ {E}_{n}\right\} \) of the set \( E \), then the improper integral of \( f \) over \( E \) converges. | Proof Let \( \left\{ {E}_{k}^{\prime }\right\} \) be a second exhaustion of \( E \) into elements on which \( f \) is integrable. The sets \( {E}_{n}^{k} \mathrel{\text{:=}} {E}_{k}^{\prime } \cap {E}_{n}, n = 1,2,\ldots \) form an exhaustion of the set \( {E}_{k}^{\prime } \), and so it follows from part b) of the lem... | Yes |
Let us find the improper integral \( {\iint }_{{\mathbb{R}}^{2}}{e}^{-\left( {{x}^{2} + {y}^{2}}\right) }\mathrm{d}x\mathrm{\;d}y \) . | We shall exhaust the plane \( {\mathbb{R}}^{2} \) by the sequence of disks \( {E}_{n} = \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} \mid }\right. \) \( \left. {{x}^{2} + {y}^{2} < {n}^{2}}\right\} \) . After passing to polar coordinates we find easily that\n\n\[ \n{\iint }_{{E}_{n}}{\mathrm{e}}^{-\left( {{x}^{2}... | Yes |
Proposition 2 Let \( f \) and \( g \) be functions defined on the set \( E \) and integrable over exactly the same measurable subsets of it, and suppose \( \left| {f\left( x\right) }\right| \leq g\left( x\right) \) on \( E \) . If the improper integral \( {\int }_{E}g\left( x\right) \mathrm{d}x \) converges, then the i... | Proof Let \( \left\{ {E}_{n}\right\} \) be an exhaustion of \( E \) on whose elements both \( g \) and \( f \) are integrable. It follows from the Lebesgue criterion that the function \( \left| f\right| \) is integrable on the sets \( {E}_{n}, n \in \mathbb{N} \), and so we can write\n\n\[ \n{\int }_{{E}_{n + k}}\left|... | Yes |
In the deleted \( n \) -dimensional ball of radius \( 1, B \subset {\mathbb{R}}^{n} \) with its center at 0 removed, consider the function \( 1/{r}^{\alpha } \), where \( r = d\left( {0, x}\right) \) is the distance from the point \( x \in B \smallsetminus 0 \) to the point 0 . Let us determine the values of \( \alpha ... | To do this we construct an exhaustion of the domain by the annular regions \( B\left( \varepsilon \right) = \{ x \in B \mid \varepsilon < d\left( {0, x}\right) < 1\} \) . Passing to polar coordinates with center at 0 , by Fubini's theorem, we obtain \[ {\int }_{B\left( \varepsilon \right) }\frac{\mathrm{d}x}{{r}^{\alph... | Yes |
On the set \( I \smallsetminus {I}_{k} \) we consider the function \( \frac{1}{{d}^{\alpha }\left( x\right) } \), where \( d\left( x\right) \) is the distance from \( x \in I \smallsetminus {I}_{k} \) to the face \( {I}_{k} \). Let us determine the values of \( \alpha \in \mathbb{R} \) for which the integral of this fu... | We remark that if \( x = \left( {{x}^{1},\ldots ,{x}^{k},{x}^{k + 1},\ldots ,{x}^{n}}\right) \) then\n\n\[ d\left( x\right) = \sqrt{{\left( {x}^{k + 1}\right) }^{2} + \cdots + {\left( {x}^{n}\right) }^{2}}.\]\n\nLet \( I\left( \varepsilon \right) \) be the cube \( I \) from which the \( \varepsilon \) -neighborhood of ... | Yes |
Example 4 Let the function \( f : {\mathbb{R}}_{ + } \rightarrow \mathbb{R} \) be defined on the set \( {\mathbb{R}}_{ + } \) of nonnegative numbers by the following conditions: \( f\left( x\right) = \frac{{\left( -1\right) }^{n - 1}}{n} \), if \( n - 1 \leq x < n, n \in \mathbb{N} \). | Since the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{\left( -1\right) }^{n - 1}}{n} \) converges, the integral \( {\int }_{0}^{A}f\left( x\right) \mathrm{d}x \) has a limit as \( A \rightarrow \infty \) equal to the sum of this series. However, this series does not converge absolutely, and one can make i... | Yes |
Theorem 1 Let \( \varphi : {D}_{t} \rightarrow {D}_{x} \) be a diffeomorphism of the open set \( {D}_{t} \subset {\mathbb{R}}_{t}^{n} \) onto the set \( {D}_{x} \subset {\mathbb{R}}_{x}^{n} \) of the same type, and let \( f : {D}_{x} \rightarrow \mathbb{R} \) be integrable on all measurable compact subsets of \( {D}_{x... | Proof The open set \( {D}_{t} \subset {\mathbb{R}}_{t}^{n} \) can be exhausted by a sequence of compact sets \( {E}_{t}^{k} \) , \( k \in \mathbb{N} \), contained in \( \mathbb{N} \), each of which is the union of a finite number of intervals in \( {\mathbb{R}}_{t}^{n} \) (in this connection, see the beginning of the p... | Yes |
Theorem 2 Let \( \varphi : {D}_{t} \rightarrow {D}_{x} \) be a mapping of the open sets \( {D}_{t} \) and \( {D}_{x} \) . Assume that there are subsets \( {S}_{t} \) and \( {S}_{x} \) of measure zero contained in \( {D}_{t} \) and \( {D}_{x} \) respectively such that \( {D}_{t} \smallsetminus {S}_{t} \) and \( {D}_{x} ... | Proof The assertion is a direct corollary of Theorem 1 and Theorem 2 of Sect. 11.5, provided we take account of the fact that when finding an improper integral over an open set one may restrict consideration to exhaustions that consist of measurable compact sets (see Remark 3). | Yes |
Let us compute the integral \( {\iint }_{{x}^{2} + {y}^{2} < 1}\frac{\mathrm{d}x\mathrm{\;d}y}{{\left( 1 - {x}^{2} - {y}^{2}\right) }^{\alpha }} \), which is an improper integral when \( \alpha > 0 \), since the integrand is unbounded in that case in a neighborhood of the disk \( {x}^{2} + {y}^{2} = 1 \) . | Passing to polar coordinates, we obtain from Theorem 2\n\n\[ \n{\iint }_{{x}^{2} + {y}^{2} < 1}\frac{\mathrm{d}x\mathrm{\;d}y}{{\left( 1 - {x}^{2} - {y}^{2}\right) }^{\alpha }} = {\iint }_{\begin{matrix} {0 < \varphi < {2\pi }} \\ {0 < r < 1} \end{matrix}}\frac{r\mathrm{\;d}r\mathrm{\;d}\varphi }{{\left( 1 - {r}^{2}\ri... | Yes |
We recall that if \( {F}^{i} \in {C}^{\left( m\right) }\left( {{\mathbb{R}}^{n},\mathbb{R}}\right), i = 1,\ldots, n - k \), is a set of smooth functions such that the system of equations\n\n\[ \left\{ \begin{array}{l} {F}^{1}\left( {{x}^{1},\ldots ,{x}^{k},{x}^{k + 1},\ldots ,{x}^{n}}\right) = 0, \\ \vdots \\ {F}^{n - ... | Proof We shall verify that if \( S \neq \varnothing \), then \( S \) does indeed satisfy Definition 4. This follows from the implicit function theorem, which says that in some neighborhood of each point \( {x}_{0} \in S \) the system (12.2) is equivalent, up to a relabeling of the variables, to a system\n\n\[ \left\{ \... | Yes |
The cylinder\n\n\[ \n{\left( {x}^{1}\right) }^{2} + \cdots + {\left( {x}^{k}\right) }^{2} = {r}^{2}\;\left( {r > 0}\right) ,\n\]\n\nfor \( k < n \) is an \( \left( {n - 1}\right) \) -dimensional surface in \( {\mathbb{R}}^{n} \) that is the direct product of the \( \left( {k - 1}\right) \) -dimensional sphere in the pl... | A local parametrization of this surface can obviously be obtained if we take the first \( k - 1 \) of the \( n - 1 \) parameters \( \left( {{t}^{1},\ldots ,{t}^{n - 1}}\right) \) to be the polar coordinates \( {\theta }_{1},\ldots ,{\theta }_{k - 1} \) of a point of the \( \left( {k - 1}\right) \) -dimensional sphere i... | Yes |
If we take a curve (a one-dimensional surface) in the plane \( x = 0 \) of \( {\mathbb{R}}^{3} \) endowed with Cartesian coordinates \( \left( {x, y, z}\right) \), and the curve does not intersect the \( z \) -axis, we can rotate the curve about the \( z \) -axis and obtain a 2-dimensional surface. The local coordinate... | In particular, if the original curve is a circle of radius \( a \) with center at \( \left( {b,0,0}\right) \) , for \( a < b \) we obtain the two-dimensional torus (Fig. 12.1). Its parametric equation can be represented in the form\n\n\[ \left\{ \begin{array}{l} x = \left( {b + a\cos \psi }\right) \cos \varphi , \\ y =... | Yes |
Example 6 Comparing the results of Examples 4 and 5 in accordance with the natural analogy, one can now prescribe how to glue a rectangle (Fig. 12.5a) that combines elements of the torus and elements of the Möbius band. But, just as it was necessary to go outside \( {\mathbb{R}}^{2} \) in order to glue the Möbius band ... | An attempt to depict this surface has been undertaken in Fig. 12.5b. | No |
Proposition 1 The mutual transitions from one curvilinear coordinate system to another on a smooth surface \( S \subset {\mathbb{R}}^{n} \) are diffeomorphisms of the same degree of smoothness as the charts of the surface. | Proof In fact, by the proposition in Sect. 12.1, we can regard any chart \( {I}^{k} \rightarrow U \subset \) \( S \) locally as the restriction to \( {I}^{k} \cap O\left( t\right) \) of a diffeomorphism \( \mathcal{F} : O\left( t\right) \rightarrow O\left( x\right) \) from some \( n \) -dimensional neighborhood \( O\le... | Yes |
The Klein bottle is also a nonorientable surface, since it contains a Möbius band. | This last fact can be seen immediately from the construction of the Klein bottle shown in Fig. 12.5. | No |
The two-dimensional torus studied in Example 4 of Sect. 12.1 is also an orientable surface. | Indeed, using the parametric equations of the torus exhibited in Example 4 of Sect. 12.1, one can easily exhibit an orienting atlas for it. | No |
Proposition 2 There exist precisely two orientations on a connected orientable surface. | The proof of Proposition 2 will be given in Sect. 15.2.3. | No |
Proposition 1 The boundary of a \( k \) -dimensional surface of class \( {C}^{\left( m\right) } \) is itself a surface of the same smoothness class, and is a surface without boundary having dimension one less than the dimension of the original surface with boundary. | Proof Indeed, if \( A\left( S\right) = \left\{ \left( {{H}^{k},{\varphi }_{i},{U}_{i}}\right) \right\} \cup \left\{ \left( {{\mathbb{R}}^{k},{\varphi }_{j},{U}_{j}}\right) \right\} \) is an atlas for the surface \( S \) with boundary, then \( A\left( {\partial S}\right) = \left\{ \left( {{\left. {\mathbb{R}}^{k - 1},{\... | Yes |
A closed \( n \) -dimensional ball \( {\bar{B}}^{n} \) in \( {\mathbb{R}}^{n} \) is an \( n \) -dimensional surface with boundary. Its boundary \( \partial {\bar{B}}^{n} \) is the \( \left( {n - 1}\right) \) -dimensional sphere (see Figs. 12.8 and 12.9a). | The ball \( {\bar{B}}^{n} \), which is often called in analogy with the two-dimensional case an \( n \) -dimensional disk, can be homeomorphically mapped to half of an \( n \) -dimensional sphere whose boundary is the equatorial \( \left( {n - 1}\right) \) -dimensional sphere (Fig. 12.9b). | No |
The closed cube \( {\bar{I}}^{n} \) in \( {\mathbb{R}}^{n} \) can be homeomorphically mapped to the closed ball \( \partial {\bar{B}}^{n} \) along rays emanating from its center. | Consequently \( {\bar{I}}^{n} \), like \( {\bar{B}}^{n} \) is an \( n \) -dimensional surface with boundary, which in this case is formed by the faces of the cube (Fig. 12.10). We note that on the edges, which are the intersections of the faces, it is obvious that no mapping of the cube onto the ball can be regular (th... | No |
Example 5 The surface of a three-dimensional cube, as one can easily verify, is an orientable piecewise-smooth surface. | In general, all the piecewise-smooth surfaces exhibited in Example 4 are orientable. | No |
The mapping \( \rbrack 0,{2\pi }\lbrack \ni t \mapsto \left( {R\cos t, R\sin t}\right) \in {\mathbb{R}}^{2} \) is a chart for the arc \( \widetilde{S} \) of the circle \( {x}^{2} + {y}^{2} = {R}^{2} \) obtained by removing the single point \( E = \left( {R,0}\right) \) from that circle. | Since \( E \) is a set of measure zero on \( S \), we can write\n\n\[ \n{V}_{1}\left( S\right) = {V}_{1}\left( \widetilde{S}\right) = {\int }_{0}^{2\pi }\sqrt{{R}^{2}{\sin }^{2}t + {R}^{2}{\cos }^{2}t}\mathrm{\;d}t = {2\pi R}.\n\] | No |
In Example 4 of Sect. 12.1 we exhibited the following parametric representation of the two-dimensional torus \( S \) in \( {\mathbb{R}}^{3} \) :\n\n\[ \n\mathbf{r}\left( {\varphi ,\psi }\right) = \left( {\left( {b + a\cos \psi }\right) \cos \varphi ,\left( {b + a\cos \psi }\right) \sin \varphi, a\sin \psi }\right) .\n\... | Let us carry out the necessary computations:\n\n\[ \n{\dot{\mathbf{r}}}_{\varphi } = \left( {-\left( {b + a\cos \psi }\right) \sin \varphi ,\left( {b + a\cos \psi }\right) \cos \varphi ,0}\right) ,\n\]\n\n\[ \n{\dot{\mathbf{r}}}_{\psi } = \left( {-a\sin \psi }\right) \cos \varphi , - a\sin \psi \sin \varphi, a\cos \psi... | Yes |
Example 3 Let \( {\pi }^{i} \in \mathcal{L}\left( {{\mathbb{R}}^{n},\mathbb{R}}\right), i = 1,\ldots, n \), be the projections. More precisely, the linear function \( {\pi }^{i} : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) is such that on each vector \( \xi = \left( {{\xi }^{1},\ldots ,{\xi }^{n}}\right) \in {\mathbb{R... | \[ {\pi }^{{i}_{1}} \land \cdots \land {\pi }^{{i}_{k}}\left( {{\mathbf{\xi }}_{1},\ldots ,{\mathbf{\xi }}_{k}}\right) = \left| \begin{matrix} {\xi }_{1}^{{i}_{1}} & \cdots & {\xi }_{1}^{{i}_{k}} \\ \vdots & \ddots & \vdots \\ {\xi }_{k}^{{i}_{1}} & \cdots & {\xi }_{k}^{{i}_{k}} \end{matrix}\right| . \] | Yes |
The Cartesian coordinates of the vector product \( \left\lbrack {{\xi }_{1},{\xi }_{2}}\right\rbrack \) of the vectors \( {\xi }_{1} = \left( {{\xi }_{1}^{1},{\xi }_{1}^{2},{\xi }_{1}^{3}}\right) \) and \( {\xi }_{2} = \left( {{\xi }_{2}^{1},{\xi }_{2}^{2},{\xi }_{2}^{3}}\right) \) in the Euclidean space \( {\mathbb{R}... | Thus, in accordance with the result of Example 3 we can write\n\n\[ {\pi }^{1}\left( \left\lbrack {{\xi }_{1},{\xi }_{2}}\right\rbrack \right) = {\pi }^{2} \land {\pi }^{3}\left( {{\xi }_{1},{\xi }_{2}}\right) \]\n\n\[ {\pi }^{2}\left( \left\lbrack {{\xi }_{1},{\xi }_{2}}\right\rbrack \right) = {\pi }^{3} \land {\pi }^... | Yes |
Example 5 Let \( f : D \rightarrow \mathbb{R} \) be a function that is defined in a domain \( D \subset {\mathbb{R}}^{n} \) and differentiable at \( {x}_{0} \in D \) . As is known, the differential \( \mathrm{d}f\left( {x}_{0}\right) \) of the function at a point is a linear function defined on displacement vectors \( ... | \[ \mathrm{d}f\left( {x}_{0}\right) \left( \xi \right) = \frac{\partial f}{\partial {x}^{1}}\left( {x}_{0}\right) {\xi }^{1} + \cdots + \frac{\partial f}{\partial {x}^{n}}\left( {x}_{0}\right) {\xi }^{n} = {D}_{\xi }f\left( {x}_{0}\right) . \] | Yes |
Suppose a vector field \( D \subset {\mathbb{R}}^{n} \) is defined, that is, a vector \( \mathbf{F}\left( x\right) \) is attached to each point \( x \in D \) . When there is a Euclidean structure in \( {\mathbb{R}}^{n} \) this vector field generates the following differential 1-form \( {\omega }_{\mathbf{F}}^{1} \) in ... | If \( \xi \) is a vector attached to \( x \in D \), that is, \( \xi \in T{D}_{x} \), we set\n\n\[ \n{\omega }_{\mathbf{F}}^{1}\left( x\right) \left( \xi \right) = \langle \mathbf{F}\left( x\right) ,\xi \rangle \n\]\n\nIt follows from properties of the inner product that \( {\omega }_{\mathbf{F}}^{1}\left( x\right) = \l... | Yes |
Example 8 A vector field \( \mathbf{V} \) defined in a domain \( D \) of the Euclidean space \( {\mathbb{R}}^{n} \) can also be regarded as a differential form \( {\omega }_{\mathbf{V}}^{n - 1} \) of degree \( n - 1 \) . If at a point \( x \in D \) we take the vector field \( \mathbf{V}\left( x\right) \) and \( n - 1 \... | For \( n = 3 \) the form \( {\omega }_{\mathbf{V}}^{2} \) is the usual scalar triple product \( \left( {\mathbf{V}\left( x\right) ,{\mathbf{\xi }}_{1},{\mathbf{\xi }}_{2}}\right) \) of vectors, one of which \( \mathbf{V}\left( x\right) \) is given, resulting in a skew-symmetric 2-form \( {\omega }_{\mathbf{V}}^{2} = \l... | Yes |
Example 9 For a 0 -form \( \omega = f\left( {x, y, z}\right) - \) a differentiable function - defined in a domain \( D \subset {\mathbb{R}}^{3} \), we obtain | \[ \mathrm{d}\omega = \frac{\partial f}{\partial x}\mathrm{\;d}x + \frac{\partial f}{\partial y}\mathrm{\;d}y + \frac{\partial f}{\partial z}\mathrm{\;d}z. \] | Yes |
Example 10 Let\n\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\nbe a differential 1-form in a domain \( D \) of \( {\mathbb{R}}^{2} \) endowed with coordinates \( \left( {x, y}\right) \) . Assuming that \( P \) and \( Q \) are differentiable in \( D \),... | \[ \n\mathrm{d}\omega \left( {x, y}\right) = \mathrm{d}P \land \mathrm{d}x + \mathrm{d}Q \land \mathrm{d}y = \n\]\n\n\[ \n= \left( {\frac{\partial P}{\partial x}\mathrm{\;d}x + \frac{\partial P}{\partial y}\mathrm{\;d}y}\right) \land \mathrm{d}x + \left( {\frac{\partial Q}{\partial x}\mathrm{\;d}x + \frac{\partial Q}{\... | Yes |
Computing the differential of the 2-form\n\n\[ \omega = P\mathrm{\;d}y \land \mathrm{d}z + Q\mathrm{\;d}z \land \mathrm{d}x + R\mathrm{\;d}x \land \mathrm{d}y \] \n\nwhere \( P, Q \), and \( R \) are differentiable in the domain \( D \subset {\mathbb{R}}^{3} \), leads to the relation | \[ \mathrm{d}\omega = \left( {\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}}\right) \mathrm{d}x \land \mathrm{d}y \land \mathrm{d}z. \] | Yes |
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