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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 \( \mathbf{\xi } \) is a vector attached to \( x \in D \), that is, \( \mathbf{\xi } \in T{D}_{x} \), we set\n\n\[ \n{\omega }_{\mathbf{F}}^{1}\left( x\right) \left( \mathbf{\xi }\right) = \langle \mathbf{F}\left( x\right) ,\mathbf{\xi }\rangle \n\]\n\nIt follows from properties of the inner product that \( {\omega ... | Yes |
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 \( n - 1 \) addition... | 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 |
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 |
Let\n\n\[ \omega \left( {x, y}\right) = P\left( {x, y}\right) \mathrm{d}x + Q\left( {x, y}\right) \mathrm{d}y \]\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 \( D \), by Definition 2... | \n\[ \mathrm{d}\omega \left( {x, y}\right) = \mathrm{d}P \land \mathrm{d}x + \mathrm{d}Q \land \mathrm{d}y = \]\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}{\part... | Yes |
In particular, if \( m = n = p \), relation (12.31) reduces to the equality\n\n\[ \n{\varphi }^{ * }\left( {\mathrm{\;d}{x}^{1} \land \cdots \land \mathrm{d}{x}^{n}}\right) = \det {\varphi }^{\prime }\left( t\right) \mathrm{d}{t}^{1} \land \cdots \land \mathrm{d}{t}^{n}. \n\]\n\n(12.32)\n\nHence, if we write \( f\left(... | We remark in conclusion that if the degree \( p \) of the form \( \omega \) in the domain \( V \subset {\mathbb{R}}_{x}^{n} \) is larger than the dimension \( m \) of the domain \( U \subset {\mathbb{R}}^{m} \) that is mapped into \( V \) via \( \varphi : U \rightarrow V \), then the form \( {\varphi }^{ * }\omega \) o... | Yes |
If the smooth surface \( S \) is contained in the domain \( D \subset {\mathbb{R}}^{n} \) in which a form \( \omega \) is defined, then, since the inclusion \( T{S}_{x} \subset T{D}_{x} \) holds at each point \( x \in S \), one can consider the restriction of \( \omega \left( x\right) \) to \( T{S}_{x} \) . In this way... | As we know, a surface can be defined parametrically, either locally or globally. Let \( \varphi : U \rightarrow S = \varphi \left( U\right) \subset D \) be a parametrized smooth surface in the domain \( D \) and \( \omega \) a form on \( D \) . Then we can transfer the form \( \omega \) to the domain \( U \) of paramet... | Yes |
Let \( {\omega }_{\mathbf{V}}^{2} \) be the flux form considered in Example 8, generated by the velocity field \( \mathbf{V} \) of a flow in the domain \( D \) of the oriented Euclidean space \( {\mathbb{R}}^{3} \). If \( S \) is a smooth oriented surface in \( D \), one may consider the restriction \( {\left. {\omega ... | If \( \varphi : I \rightarrow S \) is a local chart of the surface \( S \), then, making the change of variable \( x = \varphi \left( t\right) \) in the coordinate expression (12.22) for the form \( {\omega }_{\mathbf{V}}^{2} \), we obtain the coordinate expression for the form \( {\varphi }^{ * }{\omega }_{\mathbf{V}}... | No |
Consider the force field \( \mathbf{F} = \left( {-\frac{y}{{x}^{2} + {y}^{2}},\frac{x}{{x}^{2} + {y}^{2}}}\right) \) defined at all points of the plane \( {\mathbb{R}}^{2} \) except the origin. Let us compute the work of this field along the curve \( {\gamma }_{1} \) defined as \( x = \cos t, y = \sin t,0 \leq t \leq {... | According to formulas (13.1),(13.2), and \( \left( {13.2}^{\prime }\right) \), we find\n\n\[ \n{\int }_{{\gamma }_{1}}{\omega }_{\mathbf{F}}^{1} = {\int }_{{\gamma }_{1}} - \frac{y}{{x}^{2} + {y}^{2}}\mathrm{\;d}x + \frac{x}{{x}^{2} + {y}^{2}}\mathrm{\;d}y = \n\]\n\n\[ \n= {\int }_{0}^{2\pi }\left( {-\frac{\sin t \cdot... | Yes |
Let \( \mathbf{r} \) be the radius vector of a point \( \left( {x, y, z}\right) \in {\mathbb{R}}^{3} \) and \( r = \left| \mathbf{r}\right| \) . Suppose a force field \( \mathbf{F} = f\left( r\right) \mathbf{r} \) is defined everywhere in \( {\mathbb{R}}^{3} \) except at the origin. This is a so-called central force fi... | Using (13.2), we find\n\n\[ \n{\int }_{\gamma }f\left( r\right) \left( {x\mathrm{\;d}x + y\mathrm{\;d}y + z\mathrm{\;d}z}\right) = \frac{1}{2}{\int }_{\gamma }f\left( r\right) \mathrm{d}\left( {{x}^{2} + {y}^{2} + {z}^{2}}\right) = \n\]\n\n\[ \n= \frac{1}{2}{\int }_{0}^{1}f\left( {r\left( t\right) }\right) \mathrm{d}{r... | Yes |
Suppose a medium is advancing with constant velocity \( \mathbf{V} = \) \( \left( {1,0,0}\right) \) . If we take any closed surface in the domain of the flow, then, since the density of the medium does not change, the amount of matter in the volume bounded by this surface must remain constant. Hence the total flux of t... | In this case, let us check formula (13.6) by taking \( S \) to be the sphere \( {x}^{2} + {y}^{2} + {z}^{2} = {R}^{2}. \)\n\nUp to a set of area zero, which is therefore negligible, this sphere can be defined parametrically\n\n\[ x = R\cos \psi \cos \varphi \]\n\[ y = R\cos \psi \sin \varphi \]\n\[ z = R\sin \psi \]\n\... | Yes |
Suppose the velocity field of a medium moving in \( {\mathbb{R}}^{3} \) is defined in Cartesian coordinates \( x, y, z \) by the equality \( \mathbf{V}\left( {x, y, z}\right) = \) \( \left( {{V}^{1},{V}^{2},{V}^{3}}\right) \left( {x, y, z}\right) = \left( {x, y, z}\right) \). Let us find the flux through the sphere \( ... | Taking the parametrization of the sphere given in the last example, and carrying out the substitution in the right-hand side of (13.6), we find that\n\n\[ \n{\int }_{0}^{2\pi }\mathrm{d}\varphi {\int }_{-\pi /2}^{\pi /2}\left| \begin{matrix} R\cos \psi \cos \varphi & R\cos \psi \sin \varphi & R\sin \psi \\ - R\cos \psi... | Yes |
The integral \( \left( {13.2}^{\prime }\right) \) of Sect. 13.1, which expresses the work on the path \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow {\mathbb{R}}^{n} \), can be written as the integral of first kind | \[ {\int }_{\gamma }\langle \mathbf{F},\mathbf{e}\rangle \mathrm{d}s \] (13.23) where \( s \) is arc length on \( \gamma \) , \( \mathrm{d}s \) is the element of length (a 1-form), and \( \mathbf{e} \) is a unit velocity vector containing all the information about the orientation of \( \gamma \) . From the point of vie... | Yes |
The flux (13.3) of Sect. 13.1 of the velocity field \( \mathbf{V} \) across a surface \( S \subset {\mathbb{R}}^{n} \) oriented by unit normals \( \mathbf{n}\left( x\right) \) can be written as the surface integral of first kind | \[ {\int }_{\gamma }\langle \mathbf{V},\mathbf{n}\rangle \mathrm{d}\sigma \] | Yes |
Faraday’s law \( {}^{4} \) asserts that the electromotive force arising in a closed conductor \( \Gamma \) in a variable magnetic field \( \mathbf{B} \) is proportional to the rate of variation of the flux of the magnetic field across a surface \( S \) bounded by \( \Gamma \) . Let \( \mathbf{E} \) be the electric fiel... | The circle in the integration sign over \( \Gamma \) is an additional reminder that the integral is being taken over a closed curve. The work of the field over a closed curve is often called the circulation of the field along this curve. Thus by Faraday's law the circulation of the electric field intensity generated in... | Yes |
Proposition 1. Let \( {\mathbb{R}}^{2} \) be the plane with a fixed coordinate grid \( x, y \), and let \( \overline{D} \) be a compact domain in this plane bounded by piecewise-smooth curves. Let \( P \) and \( Q \) be smooth functions in the closed domain \( \bar{D} \) . Then the following relation holds:\n\n\[ \n{\i... | We shall first consider the simplest version of (13.25) in which \( \bar{D} \) is the square \( I = \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} \mid 0 \leq x \leq 1,0 \leq y \leq 1}\right\} \) and \( Q \equiv 0 \) in \( I \) . Then Green’s theorem reduces to the equality\n\n\[ \n{\iint }_{I}\frac{\partial P}{\pa... | Yes |
Let us set \( P = - y, Q = x \) in (13.25). We then obtain\n\n\[ \n{\int }_{\partial D} - y\mathrm{\;d}x + x\mathrm{\;d}y = {\int }_{D}2\mathrm{\;d}x\mathrm{\;d}y = {2\sigma }\left( D\right) ,\n\]\n\nwhere \( \sigma \left( D\right) \) is the area of \( D \) . | Using Green’s formula one can thus obtain the following expression for the area of a domain on the plane in terms of line integrals over the oriented boundary of the domain:\n\n\[ \n\sigma \left( D\right) = \frac{1}{2}{\int }_{\partial D} - y\mathrm{\;d}x + x\mathrm{\;d}y = - {\int }_{\partial D}y\mathrm{\;d}x = {\int ... | Yes |
Let \( \bar{B} = \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} \mid {x}^{2} + {y}^{2} \leq 1}\right\} \) be the closed disk in the plane. We shall show that any smooth mapping \( f : \bar{B} \rightarrow \bar{B} \) of the closed disk into itself has at least one fixed point (that is, a point \( p \in \bar{B} \) suc... | Proof. Assume that the mapping \( f \) has no fixed points. Then for every point \( p \in \bar{B} \) the ray with initial point \( f\left( p\right) \) passing through the point \( p \) and the point \( \varphi \left( p\right) \in \partial B \) where this ray intersects the circle bounding \( \overline{B} \) are uniquel... | Yes |
The law of Archimedes. Let us compute the buoyant force of a homogeneous liquid on a body \( D \) immersed in it. We choose the Cartesian coordinates \( x, y, z \) in \( {\mathbb{R}}^{3} \) so that the \( {xy} \) -plane is the surface of the liquid and the \( z \) -axis is directed out of the liquid. A force \( {\rho g... | \[ \mathbf{F} = {\iint }_{S}{\rho gz}\mathbf{n}\mathrm{d}\sigma \] If \( \mathbf{n} = {\mathbf{e}}_{x}\cos {\alpha }_{x} + {\mathbf{e}}_{y}\cos {\alpha }_{y} + {\mathbf{e}}_{z}\cos {\alpha }_{z} \), then \( \mathbf{n}\mathrm{d}\sigma = {\mathbf{e}}_{x}\mathrm{\;d}y \land \mathrm{d}z + {\mathbf{e}}_{y}\mathrm{\;d}z \lan... | Yes |
Using the Gauss-Ostrogradskii formula (13.30), one can give the following formulas for the volume \( V\left( D\right) \) of a body \( D \) bounded by a surface \( \partial D \) . | \[ V\left( D\right) = \frac{1}{3}{\iint }_{\partial D}x\mathrm{\;d}y \land \mathrm{d}z + y\mathrm{\;d}z \land \mathrm{d}x + z\mathrm{\;d}x \land \mathrm{d}y = \] \[ = {\iint }_{\partial D}x\mathrm{\;d}y \land \mathrm{d}z = {\iint }_{\partial D}y\mathrm{\;d}z \land \mathrm{d}x = {\iint }_{\partial D}z\mathrm{\;d}x \land... | Yes |
Proposition 3. Let \( S \) be an oriented piecewise-smooth compact two-dimensional surface with boundary \( \partial S \) embedded in a domain \( G \subset {\mathbb{R}}^{3} \), in which a smooth \( 1 \) -form \( \omega = P\mathrm{\;d}x + Q\mathrm{\;d}y + R\mathrm{\;d}z \) is defined. Then the following relation holds:\... | Proof. If \( C \) is a standard parametrized surface \( \varphi : I \rightarrow C \) in \( {\mathbb{R}}^{3} \), where \( I \) is a square in \( {\mathbb{R}}^{2} \), relation (13.34) follows from Eqs. (13.28) taking account of what has been proved for the square and Green's formula.\n\nIf the orientable surface \( S \) ... | No |
Proposition 4. Let \( S \) be an oriented piecewise smooth \( k \) -dimensional compact surface with boundary \( \partial S \) in the domain \( G \subset {\mathbb{R}}^{n} \), in which a smooth \( \left( {k - 1}\right) \) -form \( \omega \) is defined.\n\nThen the following relation holds:\n\n\[ \n{\int }_{S}\mathrm{\;d... | Proof. Formula (13.35) can obviously be proved by the same general computations (13.28) and (13.29) as Stokes’ formula (13.34’) provided it holds for a standard \( k \) -dimensional interval \( {I}^{k} = \left\{ {x = \left( {{x}^{1},\ldots ,{x}^{k}}\right) \in {\mathbb{R}}^{k} \mid 0 \leq {x}^{i} \leq }\right. \) \( 1,... | Yes |
Let us show that every smooth mapping \( f : \bar{B} \rightarrow \bar{B} \) of a closed ball \( \bar{B} \subset {\mathbb{R}}^{m} \) into itself has at least one fixed point. | Proof. If the mapping \( f \) had no fixed points, then, as in Example 2, one could construct a smooth mapping \( \varphi : \bar{B} \rightarrow \partial \bar{B} \) that is the identity on the sphere \( \partial \bar{B} \) . In the domain \( {\mathbb{R}}^{m} \smallsetminus 0 \), we consider the vector field \( \frac{\ma... | Yes |
Proposition 1. To a linear combination of forms of the same degree there corresponds a linear combination of the vector and scalar fields corresponding to them. | Proof. Proposition 1 is of course obvious. However, let us write out the full proof, as an example, for 1-forms:\n\n\[ \n{\alpha }_{1}{\omega }_{{\mathbf{A}}_{1}}^{1} + {\alpha }_{2}{\omega }_{{\mathbf{A}}_{2}}^{1} = {\alpha }_{1}\left\langle {{\mathbf{A}}_{1}, \cdot }\right\rangle + \alpha \left\langle {{\mathbf{A}}_{... | Yes |
\[ \operatorname{div}\left( {\mathbf{A} \times \mathbf{B}}\right) = \mathbf{B} \cdot \operatorname{curl}\mathbf{A} - \mathbf{A} \cdot \operatorname{curl}\mathbf{B}. \] | Proof. We shall verify this last equality:\n\n\[ {\omega }_{\operatorname{div}\mathbf{A} \times \mathbf{B}}^{3} = \mathrm{d}{\omega }_{\mathbf{A} \times \mathbf{B}}^{2} = \mathrm{d}\left( {{\omega }_{\mathbf{A}}^{1} \land {\omega }_{\mathbf{B}}^{1}}\right) = \mathrm{d}{\omega }_{\mathbf{A}}^{1} \land {\omega }_{\mathbf... | Yes |
In the case when polar coordinates \( \left( {r,\varphi }\right) \) are defined in the same plane \( {\mathbb{R}}^{2} \), at each point \( x \in {\mathbb{R}}^{2} \smallsetminus 0 \) one can also attach unit vectors \( {\mathbf{e}}_{1}\left( x\right) = \) \( {\mathbf{e}}_{r}\left( x\right) ,{\mathbf{e}}_{2} = {\mathbf{e... | Thus, if \( \left( {{A}^{1},{A}^{2}}\right) \left( x\right) \equiv \left( {1,0}\right) \), this is a field of unit vectors in \( {\mathbb{R}}^{2} \) pointing radially away from the center 0 . The field \( \left( {{A}^{1},{A}^{2}}\right) \left( x\right) \equiv \left( {0,1}\right) \) can be obtained from the preceding fi... | Yes |
In Cartesian coordinates \( \left( {x, y, z}\right) \), cylindrical coordinates \( \left( {r,\varphi, z}\right) \), and spherical coordinates \( \left( {R,\varphi ,\theta }\right) \) on Euclidean space \( {\mathbb{R}}^{3} \) the quadratic form (14.25) has the respective forms | \[ \mathrm{d}{s}^{2} = \mathrm{d}{x}^{2} + \mathrm{d}{y}^{2} + \mathrm{d}{z}^{2} = \] \[ = \mathrm{d}{r}^{2} + {r}^{2}\mathrm{\;d}{\varphi }^{2} + \mathrm{d}{z}^{2} = \] \[ = \mathrm{d}{R}^{2} + {R}^{2}{\cos }^{2}\theta \;\mathrm{d}{\varphi }^{2} + {R}^{2}\;\mathrm{d}{\theta }^{2}\;. \] Thus, each of these coordinate s... | Yes |
It follows from formulas (14.27) and the results of Example 5 that for Cartesian, cylindrical, and spherical coordinates, the triples of unit vectors along the coordinate directions have respectively the following forms: | \[ {\mathbf{e}}_{x} = \left( {1,0,0}\right) ,\;{\mathbf{e}}_{y} = \left( {0,1,0}\right) ,\;{\mathbf{e}}_{z} = \left( {0,0,1}\right) ; \] \[ {\mathbf{e}}_{r} = \left( {1,0,0}\right) ,\;{\mathbf{e}}_{\varphi } = \left( {0,\frac{1}{r},0}\right) ,\;{\mathbf{e}}_{z} = \left( {0,0,1}\right) ; \] \[ {\mathbf{e}}_{R} = \left( ... | Yes |
Example 7. The volume element \( \mathrm{d}V \) in curvilinear coordinates \( {t}^{1},{t}^{2},{t}^{3} \), as we know, has the form\n\n\[ \mathrm{d}V = \sqrt{\det {g}_{ij}}\left( t\right) \mathrm{d}{t}^{1} \land \mathrm{d}{t}^{2} \land \mathrm{d}{t}^{3}. \]\n\nFor a triorthogonal system\n\n\[ \mathrm{d}V = \sqrt{{E}_{1}... | In particular, in Cartesian, cylindrical, and spherical coordinates, respectively, we obtain\n\n\[ \mathrm{d}V = \mathrm{d}x \land \mathrm{d}y \land \mathrm{d}z = \]\n\n\( \left( {14.28}^{\prime }\right) \)\n\n\[ = r\mathrm{\;d}r \land \mathrm{d}\varphi \land \mathrm{d}z = \]\n\n\( \left( {14.28}^{\prime \prime }\right... | Yes |
Using the notation introduced in Example 8 and formulas \( \left( {14.26}^{\prime }\right) ,\left( {14.26}^{\prime \prime }\right) \) and \( \left( {14.26}^{\prime \prime \prime }\right) \), we obtain in Cartesian, cylindrical, and spherical coordinates respectively\n\n\[ \n{\omega }_{\mathbf{B}}^{2} = {B}_{x}\mathrm{\... | \n\n\( \left( {14.32}^{\prime }\right) \)\n\n\[ \n= {B}_{r}r\mathrm{\;d}\varphi \land \mathrm{d}z + {B}_{\varphi }\mathrm{d}z \land \mathrm{d}r + {B}_{z}r\mathrm{\;d}r \land \mathrm{d}\varphi = \n\]\n\n\( \left( {14.32}^{\prime \prime }\right) \)\n\n\[ \n= {B}_{R}{R}^{2}\cos \theta \;\mathrm{d}\varphi \land \mathrm{d}\... | Yes |
If, in particular, \( \operatorname{div}\mathbf{B} \equiv 0 \), that is, there are no sources, then the flux across the boundary of the region must be zero: the amount flowing in equals the amount flowing out. | And, as formula \( \left( {14.42}^{\prime }\right) \) shows, this is indeed the case. | No |
A point electric charge of magnitude \( q \) creates an electric field in space. Suppose the charge is located at the origin. By Coulomb’s law \( {}^{6} \) the intensity \( \mathbf{E} = E\left( x\right) \) of the field at the point \( x \in {\mathbb{R}}^{3} \) (that is, the force acting on a unit test charge at the poi... | The field \( \mathbf{E} \) is defined at all points different from the origin. In spherical coordinates \( \mathbf{E} = \frac{q}{{4\pi }{\varepsilon }_{0}}\frac{1}{{R}^{2}}{\mathbf{e}}_{R} \), so that by formula \( \left( {14.36}^{\prime \prime \prime }\right) \) of the preceding section, one can see immediately that \... | Yes |
Suppose the entire space, regarded as a rigid body, is rotating with constant angular speed \( \omega \) about a fixed axis (let it be the \( x \) -axis). Let us find the curl of the field \( \mathbf{v} \) of linear velocities of the points of space. (The field is being studied at any fixed instant of time.) | In cylindrical coordinates \( \left( {r,\varphi, z}\right) \) we have the simple expression \( \mathbf{v}\left( {r,\varphi, z}\right) = {\omega r}{\mathbf{e}}_{\varphi } \) . Then by formula \( \left( {14.35}^{\prime \prime }\right) \) of Sect. 14.1, we find immediatly that \( \operatorname{curl}\mathbf{v} = {2\omega }... | Yes |
At a point of space having radius-vector \( \mathbf{r} \) the intensity \( \mathbf{F} \) of the gravitational field due to a point mass \( M \) located at the origin can be computed from Newton's law as | \[ \mathbf{F} = - {GM}\frac{\mathbf{r}}{{r}^{3}} \] where \( r = \left| \mathbf{r}\right| \). | Yes |
At a point of space having radius-vector \( \mathbf{r} \) the intensity \( \mathbf{E} \) of the electric field due to a point charge \( q \) located at the origin can be computed from Coulomb's law | \[ \mathbf{E} = \frac{q}{{4\pi }{\varepsilon }_{0}}\frac{\mathbf{r}}{{r}^{3}} \] Thus such an electrostatic field, like the gravitational field, is a potential field. Its potential \( \varphi \) in the sense of physical terminology is defined by the relation \[ \varphi = \frac{q}{{4\pi }{\varepsilon }_{0}}\frac{1}{r}. ... | Yes |
The field \( \mathbf{A} = \left( {x,{xy},{xyz}}\right) \) in Cartesian coordinates in \( {\mathbb{R}}^{3} \) cannot be a potential field, since, for example, \( \frac{\partial {xy}}{\partial x} \neq \frac{\partial x}{\partial y} \) . | since, for example, \( \frac{\partial {xy}}{\partial x} \neq \frac{\partial x}{\partial y} \) | Yes |
Proposition 2. If the necessary condition for a field to be a potential field holds in a ball, then the field has a potential in that ball. | Proof. For the sake of intuitiveness we first carry out the proof in the case of a disk \( D = \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} \mid {x}^{2} + {y}^{2} < r}\right\} \) in the plane \( {\mathbb{R}}^{2} \) . One can arrive at the point \( \left( {x, y}\right) \) of the disk from the origin along two diff... | Yes |
Proposition 3. If the 1-form \( {\omega }_{\mathbf{A}}^{1} \) in the domain \( D \) is such that \( \mathrm{d}{\omega }_{\mathbf{A}}^{1} = 0 \) , and the closed paths \( {\gamma }_{0} \) and \( {\gamma }_{1} \) are homotopic in \( D \), then\n\n\[ \n{\int }_{{\gamma }_{0}}{\omega }_{\mathbf{A}}^{1} = {\int }_{{\gamma }... | Proof. Let \( \Gamma : {I}^{2} \rightarrow D \) be a homotopy from \( {\gamma }_{0} \) to \( {\gamma }_{1} \) (see Fig. 14.4). If \( {I}_{0} \) and \( {I}_{1} \) are the bases of the square \( {I}^{2} \) and \( {J}_{0} \) and \( {J}_{1} \) its vertical sides, then by definition of a homotopy of closed paths, the restri... | Yes |
Proposition 4. If a field \( \mathbf{A} \) defined in a simply connected domain \( D \) satisfies the necessary condition (14.57) or \( \left( {14.57}^{\prime }\right) \) to be a potential field, then it is a potential field in \( D \) . | Proof. By Proposition 1 and Remark 1 it suffices to verify that Eq. (14.59) holds for every smooth path \( \gamma \) in \( D \) . The path \( \gamma \) is by hypothesis homotopic to a constant path whose support consists of a single point. The integral over such a one-point path is obviously zero. But by Proposition 3 ... | Yes |
Consider the plane \( {\mathbb{R}}^{2} \) with two points \( {p}_{1} \) and \( {p}_{2} \) removed (Fig. 14.5), and the paths \( {\gamma }_{0},{\gamma }_{1} \), and \( {\gamma }_{2} \) whose supports are shown in the figure. The path \( {\gamma }_{2} \) can be contracted to a point inside \( D \), and therefore if a clo... | Circles \( {c}_{1} \) and \( {c}_{2} \) enclosing \( {p}_{1} \) and \( {p}_{2} \) serve as a sort of basis in which every closed path \( \gamma \subset D \) has the form \( \gamma = {n}_{1}{c}_{1} + {n}_{2}{c}_{2} \), up to a homotopy, which has no effect on the integral. The quantities \( {\int }_{c}\omega = {T}_{i} \... | Yes |
Let \( {X}^{ * } \) be the dual space to \( X \) (consisting of the linear functionals on \( X \) ) and \( {e}^{1},\ldots ,{e}^{n} \) the basis of \( {X}^{ * } \) dual to the basis \( {e}_{1},\ldots ,{e}_{n} \) in \( X \) , that is, \( {e}^{i}\left( {e}_{j}\right) = {\delta }_{j}^{i} \). | Since \( {e}^{i}\left( x\right) = {e}^{i}\left( {{x}^{j}{e}_{j}}\right) = {x}^{j}{e}^{i}\left( {e}_{j}\right) = {x}^{j}{\delta }_{j}^{i} = {x}^{i} \), taking account of (15.1) and (15.9), we can write any \( k \) -form \( {F}^{k} : {X}^{k} \rightarrow \mathbb{R} \) as\n\n\[ \n{F}^{k} = {a}_{{i}_{1}\ldots {i}_{k}}{e}^{{... | Yes |
Taking account of relations (15.12) and (15.13), we find by (15.10) that\n\n\[ A{F}^{k} = {a}_{{i}_{1}\ldots {i}_{k}}A\left( {{e}^{{i}_{1}} \otimes \cdots \otimes {e}^{{i}_{k}}}\right) ,\n\]\n\nso that it is of interest to find \( A\left( {{e}^{{i}_{1}} \otimes \cdots \otimes {e}^{{i}_{k}}}\right) \) . | From Definition (15.11), taking account of the relation \( {e}^{i}\left( x\right) = {x}^{i} \), we find\n\n\[ A\left( {{e}^{{j}_{1}} \otimes \cdots \otimes {e}^{{j}_{k}}}\right) \left( {{x}_{1},\ldots ,{x}_{k}}\right) = \frac{1}{k!}{e}^{{j}_{1}}\left( {x}_{{i}_{1}}\right) \cdot \ldots \cdot {e}^{{j}_{k}}\left( {x}_{{i}... | Yes |
Example 3. Based on the result (15.14) of Example 2, we find by Definition (15.15) that\n\n\[ \n{e}^{{i}_{1}} \land {e}^{{i}_{2}}\left( {{x}_{1},{x}_{2}}\right) = \frac{2!}{1!1!}A\left( {{e}^{{i}_{1}} \otimes {e}^{{i}_{2}}}\right) \left( {{x}_{1},{x}_{2}}\right) = \n\] | \[ \n= \left| \begin{array}{ll} {e}^{{i}_{1}}\left( {x}_{1}\right) & {e}^{{i}_{2}}\left( {x}_{1}\right) \\ {e}^{{i}_{1}}\left( {x}_{2}\right) & {e}^{{i}_{2}}\left( {x}_{2}\right) \end{array}\right| = \left| \begin{array}{ll} {x}_{1}^{{i}_{1}} & {x}_{1}^{{i}_{2}} \\ {x}_{2}^{{i}_{1}} & {x}_{2}^{{i}_{2}} \end{array}\righ... | Yes |
Let \( {e}^{1},\ldots ,{e}^{m} \) and \( {\widetilde{e}}^{1},\ldots ,{\widetilde{e}}^{n} \) be the bases of the conjugate spaces \( {X}^{ * } \) and \( {Y}^{ * } \) dual to the bases in Example 5. Under the hypotheses of Example 5 we obtain | \[ \left( {{l}^{ * }{\widetilde{e}}^{j}}\right) \left( x\right) = \left( {{l}^{ * }{\widetilde{e}}^{j}}\right) \left( {{x}^{i}{e}_{i}}\right) = {\widetilde{e}}^{j}\left( {{x}^{i}l{e}_{i}}\right) = {x}^{i}{\widetilde{c}}^{j}\left( {{c}_{i}^{k}{\widetilde{e}}_{k}}\right) = \] \[ = {x}^{i}{c}_{i}^{k}{\widetilde{e}}^{j}\le... | Yes |
Example 7. Retaining the notation of Example 6 and taking account of relations (15.22) and (15.29), we now obtain | \[ {l}^{ * }\left( {{\widetilde{e}}^{{j}_{1}} \land \cdots \land {\widetilde{e}}^{{j}_{k}}}\right) = {l}^{ * }{\widetilde{e}}^{{j}_{1}} \land \cdots \land {\widetilde{l}}^{ * }{e}^{{j}_{k}} = \] \[ \left( {{c}_{{i}_{1}}^{{j}_{1}}{e}^{{i}_{1}}}\right) \land \cdots \land \left( {{c}_{{i}_{k}}^{{j}_{k}}{e}^{{i}_{k}}}\righ... | Yes |
The sphere \( {S}^{2} = \left\{ {x \in {\mathbb{R}}^{3}\left| \right| x \mid = 1}\right\} \) is a two-dimensional manifold. | If we interpret \( {S}^{2} \) as the surface of the Earth, then an atlas of geographical maps will be an atlas of the manifold \( {S}^{2} \). | No |
Proposition 1. The boundary \( \partial M \) of an \( n \) -dimensional manifold with boundary \( M \) is an \( \left( {n - 1}\right) \) -dimensional manifold without boundary. | Proof. Indeed, \( \partial {H}^{n} = {\mathbb{R}}^{n - 1} \), and the restriction to \( \partial {H}^{n} \) of a chart of the form \( {\varphi }_{i} : {H}^{n} \rightarrow {U}_{i} \) belonging to an atlas of \( M \) generates an atlas of \( \partial M \) . | Yes |
Consider the planar double pendulum (Fig. 15.2) with arm \( a \) shorter than arm \( b \), both being free to oscillate, except that the oscillations of \( b \) are limited in range by barriers. The configuration of such a system is characterized at each instant of time by the two angles \( \alpha \) and \( \beta \). I... | Under these constraints, the configuration space of the double pendulum is parametrized by the points of the cylinder \( {S}_{\alpha }^{1} \times {I}_{\beta }^{1} \), where \( {S}_{\alpha }^{1} \) is the circle, corresponding to all possible positions of the arm \( a \), and \( {I}_{\beta }^{1} = \{ \beta \in \mathbb{R... | Yes |
Proposition 2. If a manifold \( M \) is connected, it is path connected. | Proof. After fixing a point \( {x}_{0} \in M \), consider the set \( {E}_{{x}_{0}} \) of points of \( M \) that can be joined to \( {x}_{0} \) by a path in \( M \) . The set \( {E}_{{x}_{0}} \), as one can easily verify from the definition of a manifold, is both open and closed in \( M \) . But that means that \( {E}_{... | Yes |
If to each real \( n \times n \) matrix we assign the point of \( {\mathbb{R}}^{{n}^{2}} \) whose coordinates are obtained by writing out the elements of the matrix in some fixed order, then the group \( {GL}\left( {n,\mathbb{R}}\right) \) of nonsingular \( n \times n \) matrices becomes a manifold of dimension \( {n}^... | This manifold is noncompact (the elements of the matrices are not bounded) and nonconnected. This last fact follows from the fact that \( {GL}\left( {n,\mathbb{R}}\right) \) contains matrices with both positive and negative determinants. The points of \( {GL}\left( {n,\mathbb{R}}\right) \) corresponding to two such mat... | Yes |
Let \( \mathbf{a} \) be a vector in \( {\mathbb{R}}^{2} \) and \( {T}_{\mathbf{a}} \) the group of rigid motions of the plane generated by \( \mathbf{a} \) . The elements of \( {T}_{\mathbf{a}} \) are translations by vectors of the form \( n\mathbf{a} \), where \( n \in \mathbb{Z} \) . Under the action of the elements ... | In the present case we can take as a fundamental domain a strip of width \( \left| \mathbf{a}\right| \) bounded by two parallel lines orthogonal to \( \mathbf{a} \) . We need only take into account that these lines themselves are obtained from each other through translations by \( \mathbf{a} \) and \( - \mathbf{a} \) r... | Yes |
Now let \( \mathbf{a} \) and \( \mathbf{b} \) be a pair of orthogonal vectors of the plane \( {\mathbb{R}}^{2} \) and \( {T}_{\mathbf{a},\mathbf{b}} \) the group of translations generated by these vectors. In this case a fundamental domain is the rectangle with sides \( \mathbf{a} \) and \( \mathbf{b} \) . Inside this ... | After gluing the sides of this fundamental rectangle together, we verify that the resulting manifold \( {\mathbb{R}}^{2}/{T}_{\mathbf{a},\mathbf{b}} \) is homeomorphic to the two-dimensional torus. | Yes |
Now consider the group \( {G}_{a, b} \) of rigid motions of the plane \( {\mathbb{R}}^{2} \) generated by the transformations \( a\left( {x, y}\right) = \left( {x + 1,1 - y}\right) \) and \( b\left( {x, y}\right) = \left( {x, y + 1}\right) \) . | A fundamental domain for the group \( {G}_{a, b} \) is the unit square whose horizontal sides are identified at points lying on the same vertical line, but whose vertical sides are identified at points symmetric about the center. Thus the resulting manifold \( {\mathbb{R}}^{2}/{G}_{a, b} \) turns out to be homeomorphic... | No |
An atlas consisting of a single chart can be regarded as having any desired smoothness. Consider in this connection the atlas on the line \( {\mathbb{R}}^{1} \) generated by the identity mapping \( {\mathbb{R}}^{1} \ni x \mapsto \varphi \left( x\right) = x \in {\mathbb{R}}^{1} \), and a second atlas \( - \) generated b... | In particular, if \( \widetilde{\varphi }\left( x\right) = {x}^{3} \), then the atlas consisting of the two charts \( \left\{ {x,{x}^{3}}\right\} \) is not smooth, since \( {\widetilde{\varphi }}^{-1}\left( x\right) = {x}^{1/3} \) . Using what has just been said, we can construct infinitely smooth atlases in \( {\mathb... | Yes |
The one-dimensional manifold \( {\mathbb{{RP}}}^{1} \) called the real projective line, is the pencil of lines in \( {\mathbb{R}}^{2} \) passing through the origin, with the natural notion of distance between two lines (measured, for example, by the magnitude of the smaller angle between them). Each line of the pencil ... | It is useful to keep in mind the following interpretation of the manifold \( {\mathbb{{RP}}}^{1} \) . Each line of the original pencil of lines is completely determined by its intersection with the unit circle. But there are exactly two such points, diametrically opposite to each other. Lines are near if and only if th... | Yes |
If we now consider the pencil of lines passing through the origin in \( {\mathbb{R}}^{3} \), or, what is the same, the set of equivalence classes of ordered triples of points \( \left( {{x}^{1},{x}^{2},{x}^{3}}\right) \) of real numbers that are not all three zero, we obtain the real projective plane \( {\mathbb{{RP}}}... | In the regions \( {U}_{1},{U}_{2} \), and \( {U}_{3} \) where \( {x}^{1} \neq 0,{x}^{2} \neq 0,{x}^{3} \neq 0 \) respectively, we introduce local coordinate systems \( \left( {1,\frac{{x}^{2}}{{x}^{1}},\frac{{x}^{3}}{{x}^{1}}}\right) = \left( {1,{t}_{1}^{2},{t}_{1}^{3}}\right) \sim \left( {{t}_{1}^{2},{t}_{1}^{3}}\righ... | Yes |
Example 14. The set of lines in the plane \( {\mathbb{R}}^{2} \) can be parititioned into two sets: \( U \), the nonvertical lines, and \( V \), the nonhorizontal lines. Each line in \( U \) has an equation of the form \( y = {u}_{1}x + {u}_{2} \), and hence is characterized by the coordinates \( \left( {{u}_{1},{u}_{2... | Every line in the plane has an equation \( {ax} + {by} + c = 0 \) and is characterized by a triple of numbers \( \left( {a, b, c}\right) \), proportional triples defining the same line. For that reason, it might appear that we are again dealing with the projective plane \( {\mathbb{{RP}}}^{2} \) considered in Example 1... | Yes |
Proposition 5. The boundary of an orientable smooth \( n \) -dimensional manifold is an orientable \( \left( {n - 1}\right) \) -dimensional manifold admitting a structure of the same smoothness as the original manifold. | Proof. The proof of Proposition 5 is a verbatim repetition of the proof of the analogous Proposition 2 of Subsect. 12.3.2 for surfaces embedded in \( {\mathbb{R}}^{n} \) . | No |
Corollary 1. If \( M \) is a compact manifold and \( A \) a \( {C}^{\left( k\right) } \) atlas on \( M \), then there exists a finite partition of unity \( \left\{ {{e}_{1},\ldots ,{e}_{l}}\right\} \) on \( M \) subordinate to a covering of the manifold by the ranges of the charts of \( A \) . | Proof. Since \( M \) is compact, the atlas \( A \) can be regarded as finite. We now have the hypotheses of Proposition 6, if we set \( K = M \) in it. | No |
Corollary 2. For every compact set \( K \) contained in a manifold \( M \) and every open set \( G \subset M \) containing \( K \), there exists a function \( f : M \rightarrow \mathbb{R} \) with smoothness equal to that of the manifold and such that \( f\left( x\right) \equiv 1 \) on \( K \) and \( \operatorname{supp}... | Proof. Cover each point \( x \in K \) by a neighborhood \( U\left( x\right) \) contained in \( G \) and inside the range of some chart of the manifold \( M \) . From the open covering \( \{ U\left( x\right), x \in K\} \) of the compact set \( K \) extract a finite covering, and construct a partition of unity \( \left\{... | Yes |
Theorem 1. (Poincaré). Every closed \( \\left( {p + 1}\\right) \) -form \( \\left( {p \\geq 0}\\right) \) on a manifold that is contractible to a point is exact. | Proof. The nontrivial part of the proof consists of the following \ | No |
Let \( A, B \), and \( C \) be smooth real-valued functions of the variables \( x, y, z \in {\mathbb{R}}^{3} \). We ask how to solve the following system of equations for \( P, Q \), and \( R \): \[ \left\{ \begin{array}{l} \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = A, \\ \frac{\partial P}{\partial... | An obvious necessary condition for the consistency of the system (15.54) is that the functions \( A, B \), and \( C \) satisfy the relation \[ \frac{\partial A}{\partial x} + \frac{\partial B}{\partial y} + \frac{\partial C}{\partial z} = 0 \] which is equivalent to saying that the form \[ \omega = A\mathrm{\;d}y \land... | Yes |
Theorem 2. a) The integral of an exact form over a cycle equals zero. | Proof. a) By Stokes’ formula \( {\int }_{z}\omega \mathrm{d}z = {\int }_{\partial z}\omega = 0 \), since \( \partial z = 0 \) . | Yes |
Let \( X = \{ x \in \mathbb{R} \mid x \geq 0\} \) and let the functions \( {f}_{n} : X \rightarrow \mathbb{R} \) be given by the relation \( {f}_{n}\left( x\right) = {x}^{n}, n \in \mathbb{N} \). The convergence set of this sequence of functions is obviously the closed interval \( I = \left\lbrack {0,1}\right\rbrack \)... | \[ f\left( x\right) = \left\{ \begin{array}{ll} 0, & \text{ if }0 \leq x < 1, \\ 1, & \text{ if }x = 1. \end{array}\right. \] | Yes |
Example 4. Consider the sequence \( {f}_{n}\left( x\right) = 2\left( {n + 1}\right) x{\left( 1 - {x}^{2}\right) }^{n} \) on the closed interval \( I = \left\lbrack {0,1}\right\rbrack \) . Since \( n{q}^{n} \rightarrow 0 \) for \( \left| q\right| < 1 \), this sequence tends to zero on the entire closed interval \( I \) ... | Since \( n{q}^{n} \rightarrow 0 \) for \( \left| q\right| < 1 \), this sequence tends to zero on the entire closed interval \( I \). | Yes |
Let \( m, n \in \mathbb{N} \), and let \( {f}_{m}\left( x\right) \mathrel{\text{:=}} \mathop{\lim }\limits_{{n \rightarrow \infty }}{\left( \cos m!\pi x\right) }^{2n} \) . If \( m!x \) is an integer, then \( {f}_{m}\left( x\right) = 1 \), and if \( m!x \notin \mathbb{Z} \), obviously \( {f}_{m}\left( x\right) = 0 \) . | We shall now consider the sequence \( \left\{ {{f}_{m};m \in \mathbb{N}}\right\} \) and show that it converges on the entire real line to the Dirichlet function\n\n\[ \mathcal{D}\left( x\right) = \left\{ \begin{array}{ll} 0, & \text{ if }x \notin \mathbb{Q}, \\ 1, & \text{ if }x \in \mathbb{Q}. \end{array}\right. \]\n\... | Yes |
We know that for any \( x \in \mathbb{R} \)\n\n\[ \sin x = x - \frac{1}{3!}{x}^{3} + \frac{1}{5!}{x}^{5} - \cdots + \frac{{\left( -1\right) }^{m}}{\left( {{2m} + 1}\right) !}{x}^{{2m} + 1} + \cdots ,\] \n\nbut after the examples we have just considered, we understand that the relations \n\n\[ {\sin }^{\prime }x = \math... | Indeed, if the equality \n\n\[ S\left( x\right) = {a}_{1}\left( x\right) + {a}_{2}\left( x\right) + \cdots + {a}_{m}\left( x\right) + \cdots \] \n\nis understood in the sense that \( S\left( x\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{S}_{n}\left( x\right) \), where \( {S}_{n}\left( x\right) = \mathop{\s... | Yes |
Example 7. Let \( {f}_{t}\left( x\right) = {\mathrm{e}}^{-{\left( x/t\right) }^{2}}, x \in X = \mathbb{R}, t \in T = \mathbb{R} \smallsetminus 0 \), and let \( \mathcal{B} \) be the base \( t \rightarrow 0 \) . This family converges on the entire set \( \mathbb{R} \), and | \[ \mathop{\lim }\limits_{{t \rightarrow \infty }}{f}_{t}\left( x\right) = \left\{ \begin{array}{ll} 1, & \text{ if }x = 0, \\ 0, & \text{ if }x \neq 0. \end{array}\right. \] | No |
Let us consider the family of functions \( {f}_{t} : I \rightarrow \mathbb{R} \) defined on the closed interval \( I = \{ x \in \mathbb{R} \mid 0 \leq x \leq 1\} \) and depending on the parameter \( t \in \rbrack 0,1\rbrack \). The graph of the functions \( y = {f}_{t}\left( x\right) \) is shown in Fig. 16.1. It is cle... | In such cases we shall say for convenience that the family converges nonuniformly to the limit function. | Yes |
The sequence of functions \( {f}_{n}\left( x\right) = {x}^{n} - {x}^{2n} \) defined on the closed interval \( 0 \leq x \leq 1 \), as one can see, converges to zero at each point as \( n \rightarrow \infty \) . | To determine whether this convergence is uniform, we find the quantity \( {\Delta }_{n} = \) \( \mathop{\max }\limits_{{0 \leq x \leq 1}}\left| {{f}_{n}\left( x\right) }\right| \) . Since \( {f}_{n}^{\prime }\left( x\right) = n{x}^{n - 1}\left( {1 - 2{x}^{n}}\right) = 0 \) for \( x = 0 \) and \( x = {2}^{-1/n} \) , it ... | Yes |
The sequence of functions \( {f}_{n} = {x}^{n} \) on the interval \( 0 \leq x \leq 1 \) converges to the function\n\n\[ f\left( x\right) = \left\{ \begin{array}{ll} 0, & \text{ if }0 \leq x < 1 \\ 1, & \text{ if }x = 1 \end{array}\right. \] | nonuniformly, since for each \( n \in \mathbb{N} \)\n\n\[ {\Delta }_{n} = \mathop{\sup }\limits_{{0 \leq x \leq 1}}\left| {f\left( x\right) - {f}_{n}\left( x\right) }\right| = \mathop{\sup }\limits_{{0 \leq x < 1}}\left| {f\left( x\right) - {f}_{n}\left( x\right) }\right| =\n\n= \mathop{\sup }\limits_{{0 \leq x < 1}}\l... | Yes |
The sequence of functions \( {f}_{n}\left( x\right) = \frac{\sin {n}^{2}x}{n} \) studied in Example 2 converges to zero uniformly on the entire set \( \mathbb{R} \) as \( n \rightarrow \infty \) | since in this case\n\n\[\n\left| {f\left( x\right) - {f}_{n}\left( x\right) }\right| = \left| {{f}_{n}\left( x\right) }\right| = \left| \frac{\sin {n}^{2}x}{n}\right| \leq \frac{1}{n},\n\]\n\nthat is, \( {\Delta }_{n} \leq 1/n \), and hence \( {\Delta }_{n} \rightarrow 0 \) as \( n \rightarrow \infty \). | Yes |
Earlier we defined the function \( \exp : \mathbb{C} \rightarrow \mathbb{C} \) by the relation\n\n\[ \exp z \mathrel{\text{:=}} \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{1}{n!}{z}^{n} \]\n\n(16.4)\n\nafter first verifying that the series on the right converges for every value \( z \in \mathbb{C} \) . | In the language of Definitions 1-3 one can now say that the series (16.4) of functions \( {a}_{n}\left( z\right) = \frac{1}{n!}{z}^{n} \) converges on the entire complex plane, and the function \( \exp z \) is its sum. | Yes |
Theorem 1. (Cauchy criterion for uniform convergence of a series). The series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}\left( x\right) \) converges uniformly on a set \( E \) if and only if for every \( \varepsilon > 0 \) there exists \( N \in \mathbb{N} \) such that\n\n\[ \left| {{a}_{n}\left( x\right) + \cd... | Proof. Indeed, setting \( {n}_{1} = m,{n}_{2} = n - 1 \) in (16.5) and assuming that \( {s}_{n}\left( x\right) \) is the partial sum of the series, we obtain inequality (16.6), from which relation (16.5) in turn follows with the same notation and hypotheses of the theorem. | No |
Corollary 1. (Necessary condition for uniform convergence of a series). \( A \) necessary condition for the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}\left( x\right) \) to converge uniformly on a set \( E \) is that \( {a}_{n} \rightrightarrows 0 \) on \( E \) as \( n \rightarrow \infty \) . | Proof. This follows from the definition of uniform convergence of a sequence to zero and inequality (16.6) if we set \( m = n \) in it. | No |
The series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{z}^{n}}{n} \) converges in the unit disk \( K = \) \( \{ z \in \mathbb{C}\left| \right| z \mid < 1\} \). | Since \( \left| \frac{{z}^{n}}{n}\right| < \frac{1}{n} \) for \( z \in K \), we have \( \frac{{z}^{n}}{n} \rightrightarrows 0 \) on \( K \) as \( n \rightarrow \infty \) . The necessary condition for uniform convergence is satisfied; however, this series converges nonuniformly on \( K \) . In fact, for any fixed \( n \... | Yes |
Proposition 1. If the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}\left( x\right) \) and \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{b}_{n}\left( x\right) \) are such that \( \left| {{a}_{n}\left( x\right) }\right| \leq \) \( {b}_{n}\left( x\right) \) for every \( x \in E \) and for all sufficiently larg... | Proof. Under these assumptions for all sufficiently large indices \( n \) and \( m \) (let \( n \leq m) \) at each point \( x \in E \) we have\n\n\[ \left| {{a}_{n}\left( x\right) + \cdots + {a}_{m}\left( x\right) }\right| \leq \left| {{a}_{n}\left( x\right) }\right| + \cdots + \left| {{a}_{m}\left( x\right) }\right| \... | Yes |
Corollary 2. (Weierstrass’ \( M \) -test for uniform convergence of a series). If for the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}\left( x\right) \) one can exhibit a convergent numerical series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{M}_{n} \) such that \( \mathop{\sup }\limits_{{x \in E}}\left|... | Proof. The convergent numerical series can be regarded as a series of constant functions on the set \( E \), which by the Cauchy criterion converges uniformly on \( E \). Hence the Weierstrass test follows from Proposition 1 if we set \( {b}_{n}\left( x\right) = {M}_{n} \) in it. | Yes |
Proposition 2. If a power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n}{\left( z - {z}_{0}\right) }^{n} \) converges at a point \( \zeta \neq {z}_{0} \) , then it converges absolutely and uniformly in any disk \( {K}_{q} = \{ z \in \mathbb{C}\left| {z - {z}_{0}}\right| < \) \( \left. {q\left| {\zeta - {z}_... | Proof. By the necessary condition for convergence of a numerical series it follows from the convergence of the series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n}{\left( \zeta - {z}_{0}\right) }^{n} \) that \( {c}_{n}{\left( \zeta - z\right) }^{n} \rightarrow 0 \) as \( n \rightarrow \infty \) . Hence for all su... | Yes |
The radius of convergence of the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{z}^{n}}{{n}^{2}} \) is 1. | But if \( \left| z\right| \leq 1 \) , then \( \left| \frac{{z}^{n}}{{n}^{2}}\right| \leq \frac{1}{{n}^{2}} \), and by the Weierstrass \( M \) -test this series converges absolutely and uniformly in the closed disk \( \bar{K} = \{ z \in \mathbb{C}\left| \right| z \mid \leq 1\} \) . | Yes |
Proposition 3. (The Abel-Dirichlet test for uniform convergence). A sufficient condition for uniform convergence on \( E \) of a series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}\left( x\right) {b}_{n}\left( x\right) \) whose terms are products of complex-valued functions \( {a}_{n} : X \rightarrow \mathbb{C} ... | Proof. The monotonicity of the sequence \( {b}_{n}\left( x\right) \) allows us to to write an estimate analogous to (16.8) for each \( x \in E \) :\n\n\[ \left| {\mathop{\sum }\limits_{{k = n}}^{m}{a}_{k}\left( x\right) {b}_{k}\left( x\right) }\right| \leq 4\mathop{\max }\limits_{{n - 1 \leq k \leq m}}\left| {{A}_{k}\l... | Yes |
Proposition 4. (The so-called second Abel theorem on power series.) If \( a \) power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n}{\left( z - {z}_{0}\right) }^{n} \) converges at a point \( \zeta \in \mathbb{C} \), then it converges uniformly on the closed interval with endpoints \( {z}_{0} \) and \( \zeta... | Proof. We represent the points of this interval in the form \( {z}_{0} + \left( {\zeta - {z}_{0}}\right) t \) , where \( 0 \leq t \leq 1 \) . Substituting this expression in the power series, we obtain the series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n}{\left( \zeta - {z}_{0}\right) }^{n}{t}^{n} \) . By hypo... | Yes |
Theorem 1. Let \( \left\{ {{F}_{t};t \in T}\right\} \) be a family of functions \( {F}_{t} : X \rightarrow \mathbb{C} \) depending on a parameter \( t \) ; let \( {\mathcal{B}}_{X} \) be a base in \( X \) and \( {\mathcal{B}}_{T} \) a base in \( T \) . If the family converges uniformly on \( X \) over the base \( {\mat... | Proof. Since \( {F}_{t} \rightrightarrows F \) on \( X \) over \( {\mathcal{B}}_{T} \), by the Cauchy criterion, for every \( \varepsilon > 0 \) there exists \( {B}_{T} \) in \( {\mathcal{B}}_{T} \) such that\n\n\[ \left| {{F}_{{t}_{1}}\left( x\right) - {F}_{{t}_{2}}\left( x\right) }\right| < \varepsilon \]\n\nfor ever... | Yes |
Theorem 2. Let \( \left\{ {{f}_{t};t \in T}\right\} \) be a family of functions \( {f}_{t} : X \rightarrow \mathbb{C} \) depending on the parameter \( t \) ; let \( \mathcal{B} \) be a base in \( T \) . If \( {f}_{t} \rightrightarrows f \) on \( X \) over the base \( \mathcal{B} \) and the functions \( {f}_{t} \) are c... | Proof. In this case the diagram (16.17) assumes the following specific form:\n\n\n\nHere all the limiting passages except the vertical passage on the right are defined by the hypotheses of Theorem 2 itself. The nontr... | Yes |
Proposition 1. If a power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n}{\left( z - {z}_{0}\right) }^{n} \) converges at a point \( \zeta \), it converges uniformly on the closed interval \( \left\lbrack {{z}_{0},\zeta }\right\rbrack \) from \( {z}_{0} \) to \( \zeta \), and the sum of the series is continu... | In particular, this means that if a numerical series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n} \) converges, then the power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n}{x}^{n} \) converges uniformly on the closed interval \( 0 \leq x \leq 1 \) of the real axis and its sum \( s\left( x\right) = \... | Yes |
We can verify that for \( \alpha > 0 \) the numerical series\n\n\[ 1 + \frac{\alpha }{1!} + \frac{\alpha \left( {\alpha - 1}\right) }{2!} + \cdots + \frac{\alpha \left( {\alpha - 1}\right) \cdots \left( {\alpha - n + 1}\right) }{n!} + \cdots \]\n\nconverges. | Hence by Abel’s theorem, if \( \alpha > 0 \), the series (16.24) converges uniformly on the closed interval \( 0 \leq x \leq 1 \) . But the function \( {\left( 1 + x\right) }^{\alpha } \) is continuous at \( x = 1 \), and so one can assert that if \( \alpha > 0 \), then Eq. (16.24) holds also for \( x = 1 \) . | No |
Proposition 2. (Dini’s \( {}^{2} \) theorem). If a sequence of continuous functions on a compact set converges monotonically to a continuous function, then the convergence is uniform. | Proof. For definiteness suppose that \( {f}_{n} \) is a nondecreasing sequence converging to \( f \) . We fix an arbitrary \( \varepsilon > 0 \), and for every point \( x \) of the compact set \( K \) we find an index \( {n}_{x} \) such that \( 0 \leq f\left( x\right) - {f}_{{n}_{x}}\left( x\right) < \varepsilon \) . S... | Yes |
Corollary 3. If the terms of the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}\left( x\right) \) are nonnegative functions \( {a}_{n} : K \rightarrow \mathbb{R} \) that are continuous on a compact set \( K \) and the series converges to a continuous function on \( K \), then it converges uniformly on \( K ... | Proof. The partial sums \( {s}_{n}\left( x\right) = \mathop{\sum }\limits_{{k = 1}}^{n}{a}_{k}\left( x\right) \) of this series satisfy the hypotheses of Dini's theorem. | No |
Example 3. We shall show that the sequence of functions \( {f}_{n}\left( x\right) = n\left( {1 - {x}^{1/n}}\right) \) tends to \( f\left( x\right) = \ln \frac{1}{x} \) as \( n \rightarrow + \infty \) uniformly on each closed interval \( \left\lbrack {a, b}\right\rbrack \) contained in the interval \( 0 < x < \infty \) ... | Proof. For fixed \( x > 0 \) the function \( {x}^{t} = {\mathrm{e}}^{t\ln x} \) is convex with respect to \( t \), so that the ratio \( \frac{{x}^{t} - {x}^{0}}{t - 0} \) (the slope of the chord) is nonincreasing as \( t \rightarrow + 0 \) and tends to \( \ln x \) .\n\nHence \( {f}_{n}\left( x\right) \nearrow \ln \frac... | Yes |
Theorem 3. Let \( \left\{ {{f}_{t};t \in T}\right\} \) be a family of functions \( {f}_{t} : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) defined on a closed interval \( a \leq x \leq b \) and depending on the parameter \( t \in T \), and let \( \mathcal{B} \) be a base in \( T \) . If the functions of th... | Proof. Let \( p = \left( {P,\xi }\right) \) be a partition \( P \) of the closed interval \( \left\lbrack {a, b}\right\rbrack \) with distinguished points \( \xi = \left\{ {{\xi }_{1},\ldots ,{\xi }_{n}}\right\} \) . Consider the Riemann sums \( {F}_{t}\left( p\right) = \) \( \mathop{\sum }\limits_{{i = 1}}^{n}{f}_{t}\... | Yes |
Corollary 4. If the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{f}_{n}\left( x\right) \) consisting of integrable functions on a closed interval \( \left\lbrack {a, b}\right\rbrack \subset \mathbb{R} \) converges uniformly on that closed interval, then its sum is also integrable on \( \left\lbrack {a, b}\right\... | \[ {\int }_{a}^{b}\left( {\mathop{\sum }\limits_{{n = 1}}^{\infty }{f}_{n}\left( x\right) }\right) \mathrm{d}x = \mathop{\sum }\limits_{{n = 1}}^{\infty }{\int }_{a}^{b}{f}_{n}\left( x\right) \mathrm{d}x. \] | Yes |
We have noted earlier that the function \( \operatorname{Si}\left( x\right) = {\int }_{0}^{x}\frac{\sin t}{t}\mathrm{\;d}t \) is not an elementary function. Using the theorem just proved, we can nevertheless obtain a very simple representation of this function as a power series. | To do this, we remark that\n\n\[ \frac{\sin t}{t} = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{\left( -1\right) }^{n}}{\left( {{2n} + 1}\right) !}{t}^{2n} \]\n\n(16.28)\n\nand the series on the right-hand side converges uniformly on every closed interval \( \left\lbrack {-a, a}\right\rbrack \subset \mathbb{R} \) .... | Yes |
Corollary 5. If the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{f}_{n}\left( x\right) \) of functions \( {f}_{n} : X \rightarrow \mathbb{C} \) that are differentiable on a bounded convex subset \( X \) (contained in \( \mathbb{R},\mathbb{C} \), or any other normed vector space) converges at even one point \( x ... | \[ {\left( \mathop{\sum }\limits_{{n = 1}}^{\infty }{f}_{n}\left( x\right) \right) }^{\prime }\left( x\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{f}_{n}^{\prime }\left( x\right) . \] This follows from Theorem 4 and the definitions of the sum and uniform convergence of a series, together with the linearity of th... | Yes |
Proposition 3. Let \( K \subset \mathbb{C} \) be the convergence disk for a power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{c}_{n}{\left( z - {z}_{0}\right) }^{n} \) . If \( K \) contains more than just the point \( {z}_{0} \), then the sum of the series \( f\left( z\right) \) is differentiable inside \( K \)... | Proof. Since \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\sqrt[{n - 1}]{n\left| {c}_{n}\right| } = \mathop{\lim }\limits_{{n \rightarrow \infty }}\sqrt[n]{\left| {c}_{n}\right| } \), it follows from the Cauchy-Hadamard formula (Theorem 2 of Subsect. 16.2.2 that the power series \( \mathop{\sum }\limits_{{n = 1}}^... | Yes |
The Bessel function \( {J}_{n}\left( x\right), n \in \mathbb{N} \), is a solution of Bessel’s \( {}^{3} \) equation\n\n\[ \n{x}^{2}{y}^{\prime \prime } + x{y}^{\prime } + \left( {{x}^{2} - {n}^{2}}\right) y = 0.\n\] | Let us attempt to solve this equation, for example, for \( n = 0 \), as a power series \( y = \mathop{\sum }\limits_{{k = 0}}^{\infty }{c}_{k}{x}^{k} \) . Applying formula (16.29) successively, after elementary transformations, we arrive at the relation\n\n\[ \n{c}_{1} + \mathop{\sum }\limits_{{k = 0}}^{\infty }\left( ... | Yes |
If the family \( \left\{ {{f}_{\alpha } : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R};\alpha \in A}\right\} \) of differentiable functions \( {f}_{\alpha } \) is such that the family \( \left\{ {{f}_{\alpha }^{\prime };\alpha \in A}\right\} \) of their derivatives is uniformly bounded by a constant, then \(... | as follows from the mean-value theorem, and hence the original family is equicontinuous on the closed interval \( \left\lbrack {a, b}\right\rbrack \) . | Yes |
Lemma 1. Let \( K \) and \( Y \) be metric spaces, with \( K \) compact. A necessary condition for the sequence \( \left\{ {{f}_{n};n \in \mathbb{N}}\right\} \) of continuous functions \( {f}_{n} : K \rightarrow Y \) to converge uniformly on \( K \) is that the family \( \left\{ {{f}_{n};n \in \mathbb{N}}\right\} \) be... | Proof. Let \( {f}_{n} \rightrightarrows f \) on \( K \) . By Theorem 2 of Sect. 16.3, we conclude that \( f \in C\left( {K, Y}\right) \) . It follows from the uniform continuity of \( f \) on the compact set \( K \) that for every \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that \( \left( {{d}_{K}\left( {{... | Yes |
Theorem 1. (Arzelà-Ascoli). Let \( \mathcal{F} \) be a family of functions \( f : K \rightarrow Y \) defined on a compact metric space \( K \) with values in a complete metric space \( Y \) . A necessary and sufficient condition for every sequence \( \left\{ {{f}_{n} \in \mathcal{F};n \in \mathbb{N}}\right\} \) to cont... | Proof. Necessity. If \( \mathcal{F} \) were not a totally bounded family, one could obviously construct a sequence \( \left\{ {{f}_{n};n \in \mathbb{N}}\right\} \) of functions \( {f}_{n} \in \mathcal{F} \) that would not be totally bounded and from which (see the lemma) one could not extract a uniformly convergence su... | No |
Although Theorem 2 still requires a nontrivial proof (given below), one can at least conclude from the uniform continuity of any function \( f \in C\left( {\left\lbrack {a, b}\right\rbrack ,\mathbb{R}}\right) \) that the piecewise-linear continuous real-valued functions on the interval \( \left\lbrack {a, b}\right\rbra... | Proof. We first remark that if \( f, g \in C\left( {\left\lbrack {a, b}\right\rbrack ,\mathbb{R}}\right) ,\alpha \in \mathbb{R} \), and the functions \( f \) and \( g \) admit a uniform approximation (with arbitrary accuracy) by polynomials, then the continuous functions \( f + g, f \cdot g \), and \( {\alpha f} \) als... | Yes |
Example 6. Let \( X \subset \mathbb{C} \) . The polynomials \( P\left( z\right) = {c}_{0} + {c}_{1}z + {c}_{2}{z}^{2} + \cdots + {c}_{n}{z}^{n} \) , \( n \in \mathbb{N} \), obviously form a complex algebra of functions on \( X \) . | If we take \( X = \left\lbrack {a, b}\right\rbrack \subset \mathbb{R} \), and take only polynomials with real coefficients, we obtain a real algebra of functions on the closed interval \( \left\lbrack {a, b}\right\rbrack \) . | Yes |
The real polynomials together form a set of functions that separates the points of every closed interval \( \left\lbrack {a, b}\right\rbrack \), since the polynomial \( P\left( x\right) = x \) does that all by itself. | What has just been said can be repeated for a set \( X \subset \mathbb{C} \) and the set of complex polynomials on \( X \) . As a single separating function, one can take \( P\left( z\right) = z \) . | No |
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