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# empymod, revisited

#### 20 June 2017

My postdoc project around the electromagnetic Python modeller empymod grew bigger than anticipated, and lives on even after the postdoc finished last January. Here a short summary of what happened since my previous blog entry.

## Website

Most importantly, empymod got its own home at empymod.github.io, with its own GitHub repository available under github.com/empymod.

## Features

The feature list grew a lot bigger. Current features:

• Calculates the complete (diffusion and wave phenomena) 3D electromagnetic field in a layered-earth model including vertical transverse isotropic (VTI) resistivity, VTI electric permittivity, and VTI magnetic permeability, for electric and magnetic sources as well as electric and magnetic receivers.
• Modelling routines:
• bipole: arbitrary oriented, finite length bipoles with given source strength; space-frequency and space-time domains.
• dipole: infinitesimal small dipoles oriented along the principal axes, normalized field; space-frequency and space-time domains.
• wavenumber: as dipole, but returns the wavenumber-frequency domain response.
• gpr: calculates the ground-penetrating radar response for given central frequency, using a Ricker wavelet (experimental).
• analytical: interface to the analytical, space-frequency and space-time domain solutions.
• Hankel transforms (wavenumber-frequency to space-frequency transform)
• Fast Hankel transform (includes 9 different filters between 51 and 801 points; own ones can be provided)
• Fourier transforms (space-frequency to space-time transform)
• Sine/Cosine-transforms (includes 5 different filters between 81 and 241 points; own ones can be provided)
• Fast Fourier Transform
• Logarithmic Fast Fourier Transform
• Analytical, space-frequency and space-time domain solutions
• Complete full-space (electric and magnetic sources and receivers); space-frequency domain
• Diffusive half-space (electric sources and receivers); space-frequency and space-time domains
• Direct wave (= diffusive full-space solution)
• Reflected wave
• Airwave (semi-analytical in the case of step responses)

More is planned for the future, have a look at the Roadmap if you are interested.

## Publications

There are two publications regarding empymod. The first one is in the journal Geophysics, about empymod as such:

Werthmüller, D., 2017, An open-source full 3D electromagnetic modeler for 1D VTI media in Python: empymod: Geophysics, 82, WB9-WB19; DOI: 10.1190/geo2016-0626.1.

The second one is a Geophysical Tutorial using empymod as forward modeller in The Leading Edge.

Werthmüller, D., 2017, Getting started with controlled-source electromagnetic 1D modeling: The Leading Edge, 36, 352-355; 10.1190/tle36040352.1.

The second article is Open Access, and freely available. Either way, you find both articles either on my research page or on the empymod project-page if you are interested.

## Logo

Together with its own website and its own GitHub page empymod got as well its own, simple logo in two versions:

## Final words

Three things remained the same: The manual can still be found on empymod.readthedocs.io, and the notebook repository at empymod/example-notebooks contains more and more examples of the usage of empymod. And, of course, it remained open-source under the lax permissive Apache License, Version 2.0.

Please get in touch if you would like to contribute some additional code to empymod or additional examples of its usage, encountered a bug, or simply have a question regarding its usage. Just make a pull-request or open an issue on GitHub. Have a look at the Roadmap if you need an idea on what to work.

format: plain


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Maplets[Examples][LinearAlgebra] - Maple Help

Maplets[Examples][LinearAlgebra]

 VectorNorm
 display a graphic interface to the VectorNorm function

 Calling Sequence VectorNorm(V)

Parameters

 V - Vector

Description

 • The VectorNorm(V) calling sequence displays a Maplet application that returns the vector norm of V.
 • A definition of the vector norm is given in the Maplet application.
 • By using the Norm (Select from list or enter non-negative number) combo box, control the vector norm returned.
 • By using the Vector has real values check box, control the algorithm used to calculate the vector norm. If the vector has real entries, a faster algorithm can be used. The default value is determined by using the value of $\mathrm{has}\left(V,I\right)$, which returns true if V has complex entries. That is, it is selected, by default, only if V has no complex entries.
 Note: If Vector has real values is selected and the vector has any complex entries, an incorrect result may be returned.
 • By using the Evaluate result check box, control whether the Maplet application returns the vector norm of V or the calling sequence required to calculate the vector norm in the worksheet. The default behavior is to evaluate the result, that is, return the norm.
 • The VectorNorm sample Maplet worksheet demonstrates how to write a Maplet application that functions similarly to the Maplet application displayed by this routine.

Examples

 > $\mathrm{with}\left({\mathrm{Maplets}[\mathrm{Examples}]}_{\mathrm{LinearAlgebra}}\right):$
 > $\mathrm{VectorNorm}\left(⟨1,3,2,5⟩\right)$


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# American Institute of Mathematical Sciences

July  2017, 16(4): 1331-1372. doi: 10.3934/cpaa.2017065

## Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat

 1 Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland 2 Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, S. Kaliskiego 2, 00-908 Warsaw, Poland 3 Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland 4 Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology, S. Kaliskiego 2, 00-908 Warsaw, Poland

Received  April 2016 Revised  September 2016 Published  April 2017

A three-dimensional thermo-visco-elastic system for Kelvin-Voigt type material at small strains is considered. The system involves nonlinear temperature-dependent specific heat relevant in the limit of low temperature range. The existence of a unique global regular solution is proved without small data assumptions. The proof consists of two parts. First the existence of a local in time solution is proved by the Banach successive approximations method. Then a lower bound on temperature and global a priori estimates are derived with the help of the theory of anisotropic Sobolev spaces with a mixed norm. Such estimates allow to extend the local solution step by step in time. The paper generalizes the results of the previous author's publication in SIAM J. Math. Anal. 45, No. 4 (2013), pp. 1997–2045.

Citation: Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065
##### References:

show all references

##### References:
 [1] Miroslav Bulíček, Josef Málek, K. R. Rajagopal. On Kelvin-Voigt model and its generalizations. Evolution Equations & Control Theory, 2012, 1 (1) : 17-42. doi: 10.3934/eect.2012.1.17 [2] Irena Pawłow, Wojciech M. Zajączkowski. Unique solvability of a nonlinear thermoviscoelasticity system in Sobolev space with a mixed norm. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 441-466. doi: 10.3934/dcdss.2011.4.441 [3] Louis Tebou. Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7117-7136. doi: 10.3934/dcds.2016110 [4] Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021 [5] Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029 [6] Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485 [7] Xiaobin Yao, Qiaozhen Ma, Tingting Liu. Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1889-1917. doi: 10.3934/dcdsb.2018247 [8] Maria-Magdalena Boureanu, Andaluzia Matei, Mircea Sofonea. Analysis of a contact problem for electro-elastic-visco-plastic materials. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1185-1203. doi: 10.3934/cpaa.2012.11.1185 [9] Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51. [10] Khalid Addi, Oanh Chau, Daniel Goeleven. On some frictional contact problems with velocity condition for elastic and visco-elastic materials. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1039-1051. doi: 10.3934/dcds.2011.31.1039 [11] Roger Grimshaw, Dmitry Pelinovsky. Global existence of small-norm solutions in the reduced Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 557-566. doi: 10.3934/dcds.2014.34.557 [12] Irena Pawłow, Wojciech M. Zajączkowski. Regular weak solutions to 3-D Cahn-Hilliard system in elastic solids. Conference Publications, 2007, 2007 (Special) : 824-833. doi: 10.3934/proc.2007.2007.824 [13] Ping Li, Pablo Raúl Stinga, José L. Torrea. On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 855-882. doi: 10.3934/cpaa.2017041 [14] Vy Khoi Le. On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 809-818. doi: 10.3934/dcdss.2012.5.809 [15] Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209 [16] Shuji Yoshikawa, Irena Pawłow, Wojciech M. Zajączkowski. A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1093-1115. doi: 10.3934/cpaa.2009.8.1093 [17] Michela Eleuteri, Jana Kopfová, Pavel Krejčí. Fatigue accumulation in a thermo-visco-elastoplastic plate. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2091-2109. doi: 10.3934/dcdsb.2014.19.2091 [18] Xie Li, Zhaoyin Xiang. Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1465-1480. doi: 10.3934/cpaa.2014.13.1465 [19] Tomás Caraballo, I. D. Chueshov, Pedro Marín-Rubio, José Real. Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 253-270. doi: 10.3934/dcds.2007.18.253 [20] Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057

2018 Impact Factor: 0.925


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# Documentation

### This is machine translation

Translated by
Mouse over text to see original. Click the button below to return to the English verison of the page.

## Spherical Coordinates

### Support for Spherical Coordinates

Spherical coordinates describe a vector or point in space with a distance and two angles. The distance, R, is the usual Euclidean norm. There are multiple conventions regarding the specification of the two angles. They include:

• Azimuth and elevation angles

• Phi and theta angles

• u and v coordinates

Phased Array System Toolbox™ software natively supports the azimuth/elevation representation. The software also provides functions for converting between the azimuth/elevation representation and the other representations. See Phi and Theta Angles and U and V Coordinates.

### Azimuth and Elevation Angles

In Phased Array System Toolbox software, the predominant convention for spherical coordinates is as follows:

• Use the azimuth angle, az, and the elevation angle, el, to define the location of a point on the unit sphere.

• Specify all angles in degrees.

• List coordinates in the sequence (az,el,R).

The azimuth angle is the angle from the positive x-axis toward the positive y-axis, to the vector's orthogonal projection onto the xy plane. The azimuth angle is between –180 and 180 degrees. The elevation angle is the angle from the vector's orthogonal projection onto the xy plane toward the positive z-axis, to the vector. The elevation angle is between –90 and 90 degrees. These definitions assume the boresight direction is the positive x-axis.

 Note:   The elevation angle is sometimes defined in the literature as the angle a vector makes with the positive z-axis. The MATLAB® and Phased Array System Toolbox products do not use this definition.

This figure illustrates the azimuth angle and elevation angle for a vector that appears as a green solid line. The coordinate system is relative to the center of a uniform linear array, whose elements appear as blue circles.

### Phi and Theta Angles

As an alternative to azimuth and elevation angles, you can use angles denoted by φ and θ to express the location of a point on the unit sphere. To convert the φ/θ representation to and from the corresponding azimuth/elevation representation, use coordinate conversion functions, `phitheta2azel` and `azel2phitheta`.

The φ angle is the angle from the positive y-axis toward the positive z-axis, to the vector's orthogonal projection onto the yz plane. The φ angle is between 0 and 360 degrees. The θ angle is the angle from the x-axis toward the yz plane, to the vector itself. The θ angle is between 0 and 180 degrees.

The figure illustrates φ and θ for a vector that appears as a green solid line. The coordinate system is relative to the center of a uniform linear array, whose elements appear as blue circles.

The coordinate transformations between φ/θ and az/el are described by the following equations

`$\begin{array}{l}\mathrm{sin}\left(\text{el}\right)=\mathrm{sin}\varphi \mathrm{sin}\theta \hfill \\ \mathrm{tan}\left(\text{az}\right)=\mathrm{cos}\varphi \mathrm{tan}\theta \hfill \\ \hfill \\ \mathrm{cos}\theta =\mathrm{cos}\left(\text{el}\right)\mathrm{cos}\left(\text{az}\right)\hfill \\ \mathrm{tan}\varphi =\mathrm{tan}\left(\text{el}\right)/\mathrm{sin}\left(\text{az}\right)\hfill \end{array}$`

### U and V Coordinates

In radar applications, it is often useful to parameterize the hemisphere x ≥ 0 using coordinates denoted by u and v.

You can define u and v in terms of φ and θ:

`$\begin{array}{l}u=\mathrm{sin}\theta \mathrm{cos}\varphi \\ v=\mathrm{sin}\theta \mathrm{sin}\varphi \end{array}$`

In these expressions, φ and θ are the phi and theta angles, respectively.

In terms of azimuth and elevation, the u and v coordinates are

`$\begin{array}{l}u=\mathrm{cos}el\mathrm{sin}az\\ v=\mathrm{sin}el\end{array}$`

The values of u and v satisfy the inequalities

`$\begin{array}{l}-1\le u\le 1\\ -1\le v\le 1\\ {u}^{2}+{v}^{2}\le 1\end{array}$`

Conversely, the phi and theta angles can be written in terms of u and v using

`$\begin{array}{l}\mathrm{tan}\varphi =u/v\\ \mathrm{sin}\theta =\sqrt{{u}^{2}+{v}^{2}}\end{array}$`

The azimuth and elevation angles can also be written in terms of u and v

`$\begin{array}{l}\mathrm{sin}el=v\\ \mathrm{tan}az=\frac{u}{\sqrt{1-{u}^{2}-{v}^{2}}}\end{array}$`

### Conversion between Rectangular and Spherical Coordinates

The following equations define the relationships between rectangular coordinates and the (az,el,R) representation used in Phased Array System Toolbox software.

To convert rectangular coordinates to (az,el,R):

`$\begin{array}{l}R=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}\\ az={\mathrm{tan}}^{-1}\left(y/x\right)\\ el={\mathrm{tan}}^{-1}\left(z/\sqrt{{x}^{2}+{y}^{2}}\right)\end{array}$`

To convert (az,el,R) to rectangular coordinates:

`$\begin{array}{l}x=R\mathrm{cos}\left(el\right)\mathrm{cos}\left(az\right)\\ y=R\mathrm{cos}\left(el\right)\mathrm{sin}\left(az\right)\\ z=R\mathrm{sin}\left(el\right)\end{array}$`

When specifying a target's location with respect to a phased array, it is common to refer to its distance and direction from the array. The distance from the array corresponds to R in spherical coordinates. The direction corresponds to the azimuth and elevation angles.

 Tip   To convert between rectangular coordinates and (az,el,R), use the MATLAB functions `cart2sph` and `sph2cart`. These functions specify angles in radians. To convert between degrees and radians, use `deg2rad` and `rad2deg`.

The special case of the uniform linear arrays (ULA) uses the concept of the broadside angle. The broadside angle is the angle measured from array normal direction projected onto the plane determined by the signal incident direction and the array axis to the signal incident direction. Broadside angles assume values in the interval [–90,90] degrees. The following figure illustrates the definition of the broadside angle.

The shaded gray area in the figure is the plane determined by the signal incident direction and the array axis. The broadside angle is positive when measured toward the positive direction of the array axis. A number of algorithms for ULA's use the broadside angle instead of the azimuth and elevation angles. The algorithms do so because the broadside angle more accurately describes the ability to discern direction of arrival with this geometry.

Phased Array System Toolbox software provides functions `az2broadside` and `broadside2az` for converting between azimuth and broadside angles. The following equation determines the broadside angle, β, from the azimuth and elevation angles, az and el:

`$\beta ={\mathrm{sin}}^{-1}\left(\mathrm{sin}\left(az\right)\mathrm{cos}\left(el\right)\right)$`

Expressing the broadside angle in terms of the azimuth and elevation angles reveals a number of important characteristics, including:

• For an elevation angle of zero degrees, the broadside angle is equal to the azimuth angle.

• Elevation angles equally above and below the xy plane result in identical broadside angles.

The following figure depicts a ULA with elements spaced d meters apart. The ULA is illuminated by a plane wave emitted from a point source in the far field. For convenience, the elevation angle is zero degrees. The plane determined by the signal incident direction and the array axis is the xy plane. The broadside angle reduces to the azimuth angle.

Because of the angle of arrival, the array elements are not simultaneously illuminated by the plane wave. The additional distance the incident wave travels between array elements is d sin(β) where d is the distance between array elements. Therefore, the constant time delay between array elements is:

`$\tau =\frac{d\mathrm{sin}\left(\beta \right)}{c},$`

where c is the speed of the wave.

For broadside angles of ±90 degrees, the plane wave is incident on the array along the array axis and the time delay between sensors reduces to ±d/c. For a broadside angle of 0 degrees, the plane wave illuminates all elements of the ULA simultaneously and the time delay between elements is zero.

The following examples show the use of the utility functions `az2broadside` and `broadside2az`:

A target is located at an azimuth angle of 45 degrees and elevation angle of 60 degrees relative to a ULA. Determine the corresponding broadside angle:

```bsang = az2broadside(45,60) % approximately 21 degrees```

Calculate the azimuth corresponding to a broadside angle of 45 degrees and an elevation of 20 degrees:

```az = broadside2az(45,20) % approximately 49 degrees```


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Is monotonicity a necessary condition for the inverse function theorem?

A textbook I was reading, Introduction To Real Analysis By Robert G. Bartle (page 169) states that the inverse theorem is defined as:

Let $$I$$ be an interval in $$\mathbb{R}$$ and let $$f: I \rightarrow \mathbb{R}$$ be strictly monotone and continuous on $$I$$. Let $$J := f(I)$$ and let $$g: J \rightarrow \mathbb{R}$$ be the strictly montone and continuous function inverse to $$f$$. If $$f$$ is differentiable at $$c \in I$$ and $$f'(c) \neq 0$$, then $$g$$ is differentiable at $$d := f(c)$$ and $$g'(d) = \frac{1}{f'(c)} = \frac{1}{f'(g(d))}$$

However, in the proof given using Caratheodory's theorem in the book, the fact of the functions $$f$$ or $$g$$ being monotone was not used. Also, it seems a bit strange that only this should apply to only monotone functions. I can understand that if $$f$$ is monotone, then $$g$$ is monotone by continuous inverse theorem. But is this really necessary for the inverse function theorem to be used?

• Yes, it is necessary. A continuous function which has an inverse is strictly monotone, so without monotonicity the inverse is undefined. You can generalize by selecting a branch of (multivalued) inverse in a neighborhood of $d$, but this directly reduces to this version by selecting a small enough interval around $c$ where $f$ is strictly monotone. – Conifold Jun 2 at 4:09
• @Conifold So in the cases where $f$ is not strictly montone, if I would like to find the inverse of a point $c$, I need to restrict the interval of $f$ to a neighbourhood of $c$ where $f$ is strictly montone? But wouldn't this be a bit strange cause we know that non monotonic functions e.g. $f(x) = \sin(x)$ is not strictly monotone but yet we know that its inverse $g(y) = \arcsin(y)$ exists? – D. Soul Jun 2 at 4:20
• The $\arcsin(y)$ is not the inverse of $\sin(x)$, it is only the inverse of $\sin(x)$ restricted to $[-\pi/2,\pi/2]$, where it is strictly monotone. This is the "small enough interval" around $0$, say. – Conifold Jun 2 at 4:24
• @Conifold Thank you! – D. Soul Jun 2 at 7:11

If $$f$$ is a continuous strictly monotone function on an interval $$I$$, then its inverse function $$g = f^{-1}$$ is defined on the interval $$J = f(I)$$ and satisfies the relation $$g(f(x)) = x$$ for $$x \in I$$.
What this means, then, is that the inverse of $$f$$ may only be defined if $$f$$ is strictly monotone and continuous. You thus cannot claim the existence of the function $$g$$ in the claim you've provided above unless $$f$$ is strictly monotone and continuous.
• Hi Thanks! But what about functions like $f(x) = \sin (x)$ where it is not monotone but yet we know the inverse $g(y) = \arcsin(y)$ exists? – D. Soul Jun 2 at 4:21
• @D.Soul Per the definition, choose an interval $I$ in which $f$ is strictly monotone. For the $\sin$ example, by convention, we choose the interval $I = [-\pi/2, \pi/2]$, in which $f$ is continuous and strictly monotone, hence the inverse exists. – Clarinetist Jun 2 at 4:23
• @D.Soul over the interval $f(I)$ of course, and not $\mathbb{R}$. – Clarinetist Jun 2 at 4:26


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Peeter Joot's (OLD) Blog.

• 284,856

PHY450H1S. Relativistic Electrodynamics Lecture 24 (Taught by Prof. Erich Poppitz). Non-relativistic electrostatic Lagrangian.

Posted by peeterjoot on March 30, 2011

[Click here for a PDF of this post with nicer formatting]

Covering chapter 5 \S 37, and chapter 8 \S 65 material from the text [1].

Covering pp. 181-195: the Lagrangian for a system of non relativistic charged particles to zeroth order in $(v/c)$: electrostatic energy of a system of charges and .mass renormalization.

A closed system of charged particles.

Consider a closed system of charged particles $(m_a, q_a)$ and imagine there is a frame where they are non-relativistic $v_a/c \ll 1$. In this case we can describe the dynamics using a Lagrangian only for particles. i.e.

\begin{aligned}\mathcal{L} = \mathcal{L}( \mathbf{x}_1, \cdots, \mathbf{x}_N, \mathbf{v}_1, \cdots, \mathbf{v}_N)\end{aligned} \hspace{\stretch{1}}(2.1)

If we work t order $(v/c)^2$.

If we try to go to $O((v/c)^3$, it’s difficult to only use $\mathcal{L}$ for particles.

This can be inferred from

\begin{aligned}P = \frac{2}{3} \frac{e^2}{c^3} {\left\lvert{\dot{d}{\mathbf{d}}}\right\rvert}^2\end{aligned} \hspace{\stretch{1}}(2.2)

because at this order, due to radiation effects, we need to include EM field as dynamical.

Start simple

Start with a system of (non-relativistic) free particles

\begin{aligned}S \end{aligned}

So in the non-relativistic limit, after dropping the constant term that doesn’t effect the dynamics, our Lagrangian is

\begin{aligned}\mathcal{L}(\mathbf{x}_a, \mathbf{v}_a) = \frac{1}{{2}} \sum_a m_a \mathbf{v}_a^2 - \frac{1}{{8}} \frac{m_a \mathbf{v}_a^4}{c^2}\end{aligned} \hspace{\stretch{1}}(3.3)

The first term is $O((v/c)^0)$ where the second is $O((v/c)^2)$.

Next include the fact that particles are charged.

\begin{aligned}\mathcal{L}_{\text{interaction}} = \sum_a \left( \cancel{q_a \frac{\mathbf{v}_a}{c} \cdot \mathbf{A}(\mathbf{x}_a, t)} - q_a \phi(\mathbf{x}_a, t) \right)\end{aligned} \hspace{\stretch{1}}(3.4)

Here, working to $O((v/c)^0)$, where we consider the particles moving so slowly that we have only a Coulomb potential $\phi$, not $\mathbf{A}$.

HERE: these are NOT ‘EXTERNAL’ potentials. They are caused by all the charged particles.

\begin{aligned}\partial_i F^{i l} = \frac{4 \pi}{c} j^l = 4 \pi \rho\end{aligned} \hspace{\stretch{1}}(3.5)

For $l = \alpha$ we have have $4 \pi \rho \mathbf{v}/c$, but we won’t do this today (tomorrow).

To leading order in $v/c$, particles only created Coulomb fields and they only “feel” Coulomb fields. Hence to $O((v/c)^0)$, we have

\begin{aligned}\mathcal{L} = \sum_a \frac{m_a \mathbf{v}_a^2}{2} - q_a \phi(\mathbf{x}_a, t)\end{aligned} \hspace{\stretch{1}}(3.6)

What’s the $\phi(\mathbf{x}_a, t)$, the Coulomb field created by all the particles.

\paragraph{How to find?}

\begin{aligned}\partial_i F^{i 0} = \frac{4 \pi}{c} = 4 \pi \rho\end{aligned} \hspace{\stretch{1}}(3.7)

or

\begin{aligned}\boldsymbol{\nabla} \cdot \mathbf{E} = 4 \pi \rho = - \boldsymbol{\nabla}^2 \phi \end{aligned} \hspace{\stretch{1}}(3.8)

where

\begin{aligned}\rho(\mathbf{x}, t) = \sum_a q_a \delta^3 (\mathbf{x} - \mathbf{x}_a(t))\end{aligned} \hspace{\stretch{1}}(3.9)

This is a Poisson equation

\begin{aligned}\Delta \phi(\mathbf{x}) = \sum_a q_a 4 \pi \delta^3(\mathbf{x} - \mathbf{x}_a)\end{aligned} \hspace{\stretch{1}}(3.10)

(where the time dependence has been suppressed). This has solution

\begin{aligned}\phi(\mathbf{x}, t) = \sum_b \frac{q_b}{{\left\lvert{\mathbf{x} - \mathbf{x}_b(t)}\right\rvert}}\end{aligned} \hspace{\stretch{1}}(3.11)

This is the sum of instantaneous Coulomb potentials of all particles at the point of interest. Hence, it appears that $\phi(\mathbf{x}_a, t)$ should be evaluated in 3.11 at $\mathbf{x}_a$?

However 3.11 becomes infinite due to contributions of the a-th particle itself. Solution to this is to drop the term, but let’s discuss this first.

Let’s talk about the electrostatic energy of our system of particles.

\begin{aligned}\mathcal{E} &= \frac{1}{{8 \pi}} \int d^3 \mathbf{x} \left(\mathbf{E}^2 + \cancel{\mathbf{B}^2} \right) \\ &= \frac{1}{{8 \pi}} \int d^3 \mathbf{x} \mathbf{E} \cdot (-\boldsymbol{\nabla} \phi) \\ &= \frac{1}{{8 \pi}} \int d^3 \mathbf{x} \left( \boldsymbol{\nabla} \cdot (\mathbf{E} \phi) - \phi \boldsymbol{\nabla} \cdot \mathbf{E} \right) \\ &= -\frac{1}{{8 \pi}} \oint d^2 \boldsymbol{\sigma} \cdot \mathbf{E} \phi + \frac{1}{{8 \pi}} \int d^3 \mathbf{x} \phi \boldsymbol{\nabla} \cdot \mathbf{E} \\ \end{aligned}

The first term is zero since $\mathbf{E} \phi$ for a localized system of charges $\sim 1/r^3$ or higher as $V \rightarrow \infty$.

In the second term

\begin{aligned}\boldsymbol{\nabla} \cdot \mathbf{E} = 4 \pi \sum_a q_a \delta^3(\mathbf{x} - \mathbf{x}_a(t))\end{aligned} \hspace{\stretch{1}}(3.12)

So we have

\begin{aligned}\sum_a \frac{1}{{2}} \int d^3 \mathbf{x} q_a \delta^3(\mathbf{x} - \mathbf{x}_a) \phi(\mathbf{x})\end{aligned} \hspace{\stretch{1}}(3.13)

for

\begin{aligned}\mathcal{E} = \frac{1}{{2}} \sum_a q_a \phi(\mathbf{x}_a)\end{aligned} \hspace{\stretch{1}}(3.14)

Now substitute 3.11 into 3.14 for

\begin{aligned}\mathcal{E} = \frac{1}{{2}} \sum_a \frac{q_a^2}{{\left\lvert{\mathbf{x} - \mathbf{x}_a}\right\rvert}} + \frac{1}{{2}} \sum_{a \ne b} \frac{q_a q_b}{{\left\lvert{\mathbf{x}_a - \mathbf{x}_b}\right\rvert}}\end{aligned} \hspace{\stretch{1}}(3.15)

or

\begin{aligned}\mathcal{E} = \frac{1}{{2}} \sum_a \frac{q_a^2}{{\left\lvert{\mathbf{x} - \mathbf{x}_a}\right\rvert}} + \sum_{a < b} \frac{q_a q_b}{{\left\lvert{\mathbf{x}_a - \mathbf{x}_b}\right\rvert}}\end{aligned} \hspace{\stretch{1}}(3.16)

The first term is the sum of the electrostatic self energies of all particles. The source of this infinite self energy is in assuming a \underline{point like nature} of the particle. i.e. We modeled the charge using a delta function instead of using a continuous charge distribution.

Recall that if you have a charged sphere of radius $r$

PICTURE: total charge $q$, radius $r$, our electrostatic energy is

\begin{aligned}\mathcal{E} \sim \frac{q^2}{r}\end{aligned} \hspace{\stretch{1}}(3.17)

Stipulate that rest energy $m_e c^2$ is all of electrostatic origin $\sim e^2/r_e$ we get that

\begin{aligned}r_e \sim \frac{e^2}{m_e c^2}\end{aligned} \hspace{\stretch{1}}(3.18)

This is called the classical radius of the electron, and is of a very small scale $10^{-13} \text{cm}$.

As a matter of fact the applicability of classical electrodynamics breaks down much sooner than this scale since quantum effects start kicking in.

Our Lagrangian is now

\begin{aligned}\mathcal{L}_a = \frac{1}{{2}} m_a \mathbf{v}_a^2 - q_a \phi(\mathbf{x}_a, t)\end{aligned} \hspace{\stretch{1}}(3.19)

where $\phi$ is the electrostatic potential due to all \underline{other} particles, so we have

\begin{aligned}\mathcal{L}_a = \frac{1}{{2}} m_a \mathbf{v}_a^2 - \sum_{a \ne b} \frac{q_a q_b }{{\left\lvert{\mathbf{x}_a - \mathbf{x}_b}\right\rvert}}\end{aligned} \hspace{\stretch{1}}(3.20)

and for the system

\begin{aligned}\mathcal{L} = \frac{1}{{2}} \sum_a m_a \mathbf{v}_a^2 - \sum_{a < b} \frac{q_a q_b }{{\left\lvert{\mathbf{x}_a - \mathbf{x}_b}\right\rvert}}\end{aligned} \hspace{\stretch{1}}(3.21)

This is THE Lagrangian for electrodynamics in the non-relativistic case, starting with the relativistic action.

What’s next?

We continue to the next order of $v/c$ tomorrow.

References

[1] L.D. Landau and E.M. Lifshitz. The classical theory of fields. Butterworth-Heinemann, 1980.


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# IEEE Transactions on Antennas and Propagation

## Filter Results

Displaying Results 1 - 25 of 73

Publication Year: 2012
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• ### IEEE Transactions on Antennas and Propagation publication information

Publication Year: 2012, Page(s): C2
| PDF (138 KB)
• ### A Wideband Magneto-Electric Dipole Antenna

Publication Year: 2012, Page(s):4987 - 4991
Cited by:  Papers (31)
| | PDF (1344 KB) | HTML

A new wideband magneto-electric dipole antenna composed of a horizontal bowtie electric dipole and a vertically oriented folded shorted patch antenna is designed. The antenna is excited by a stair-shaped probe feed without the need of an additional balun. A prototype was fabricated and measured. Results show that an impedance bandwidth of 95.2%, with View full abstract»

• ### A Design of Dual-Frequency Dual-Sense Circularly-Polarized Slot Antenna

Publication Year: 2012, Page(s):4992 - 4997
Cited by:  Papers (19)
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A single-feed dual-frequency dual-sense circularly-polarized (CP) printed slot antenna etched on a single substrate is proposed. This antenna is composed of an annular slot and a cross slot, which produce two different frequency bands. The lower band and the upper band are controlled by the annular slot and the cross slot respectively. By introducing asymmetry on both annular slot and cross slot, ... View full abstract»

• ### A Circularly-Polarized Isoflux Antenna Based on Anisotropic Metasurface

Publication Year: 2012, Page(s):4998 - 5009
Cited by:  Papers (89)
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Theory, design, realization and measurements of an X-band isoflux circularly polarized antenna for LEO satellite platforms are presented. The antenna is based on a metasurface composed by a dense texture of sub-wavelength metal patches on a grounded dielectric slab, excited by a surface wave generated by a coplanar feeder. The antenna is extremely flat (1.57 mm) and light (less than 1 Kg) and repr... View full abstract»

• ### High-Efficiency Elliptical Slot Antennas With Quartz Superstrates for Silicon RFICs

Publication Year: 2012, Page(s):5010 - 5020
Cited by:  Papers (28)  |  Patents (2)
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To increase the efficiency of on-chip patch and slot antennas, a single-layer superstrate is proposed. An analytical model to calculate the radiation efficiency is presented, and a detailed design approach is described. This antenna was implemented at W-band in the IBM CMOS8RF (0.13 $mu {rm m}$) process. The measured antenna result... View full abstract»

• ### Compact Wideband Multimode Dielectric Resonator Antennas Fed With Parallel Standing Strips

Publication Year: 2012, Page(s):5021 - 5031
Cited by:  Papers (19)
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As the number of resonances increases, it becomes difficult to improve and maintain the performance of dielectric resonator antennas (DRAs), over the expanded impedance bandwidth. To remove unwanted modes, adjust the frequency distance between individual modes, reduce antenna size and cross polarization, and preserve radiation patterns in a wideband configuration, a dielectric resonator antenna fe... View full abstract»

• ### Superior Modes in High Permittivity Cylindrical Dielectric Resonator Antenna Excited by a Central Rectangular Slot

Publication Year: 2012, Page(s):5032 - 5038
Cited by:  Papers (12)
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High permittivity materials allow good miniaturization of microwave components, but the dielectric resonator antenna using these materials has the drawback of a reduced bandwidth. To overcome this disadvantage we wanted to obtain a relatively wideband high permittivity DRA using the superior modes. Using HFSS simulations, we identified some of the superior modes that appear in cylindrical DRAs sit... View full abstract»

• ### Circular/Linear Polarization Reconfigurable Antenna on Simplified RF-MEMS Packaging Platform in K-Band

Publication Year: 2012, Page(s):5039 - 5045
Cited by:  Papers (22)
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A linear-polarization (LP)/circular-polarization (CP) switchable reconfigurable antenna using a radio frequency micro-electro-mechanical-system (RF-MEMS) switch is proposed and a novel package platform is introduced. Since the package substrate of the novel package platform is used as a radiation aperture substrate of the antenna structure, the fabrication process can be simplified, and degradatio... View full abstract»

• ### Electronically Steerable 1-D Fabry-Perot Leaky-Wave Antenna Employing a Tunable High Impedance Surface

Publication Year: 2012, Page(s):5046 - 5055
Cited by:  Papers (51)
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A novel fixed-frequency electronically-steerable one-dimensional (1-D) leaky-wave antenna is presented. The antenna is based on a parallel-plate waveguide loaded with a planar partially reflective surface and a tunable high impedance surface (HIS), which creates a 1-D Fabry-Perot leaky-waveguide. The tunable HIS consists of printed patches loaded with varactor diodes that allow the electronic tuni... View full abstract»

• ### Multilayered Composite Right/Left-Handed Leaky-Wave Antenna With Consistent Gain

Publication Year: 2012, Page(s):5056 - 5062
Cited by:  Papers (29)
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A multilayered composite right/left-handed (CRLH) based leaky-wave antenna is proposed for the continuous beam scanning from backward to forward direction with consistent gain. The antenna comprises a substrate integrated waveguide (SIW) with an array of the CRLH unit cells. The CRLH unit cell is formed by a slot cut on the upper ground of the SIW, and a parasitic patch beneath the slot. The anten... View full abstract»

• ### Broadband High Directivity Multibeam Emission Through Transformation Optics-Enabled Metamaterial Lenses

Publication Year: 2012, Page(s):5063 - 5074
Cited by:  Papers (24)
| | PDF (2339 KB) | HTML

A new broadband two- and three-dimensional, polarization independent coordinate transformation is introduced that is capable of mapping the radiation from an embedded omnidirectional source into any desired number of highly directive beams pointed in arbitrary directions. This transformation requires anisotropic materials, yet is spatially invariant and thereby can be readily implemented by curren... View full abstract»

• ### Printed Magnetoinductive-Wave (MIW) Delay Lines for Chipless RFID Applications

Publication Year: 2012, Page(s):5075 - 5082
Cited by:  Papers (18)
| | PDF (1211 KB) | HTML

A novel fully passive and electromagnetic chipless radiofrequency identification (RFID) system is proposed. The system is based on printed tags implemented with magnetoinductive-wave (MIW) delay lines. Such lines are composed of a periodic array of coupled square split ring resonators (SSRRs) and propagate slow waves. The tag is codified by introducing reflectors (which provide the identification ... View full abstract»

• ### Dielectric Tube Antennas for Industrial Radar Level Gauging

Publication Year: 2012, Page(s):5083 - 5091
Cited by:  Papers (4)
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This paper concerns dielectric tube antenna design in the frequency range from 8.5 to 10.5 GHz for the use in the field of industrial radar level gauging for process automation industry. The application is based on monostatic radar configurations, which thus demands for highly directive endfire antennas made of chemically inert dielectrics, such as polytetrafluoroethylene (PTFE) or polypropylene (... View full abstract»

• ### Polya Elements With Application to Antennas, Thin Absorbers and Filters

Publication Year: 2012, Page(s):5092 - 5099
Cited by:  Papers (1)
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This article introduces the 2-D Polya fractal and its evolution into the modified Polya curve (MPC). The MPC is assigned a linewidth, which, together with the MPC order and shaping factor, provides optimization parameters for specific device applications. The MPCs are selected to generate useful shapes that fit within a chosen triangular envelope. These are applied as basis elements and are employ... View full abstract»

• ### Multi-Port Network Approach for the Analysis of Dual Band Fractal Microstrip Antennas

Publication Year: 2012, Page(s):5100 - 5106
Cited by:  Papers (11)
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The multiport network approach is extended to analyze the behavior of microstrip fractal antennas. The capacitively fed microstrip square ring antenna has the side opposite to the feed arm replaced with a fractal Minkowski geometry. Dual frequency operation is achieved by suitably choosing the indentation of this fractal geometry. The width of the two sides adjacent to this is increased to further... View full abstract»

• ### Efficient Combined Array Thinning and Weighting for Pattern Synthesis With a Nested Optimization Scheme

Publication Year: 2012, Page(s):5107 - 5117
Cited by:  Papers (6)
| | PDF (2332 KB) | HTML

A novel procedure to thin an antenna array which synthesizes a desired pattern with the minimum number of active elements is introduced. The proposed method yields both the active elements and their corresponding excitations of a thinned array having the minimum number of active elements needed to meet several prescribed design specifications of the radiated far-field pattern. Specifications such ... View full abstract»

• ### Novel Dual-Band Decoupling Network for Two-Element Closely Spaced Array Using Synthesized Microstrip Lines

Publication Year: 2012, Page(s):5118 - 5128
Cited by:  Papers (21)  |  Patents (1)
| | PDF (2326 KB) | HTML

We investigate an innovative decoupling network for dual-band two-element closely spaced arrays in MIMO applications. The new architecture, combining the reactive element decoupling technique and the eigenmode feed network, provides dual-band decoupling with good radiation characteristics and power balance between the modes. The core block of the decoupling network is a dual-mode 180-degree hybrid... View full abstract»

• ### Effective Beam Forming Networks for Large Arbitrary Array of Antennas

Publication Year: 2012, Page(s):5129 - 5135
Cited by:  Papers (1)
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A deterministic and effective strategy to reduce as much as possible the number of amplitude controls points in large arrays is herein proposed, and applied to cases of practical interest. The proposed strategy exploits two important concepts: the first one, coming from a special class of aperiodic tiling, and the second one based on NuFFT – Non uniform Fast Fourier Transform – techn... View full abstract»

• ### Broadband Multiple Cone-Beam 3-D IIR Digital Filters Applied to Planar Dense Aperture Arrays

Publication Year: 2012, Page(s):5136 - 5146
Cited by:  Papers (16)
| | PDF (3136 KB) | HTML

A digital beamformer that can synthesize array factors (AFs) with multiple, ultrawideband (UWB), frequency-independent beams at lower computational complexity is proposed. The beamformer is based on a novel 3-D infinite impulse response (IIR) transfer function $H_{rm MC}left({bf z}right)$ having multiple cone-shaped passbands in the... View full abstract»

• ### Efficiency Measurement of Connected Arrays Using the Improved Wheeler Cap Method

Publication Year: 2012, Page(s):5147 - 5156
Cited by:  Papers (4)
| | PDF (2202 KB) | HTML

We present a method for measuring antenna efficiency of connected arrays. The method is a modified version of the improved Wheeler cap (IWC) method where the antenna under test (AUT) is electrically connected to two walls of the Wheeler Cap which simulates array connection. Theoretical analyses based on plane wave and cylindrical wave problems and numerical simulations using HFSS and FEKO are pres... View full abstract»

• ### Comparison of Correction Techniques and Analysis of Errors for Digital Beamforming Antenna Array With Single RF Receiver

Publication Year: 2012, Page(s):5157 - 5163
Cited by:  Papers (5)
| | PDF (1039 KB) | HTML

A new correction method is proposed for the single RF channel digital beamforming (DBF) antenna array. In this array, each array element is followed by a $0/pi$ phase shifter, and the element outputs can be recovered by the single channel through the time sequence phase weighting (TSPW) technique. The outputs can be used to form des... View full abstract»

• ### Towards Experimental Perfectly-Matched Layers With Ultra-Thin Metamaterial Surfaces

Publication Year: 2012, Page(s):5164 - 5172
Cited by:  Papers (20)
| | PDF (2015 KB) | HTML

In this paper, we modify uniaxial perfectly matched layer to an entirely passive medium and demonstrate its experimental realization using a deep sub-wavelength metamaterial surface. The relative permittivity and permeability of the surface are matched, mostly imaginary-valued, and far greater than unity. These conditions allow the impedance of the metamaterial surface to be completely matched to ... View full abstract»

• ### Use of Generalized Sheet Transition Conditions to Model Guided Waves on Metasurfaces/Metafilms

Publication Year: 2012, Page(s):5173 - 5186
Cited by:  Papers (26)
| | PDF (3440 KB) | HTML

We use generalized sheet transition conditions (GSTCs) to investigate the existence of guided waves (both surface waves and complex modes) on a metafilm: a surface distribution of electrically small scatterers characterized by electric and magnetic surface susceptibilities. In this paper, excitation of a metafilm by both electric and magnetic line currents is investigated. The characteristics of t... View full abstract»

• ### Iterative On-Surface Discretized Boundary Equation Method for 2-D Scattering Problems

Publication Year: 2012, Page(s):5187 - 5194
| | PDF (1041 KB) | HTML

The recently developed on-surface discretized boundary equation (OS-DBE) method has low memory requirement and is very suitable for parallel computing because the current at each point can be independently evaluated with a matrix of much smaller order than that in the method of moments (MoM) for electrically large objects. However, repeated solutions of the matrix equation in generating the whole ... View full abstract»

## Aims & Scope

IEEE Transactions on Antennas and Propagation includes theoretical and experimental advances in antennas.

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## Meet Our Editors

Editor-in-Chief                                             Danilo Erricolo


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UPSKILL MATH PLUS

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In $$\triangle ABC$$, $$AB =$$ 2 $$cm$$, $$BC =$$ 8 $$cm$$, and $$AC =$$ 13 $$cm$$. Using the given conditions, say whether a triangle is formed or not.


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# why does contour lines intersect at saddle points?

Consider functions of two variables. A saddle point of $$f(x,y)$$ is a point in the domain of $$f$$ (or on the graph of $$f$$ by an abuse of language) where the gradient is $$\bf 0$$, but which is not a local extremum of the function. A classic example is $$(0,0)$$ for $$f(x,y)=y^2-x^2$$. However, there are two types of saddle points.

Type I: the graph looks like a saddle.

Type II: the graph does not look like a saddle. e.g. $$f(x,y)=x^3$$, and $$f(x,y)=x^2+y^3$$.

I have two questions:

Question 1: How to rigorously define saddle points of the above two types?

Question 2: A folklore theorem says $$P$$ is a saddle point of type I if and only if the contour lines (aka level curves) intersect at $$P$$. I don't find a proof of such a result, and I'm not sure if the theorem holds as "if and only if" or just holds in one direction.

• One natural classification would be to declare a saddle point $p$ of $f$ to be of type I if it is a nondegenerate critical point, that is, $\det \operatorname{Jac}(f)(p) \neq 0$ and type II if it is degenerate, but that definition would make both directions of the folklore theorem false and would exclude examples like $g(x, y) = x^4 - y^4$. As C. Falcon mentions, what you might be looking for is a special case of Morse's Theorem. Mar 10 at 23:35

A reasonable definition for saddle points of type I is the following:

Let $$f\colon\mathbf{R}^2\to\mathbf{R}$$ be a smooth function, a saddle point of type I of $$f$$ is a point $$p\in\mathbf{R}^2$$ such that $$\frac{\partial f}{\partial x}(p)=0=\frac{\partial f}{\partial y}(p)$$ and $$\begin{bmatrix}\frac{\partial^2 f}{\partial x^2}(p) & \frac{\partial^2 f}{\partial x\partial y2}(p)\\\frac{\partial^2 f}{\partial y\partial x}(p) & \frac{\partial^2 f}{\partial y^2}(p)\end{bmatrix}$$ has nonpositive determinant.

Regarding the saddle points of type II, you can always say that the previous matrix has zero determinant (it cannot have nonnegative determinant, otherwise $$p$$ would be a maximum or a minimum). However, from the differential topology view point, this is not such a great definition, since the level sets around such critical points can be very different looking.

Using a standard result called Morse's lemma, around a saddle point of type I, there exist local coordinates such that $$f(x,y)=x^2-y^2$$, in this local model, it is clear that the level curves $$f^{-1}(0)$$ intersects at $$(0,0)$$. The reverse implication is false, a counter example is given by $$f(x,y)=x^3-3xy^2$$ for which $$(0,0)$$ is a saddle point of type II, even though $$f^{1}(0)$$ consists of three transversally intersecting curves.

• In the case of $0$ determinant, it needn’t be a saddle point at all. You really need higher order Taylor info. Mar 10 at 21:45
• @TedShifrin I agree, this was my point saying that it is not a great definition, but perhaps I was unclear (I wrote type I instead of type II). I found the OP definition for saddle points way too wide. Mar 10 at 22:29
• @TravisWillse You are right for both points, thank you! I forgot the $y$-axis as part of the $0$-level set of the monkey saddle, it was late here. However, it still provides a counterexamples to the OP's statement. Mar 10 at 23:01
• I'm still somewhat confused by the last paragraph: Since $f(x, y) = x^3 - 3 x y^2$ has zero Jacobian at the origin (it is a polynomial only with terms of degree $\geq 3$), so $\det \operatorname{Jac}(f)(0, 0) = 0$ and hence that saddle point has (in the sense of the definition you proposed) type I. Mar 10 at 23:28
• @TravisWillse It does not have type I, since the Hessian at the origin is degenerate, this was precisely my point. If the level sets intersect transversally, then the critical point not necessarily has type I. Nonetheless, I wrote it has "type I" instead of "type II" in my answer as I can't get my head around this terminology. Mar 11 at 1:19


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# In the figure $\vartriangle ABC$ and $\vartriangle ABD$ are two triangles on the same base AB. If line segment CD is bisected by AB at O. Show that $area\left( {\vartriangle ABC} \right) = area\left( {\vartriangle ABD} \right)$

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Hint: Here we apply the area of triangle theorem where median divides the triangle in two equal areas.

In $\vartriangle ABC$
So,$area\left( {\vartriangle AOC} \right) = area\left( {\vartriangle AOD} \right)$ …… (1)
Also, in $\vartriangle BCD$
So, $area\left( {\vartriangle BOC} \right) = area\left( {\vartriangle BOD} \right)$ …… (2)
$area\left( {\vartriangle AOC} \right) + area\left( {\vartriangle BOC} \right) = area\left( {\vartriangle AOD} \right) = area\left( {\vartriangle BOD} \right)$
$\Rightarrow area\left( {\vartriangle ABC} \right) = area\left( {\vartriangle ABD} \right)$


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If they are moving in the same direction but at different speeds, then one person will appear to be moving slower than they are. It means that it two trains are moving relative to one another, it is impossible to tell which is âreallyâ moving, unless there is a background which is deemed to be stationary. Then the relative speed of the second person to the first person will be equal to: a-b: This is defined as the relative speed of a with respect to b. Ans- The itemâs relative speed in a different direction is given by: Let us say a person is traveling from Bombay to Pune, and the second person is traveling from Pune to Bombay in opposite directions. We do all calculations according to the reference points. We introduce relative motion in one dimension first, because the velocity vectors simplify to having only two possible directions. This value is not to exceed 5 m/s, so the maximum rotation rate of the crank wheel is w â¦ So the frame of movement must be recognized to precisely describe speed. In previous videos, we have already seen how to calculate relative velocity in 1D. And the speed of the second body is y km/hr. Relative density also known as specific gravity has no unit, no dimensions. For example, a boat crosses a river that is flowing at some rate or an airplane encountering wind during its motion. It is how fast an item is moving. If both the person moves towards each other, then they are moving faster than normal.Â. Relative Velocity Relative Motion. The special theory of relativity is based on the principle of special relativity, which states that all observers moving at constant velocities with respect to each other should find the same laws of nature operating in their frames of reference. Usually, a moving object is the frame of reference. Relative Motion in One Dimension. The well-known American author, Bill Bryson, once said: âPhysics is really nothing more than a search for ultimate simplicity, but so far all we have is a kind of elegant messiness.â Physics is indeed the most fundamental of the sciences that tries to describe the whole â¦ If youâre riding in a train moving at 10 m/s east, this velocity is measured relative to the ground on which youâre traveling. Let the speed of the first body be: x km/hr. I could leave it at that, but thatâs boring, so hereâs an example. 1. All motion, and even time, is relative to an observer. A person that is walking forward at 4km/h on a ship will be seen as moving faster by an observer on the shore. Relativity, wide-ranging physical theories formed by the German-born physicist Albert Einstein. So, their relative speed is â¦ She has taught science courses at the high school, college, and graduate levels. Imagine a ball on a conveyor belt. How do we calculate the velocity of one object as seen by another object? Here are a few problems on trains for quick solutions of train-based questions: Remember some crucial formulas of train problems for quick solutions. Youâve probably heard the saying âmotion is relative.â Or perhaps youâve heard people speak about Einsteinâs Theory of General Relativity and Einsteinâs Theory of Special Relativity. The sliding velocity V rel of the pin relative to the right side of the slot is then V rel = wd/2 + wRcosÎ¸. Suppose bodies or two trains are moving in the same direction at u m/s & v m/s, where u>v then the relative speed in the same direction = (u - v) m/s. Consider two trains of length A meters and B meters going in a similar direction at u m /s & v m/ s, then the time take by the faster train to cross the slower train. So, their relative speed is equal to (x â y) km/hr [x > y], The time after which both bodies meet = distance traveled / relative speed. The relative velocity is the velocity of an object or observer B in the rest frame of another object or the observer A. Write the position and velocity vector equations for relative motion. This is the centre-of-mass (or centre-of-momentum) frame mentioned earlier. Now, we know that the relative speed is the speed of one object with respect to another. BOOK FREE CLASS COMPETITIVE EXAMS B.A., Physics and Mathematics, Hastings College Dr. Helmenstine holds a Ph.D. in biomedical sciences and is a science writer, educator, and consultant. Relative velocity is defined as the velocity of an object from the reference point of another object. By the end of this section, you will be able to: Explain the concept of reference frames. Special relativity is limited to objects that are moving with respect to inertial frames of reference. Hydrostatics: Relative Humidity The amount of water vapor in the air at any given time is usually less than that required to saturate the air. The theory predicts that moving clocks will appear to run slower than stationary ones (see time dilation), that moving objects will appear shorter and heavier than stationary oâ¦ There is no such thing as fixed points; hence we need to look at the frames. If you are lying still on the ground, so are you in motion, or are you at rest? When two objects are moving in the same direction, then A collision between two bodies can always be described in a frame of reference in which the total momentum is zero. Suppose two people are traveling in a train or the first person is on the train; the second person is traveling through a car, but they are traveling toward the same destination. Learning Objectives. The conventional method to calculate a constant velocity of an object moving in a straight line is mentioned in the below formula. Relative density has no unit because it is the ratio of same units which gets cancelled. Take the example of the person sitting in a train moving east. When immiscible liquids of different densities are poured in a container, the liquid of highest density will be at the bottom while that of lowest density at the top and interfaces will be plane. 5. r is rate, or speed (sometimes denoted as v, for velocity), t is the time taken to complete the motion. Speed doesn't have a direction. For instance, an athlete running from the starting line is said to cover a length that is relative to the point where he began the race. Suppose two bodies or two trains are moving in the opposite direction at u m/s and v m/s, then their relative's speed = (u + v) m/s. Created by Mahesh Shenoy. Search for: 4.5 Relative Motion in One and Two Dimensions. Then we can determine the relative speed as (a+b) or the sum of two velocities. This occurs because the speed of the ship is added to the person's speed when the situation is viewed relative to the shore. In Einstein's theory of relativity, time dilation describes a difference of elapsed time between two events, as measured by observers that are either moving relative to each other, or differently, depending on their proximity to a gravitational mass. Relative density is defined as the ratio of density of the substance to the standarnd which is either air for gases and water for liquids or solids. Analyze one-dimensional and two-dimensional relative motion problems using the position and velocity vector equations. Let the speed of the first body be: x km/hr. Consider two bodies (or trains) starting at similar time from points A and B in the directionÂ  of each other and after crossing they take a and b sec in reaching B and A respectively, then (Aâs speed): (B's speed) =(âb:âa). Specific gravity for liquids is nearly always measured with respect to water at its densest (at 4 °C or 39.2 °F); for gases, the reference is air at room temperature (20 °C or 68 °F). We encounter occasions where one or more objects move in a frame which is non-stationary with respect to another observer. Here we will study the fluids in motion. Relative motion definition is - motion as observed from or referred to some material system constituting a frame of reference (as two adjacent walls and floor of a room). What is the Relative Speed of an Object Moving in the Same Direction? Pro Lite, Vedantu A fluid is a material which flows at room temperature, because its upper molecule overlaps the inner molecule, which tends to flow the liquid in forward direction. When we say something, we must know that it is always moving. If an object is moving slower, then it is considered to be at a lower speed, whereas if an object is moving faster, then it is considered to be at a higher speed. No doubt the speedo meter of each indicates motion but the two drivers will always find themselves facing each other.Relative velocity of a body A wâ¦ The relative humidity is the percent of saturation humidity, generally calculated in relation to saturated vapor density. The theory of Relative Speed is vital to most of the physicists. | Definition, Formula, Units – Hydrostatics, Density of a Mixture of Substances – Hydrostatics, MCQ Questions for Class 10 Social Science SST with Answers PDF Download Chapter Wise, MCQ Questions for Class 9 Science with Answers PDF Download Chapter Wise, MCQ Questions with Answers for Class 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1 all Subjects, MCQ Questions for Class 11 Economics with Answers Chapter Wise PDF Download, MCQ Questions for Class 11 Accountancy with Answers PDF Download Chapter Wise, MCQ Questions for Class 12 Economics with Answers Chapter Wise PDF Download, MCQ Questions for Class 11 Biology with Answers Chapter Wise PDF Download, MCQ Questions for Class 11 Chemistry with Answers Chapter Wise PDF Download, MCQ Questions for Class 11 Physics with Answers Chapter Wise PDF Download, MCQ Questions for Class 11 Maths with Answers Chapter Wise PDF Download. For a solid body, density of body = density of substance. In spite of our best efforts we could not find a fixed point with respect to which we should study absolute rest and absolute motion. Relative density, or specific gravity, is the ratio of the density (mass of a unit volume) of a substance to the density of a given reference material. Basically, it states that the faster we go, the more the time is affected. Then the second person will appear to be moving away at 2m/s from the first person. For example, an airplane usually encounters a wind - air that is moving with respect to an observer on the ground below. In motion Relative density = $$\frac{\text { Density of substance }}{\text { Density of water at } 4^{\circ} \mathrm{C}}$$ Relative Density (also referred to as specific gravity) then would be the ratio of density of a given substance to the density of a given base material. Velocity of the moving objects with respect to other moving or stationary object is called ârelative velocityâ and this motion is called ârelative motionâ. Then, for example, in the collision between two bodies of the same massâ¦ Read More; significance in motion. As another example, a motorboat in a river is moving amidst a river current - water that is moving with respect to an observer on dry land. Here, both the cases need to be taken into account, whether they move or are still. For example, you can toss and catch a ball in a moving bus if the motion is in a straight line at constant speed. Relative Motion. Max Planck, Hermann Minkowski and others did subsequent work. Relative density of a substance is defined as the ratio of its density to the density of water at 4°C. Motion does not happen in isolation. Consider two trains of length A,B are moving in opposite direction at u m/s & v m/s, then period taken by the trains to cross each other: 8. It follows from this principle that the speed of light would have to appear to be the same to every observer. 7. The laws of physics which apply when you are at rest on the earth also apply when you are in any reference frame which is moving at a constant velocity with respect to the earth. We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts. And the speed of the second body is y km/hr. Relative acceleration is the comparison of the acceleration of two bodies, or for a rigid body, the acceleration of a point on the rigid body in reference to another point on it. Sorry!, This page is not available for now to bookmark. Time taken by the train of length l meters to pass an unmoving object of length b meters is the time taken by train to cover (l + b) meters. The general formula of velocity is : Velocity of B relative to A is = $$\vec{v}_b-\vec{v}_a$$ This is the only formula that describes the concept of relative velocity. Reference point is too important in physics. How fast is the sheep moving as seen by the lion? this blog will deal with O-level physics. In hydrostatics, we will learn about the condition of fluids when it is in rest or exerted by an external force. In mechanics: Relative motion. = $$\frac{\text { Weight of substance in air }}{\text { Loss of weight in water}}$$. Speed is known as distance traveled per unit of time. When do we consider the body to be in motion? Draw the position and velocity vectors for relative motion. Consider two bodies moving at a different speed in the same direction. Relative Measures of Dispersion and Lorenz Curve, Kinetic Interpretation of Temperature and RMS Speed of Gas Molecules, Vedantu A relative velocity is a velocity that is measured in a frame of reference. The object might be traveling faster or at a slower speed at different points during the time interval. So, the study of relative rest and relative motion assumes importance.Consider two cars A and B running parallel to each other on same road with same velocities. General relativity is concerned with gravity, one of the fundamental forces in the universe. Albert Einstein published the theory of special relativity in 1905, building on many theoretical results and empirical findings obtained by Albert A. Michelson, Hendrik Lorentz, Henri Poincaré and others. The use of relative density is preferred over the term specific gravity since the latter means relative density in connection with water. Question and Answer forum for K12 Students. Relative density is a unit-less quantity. This is a place where teachers and students can get physics lesson notes,example questions and exercises to enrich both teachers and students towards better performance in teaching and learning of physics subject. 9. The answer to the query is that if you change your relative position with respect to a frame of reference, then you are in motion; otherwise, you are in the state of rest. Created by Mahesh Shenoy On occasion objects move within a medium that is moving with respect to an observer. How to solve relative velocity problems in one dimension; How to solve relative velocity problems in two dimensions; Notes: Frame of reference can be thought of as the point of view that measurements are made from. Pro Lite, Vedantu In such instances as this, the magnitude of the velocity of the moving object (whether it bâ¦ In this video let's extend this concept to the next level, to 2D motion. 4 Motion in Two and Three Dimensions. Time taken by the train of length/meters to pass a pole or a single post or a standing man is equal to the time taken by the train to cover / meters. University Physics Volume 1. In all such instances, in order to describe the complete motion of the object, we need to consider the effect that the medium is causing on the object. Relative position is essential in comparing one thing to another and in measuring the distance covered by an object. Ans- The objectâs relative speed traveling in the same direction is given by: Let us say two objects with a velocity of a and b with respect to the floor are moving in the same direction. The average speed of an object over an interval of time is determined by the above equation. Considering various occasions, we take in more than one object move in a frame which is non-stationary in respect to another viewer. Relative Density Formula in Physics: Relative density = $$\frac{\text { Density of substance }}{\text { Density of water at } 4^{\circ} \mathrm{C}}$$ = $$\frac{\text { Weight of substance in air }}{\text { Loss of weight in water}}$$ Relative Density unit: Relative density has no unit because it is the ratio of same units which gets cancelled. All these are the result of relative motion. The concept of relative motion velocity in a plane is quite similar to the whole concept of relative velocity in a straight line. For example, a person moves at a speed of 20m/s towards the east, and the second person moves at a speed of 22m/s east. This physics video tutorial explains how to solve relative velocity problems in two dimensions using a frame of reference. If one person initiates movement towards the other, the only way to know their speed is by measuring the position relative to the fixed position, which is most likely to be the other person. Pro Lite, CBSE Previous Year Question Paper for Class 10, CBSE Previous Year Question Paper for Class 12. Relative Density Definition Physics: While doing so, we calculate the relative velocity of the object considering the velocity of the particle as well as the velocity â¦ 6. Hydrostatics is a property of liquid or fluid in mechanics. Let the velocity of the first person be a and that of the second person be b. In this video, we will intuitively figure out how to calculate relative velocities with some examples. 4. This is a maximum when Î¸ = 0°, which then results in maximum V rel = w(d/2 + R). We'll then write down the general expression for calculating relative velocity. Here the average speed is the total distance traveled by an object divided by the total time taken. Consider two bodies moving at a different speed in the same direction. Vedantu academic counsellor will be calling you shortly for your Online Counselling session. With the help of the reference frame, we can only define time and space. If one moves towards you, we can calculate his/her speed by evaluating the difference between their relative spots taking a fixed point as a frame of reference with respect to you. Isaac Physics a project designed to offer support and activities in physics problem solving to teachers and students from GCSE level through to university. Speed is the scalar quantity, which is the magnitude of the velocity vector. So, if time after which both the body moves is given, Then, distance covered inâtâ hours = relative speed * time. Relative Density unit: 3. Tamilnadu Board Class 10 English Solutions, Tamilnadu Board Class 9 Science Solutions, Tamilnadu Board Class 9 Social Science Solutions, Tamilnadu Board Class 9 English Solutions, What is Density in Physics? While for a hollow body, density of body is lesser than that of substance. Save my name, email, and website in this browser for the next time I comment. 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Common Core: 5th Grade Math : Divide Fractions and Whole Numbers: CCSS.Math.Content.5.NF.B.7

Example Questions

← Previous 1 3

Example Question #1 : Divide Fractions And Whole Numbers: Ccss.Math.Content.5.Nf.B.7

Which of the following statements is FALSE?

Explanation:

Example Question #1 : Divide Fractions And Whole Numbers: Ccss.Math.Content.5.Nf.B.7

Explanation:

To divide fractions, we multiply by the reciprocal. In order to find the reciprocal, we simply flip the fraction over. The numerator becomes the denominator and the denominator becomes the numerator.

Example Question #1 : Divide Fractions And Whole Numbers: Ccss.Math.Content.5.Nf.B.7

Explanation:

To divide fractions, we multiply by the reciprocal. In order to find the reciprocal, we simply flip the fraction over. The numerator becomes the denominator and the denominator becomes the numerator.

Example Question #1 : Divide Fractions And Whole Numbers: Ccss.Math.Content.5.Nf.B.7

Explanation:

To divide fractions, we multiply by the reciprocal. In order to find the reciprocal, we simply flip the fraction over. The numerator becomes the denominator and the denominator becomes the numerator.

Example Question #2 : Divide Fractions And Whole Numbers: Ccss.Math.Content.5.Nf.B.7

Explanation:

To divide fractions, we multiply by the reciprocal. In order to find the reciprocal, we simply flip the fraction over. The numerator becomes the denominator and the denominator becomes the numerator.

Example Question #1 : Divide Fractions And Whole Numbers: Ccss.Math.Content.5.Nf.B.7

Explanation:

To divide fractions, we multiply by the reciprocal. In order to find the reciprocal, we simply flip the fraction over. The numerator becomes the denominator and the denominator becomes the numerator.

Example Question #1 : Divide Fractions And Whole Numbers: Ccss.Math.Content.5.Nf.B.7

Explanation:

To divide fractions, we multiply by the reciprocal. In order to find the reciprocal, we simply flip the fraction over. The numerator becomes the denominator and the denominator becomes the numerator.

Example Question #1 : Divide Fractions And Whole Numbers: Ccss.Math.Content.5.Nf.B.7

Explanation:

To divide fractions, we multiply by the reciprocal. In order to find the reciprocal, we simply flip the fraction over. The numerator becomes the denominator and the denominator becomes the numerator.

Example Question #1 : Divide Fractions And Whole Numbers: Ccss.Math.Content.5.Nf.B.7

Explanation:

To divide fractions, we multiply by the reciprocal. In order to find the reciprocal, we simply flip the fraction over. The numerator becomes the denominator and the denominator becomes the numerator.


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# A group of important generating functions involving harmonic number.

How to prove the following identities:

$$\small{\sum_{n=1}^\infty\frac{H_{n}}{n^2}x^{n}=\operatorname{Li}_3(x)-\operatorname{Li}_3(1-x)+\ln(1-x)\operatorname{Li}_2(1-x)+\frac12\ln x\ln^2(1-x)+\zeta(3)}\tag1$$

$$\small{\sum_{n=1}^\infty\frac{H_{n}^{(2)}}{n}x^{n}=\operatorname{Li}_3(x)+2\operatorname{Li}_3(1-x)-\ln(1-x)\operatorname{Li}_2(1-x)-\zeta(2)\ln(1-x)-2\zeta(3)}\tag2$$

$$\sum_{n=1}^\infty (H_n^2-H_n^{(2)})x^{n}=\frac{\ln^2(1-x)}{1-x}\tag3$$

$$\sum_{n=1}^\infty\frac{H_{n}^2}{n}x^{n}=\operatorname{Li}_3(x)-\ln(1-x)\operatorname{Li}_2(x)-\frac13\ln^3(1-x)\tag4$$

$$\small{\sum_{n=1}^\infty H_n^3x^n= \frac{\operatorname{Li}_3(x)+3\operatorname{Li}_3(1-x)+\frac32\ln x\ln^2(1-x)-3\zeta(2)\ln(1-x)-\ln^3(1-x)-3\zeta(3)}{1-x}}\tag5$$

$$\small{\sum_{n=1}^\infty H_nH_n^{(2)}x^n= \frac{\operatorname{Li}_3(x)+\operatorname{Li}_3(1-x)+\frac12\ln x\ln^2(1-x)-\zeta(2)\ln(1-x)-\zeta(3)}{1-x}}\tag6$$

$$\sum_{n=1}^\infty\left(H_n^3-3H_nH_n^{(2)}+2H_n^{(3)}\right)x^n=-\frac{\ln^3(1-x)}{1-x}\tag7$$

Edit:

Here is some extra identities and proofs in the answer sections

$$\sum_{n=1}^\infty \frac{H_n^{(3)}}{n}x^n=\operatorname{Li}_4(x)-\ln(1-x)\operatorname{Li}_3(x)-\frac12\operatorname{Li}_2^2(x)\tag8$$

$$\sum_{n=1}^\infty\frac{ H_n^{(2)}}{n+1}x^{n}=\frac{2\operatorname{Li}_3(1-x)-\operatorname{Li}_2(1-x)\ln(1-x)-\zeta(2)\ln(1-x)-2\zeta(3)}{x}\tag{9}$$

$$\small{\sum_{n=1}^\infty\frac{ H_n^{2}}{n+1}x^{n}=\frac{6\operatorname{Li}_3(1-x)-3\operatorname{Li}_2(1-x)\ln(1-x)-\ln^3(1-x)-3\zeta(2)\ln(1-x)-6\zeta(3)}{3x}}\tag{10}$$

Note:

Proofs for (3) and (7) should be done without using the formula of the sterling number of the first kind : $$\frac{\ln^k(1+x)}{k!}=\sum_{n=k}^\infty(-1)^{n-k} \begin{bmatrix} n \\ k \end{bmatrix}\frac{x^n}{n!}$$.

• I'm voting to close this question as off-topic because this is not a question, but instead some kind of reference in our (???) solutions? – Markus Scheuer Sep 23 '19 at 22:24
• @MarkusScheuer how have you been on MSE so long and still not get that self-answers are allowed on stack exchange – qwr Dec 28 '19 at 8:16
• @qwr: You miss the point. The initial version of this post was not formulated as question. – Markus Scheuer Dec 28 '19 at 8:36
• @qwr you can see what I edited .. it was not that different. I had to edite it to avoid the down voters. – Ali Shather Dec 28 '19 at 8:41
• @Ali Shather +1 for the very useful collection of formulas (and their proofs). There should be more of these overviews. Comment: I confess that I am not particularly happy with the "red tape" requirement that anything - even nice results which we like to share with others - has to be formulated as a question. But the remedy is simple, as you have shown in several occasions: just start the heading with the prefix "How to prove". What an exciting improvement! – Dr. Wolfgang Hintze Apr 1 at 8:42

## 5 Answers

Using the fact that

$$\sum_{n=1}^\infty H_nx^n=-\frac{\ln(1-x)}{1-x}$$

Divide both sides by $$x$$ then integrate we have

\begin{align} \sum_{n=1}^\infty\frac{H_n}{n}x^n&=-\int\frac{\ln(1-x)}{x(1-x)}\ dx\\ &=-\int\frac{\ln(1-x)}{x}\ dx-\int\frac{\ln(1-x)}{1-x}\ dx\\ &=\operatorname{Li}_2(x)+\frac12\ln^2(1-x)+C,\quad x=0\Rightarrow C=0 \end{align}

Then

$$\sum_{n=1}^\infty\frac{H_n}{n}x^n=\operatorname{Li}_2(x)+\frac12\ln^2(1-x)\tag i$$

Proof of (1):

Divide both sides of (i) by $$x$$ then integrate

\begin{align} \sum_{n=1}^\infty\frac{H_n}{n^2}x^n&=\operatorname{Li}_3(x)+\frac12\underbrace{\int\frac{\ln^2(1-x)}{x}\ dx}_{IBP}\\ &=\operatorname{Li}_3(x)+\frac12\ln x\ln^2(1-x)+\underbrace{\int\frac{\ln x\ln(1-x)}{1-x}\ dx}_{IBP}\\ &=\operatorname{Li}_3(x)+\frac12\ln x\ln^2(1-x)+\operatorname{Li}_2(1-x)\ln(1-x)+\int\frac{\operatorname{Li}_2(1-x)}{1-x}\ dx\\ &=\small{\operatorname{Li}_3(x)+\frac12\ln x\ln^2(1-x)+\operatorname{Li}_2(1-x)\ln(1-x)-\operatorname{Li}_3(1-x)+C,\quad x=0\Rightarrow C=\zeta(3)} \end{align}

Then

$$\small{\sum_{n=1}^\infty\frac{H_n}{n^2}x^n=\operatorname{Li}_2(x)-\operatorname{Li}_3(1-x)+\ln(1-x)\operatorname{Li}_2(1-x)+\frac12\ln x\ln^2(1-x)+\zeta(3)}\tag{ii}$$

Proof of (2):

By Cauchy product we have

$$-\ln(1-x)\operatorname{Li}_2(x)=2\sum_{n=1}^\infty\frac{H_n}{n^2}x^n+\sum_{n=1}^\infty\frac{H_n^{(2)}}{n}x^n-3\operatorname{Li}_3(x)\tag{iii}$$

From (ii) and (iii) we get

$$\small{\sum_{n=1}^\infty\frac{H_{n}^{(2)}}{n}x^{n}=\operatorname{Li}_3(x)+2\operatorname{Li}_3(1-x)-\ln(1-x)\operatorname{Li}_2(1-x)-\zeta(2)\ln(1-x)-2\zeta(3)}\tag{iv}$$

where I substituted, using the Dilogarithm reflection formula

$$\operatorname{Li}_2(x)=\zeta(2)-\ln x\ln(1-x)-\operatorname{Li}_2(1-x)$$

Proof of (3):

Using the generalization: (proved at the bottom)

$$\sum_{n=1}^\infty a_nx^n=\frac1{1-x}\sum_{n=1}^\infty (a_n-a_{n-1})x^n,\quad a_{0}=0\tag{v}$$

Let $$a_{n}=H_n^2$$ in (v) to have

\begin{align} \sum_{n=1}^\infty H_n^2x^n&=\frac1{1-x}\sum_{n=1}^\infty \left(H_n^2-H_{n-1}^2\right)x^n\\ &=\frac1{1-x}\sum_{n=1}^\infty \left(\frac{2H_n}{n}-\frac1{n^2}\right)x^n\\ &=\frac1{1-x}\cdot 2\sum_{n=1}^\infty\frac{H_n}{n}x^n-\frac{\operatorname{Li}_2(x)}{1-x}\\ &=\frac1{1-x}\cdot 2\left(\operatorname{Li}_2(x)+\frac12\ln^2(1-x)\right)-\frac{\operatorname{Li}_2(x)}{1-x}\\ &=\frac{\ln^2(1-x)}{1-x}+\frac{\operatorname{Li}_2(x)}{1-x}\\ &=\frac{\ln^2(1-x)}{1-x}+\sum_{n=1}^\infty H_n^{(2)}x^n \end{align}

Then

$$\sum_{n=1}^\infty (H_n^2-H_n^{(2)})x^{n}=\frac{\ln^2(1-x)}{1-x}\tag{vi}$$

Proof of (4):

Divide both sides of (vi) by $$x$$ then integrate we have

\begin{align} \sum_{n=1}^\infty (H_n^2-H_n^{(2)})\frac{x^{n}}{n}&=\int\frac{\ln^2(1-x)}{x(1-x)}\ dx\\ &=\int\frac{\ln^2(1-x)}{x}\ dx+\int\frac{\ln^2(1-x)}{1-x}\ dx \end{align}

the first integral is calculated in proof of (1) and its equal to

$$\ln x\ln^2(1-x)+2\ln(1-x)\operatorname{Li}_2(1-x)-2\operatorname{Li}_3(1-x)$$

then

$$\small{\sum_{n=1}^\infty (H_n^2-H_n^{(2)})\frac{x^{n}}{n}=\ln x\ln^2(1-x)+2\ln(1-x)\operatorname{Li}_2(1-x)-2\operatorname{Li}_3(1-x)-\frac13\ln^3(1-x)+C}$$

if we set $$x=0$$ we get $$C=2\zeta(3)$$

$$\small{\sum_{n=1}^\infty (H_n^2-H_n^{(2)})\frac{x^{n}}{n}=\ln x\ln^2(1-x)+2\ln(1-x)\operatorname{Li}_2(1-x)-2\operatorname{Li}_3(1-x)-\frac13\ln^3(1-x)+2\zeta(3)}\quad \text{(vii)}$$

from (iv) and (vii) we get

$$\sum_{n=1}^\infty\frac{H_{n}^2}{n}x^{n}=\operatorname{Li}_3(x)-\ln(1-x)\operatorname{Li}_2(x)-\frac13\ln^3(1-x)\tag{viii}$$

Proof of (5):

Let $$a_n=H_n^3$$ in (v) we get

\begin{align} \sum_{n=1}^\infty H_n^3x^n&=\frac1{1-x}\sum_{n=1}^\infty (H_n^3-H_{n-1}^3)x^n\\ &=\frac1{1-x}\sum_{n=1}^\infty\left(\frac{3H_n^2}{n}-\frac{3H_n}{n^2}+\frac1{n^3}\right)x^n\\ &=\frac1{1-x}\cdot 3\sum_{n=1}^\infty\frac{H_n^2}{n}x^n-\frac1{1-x}\cdot 3\sum_{n=1}^\infty\frac{H_n}{n^2}x^n+\frac{\operatorname{Li}_3(x)}{1-x} \end{align}

Combine the results from (ii) and (viii) we get

$$\small{\sum_{n=1}^\infty H_n^3x^n= \frac{\operatorname{Li}_3(x)+3\operatorname{Li}_3(1-x)+\frac32\ln x\ln^2(1-x)-3\zeta(2)\ln(1-x)-\ln^3(1-x)-3\zeta(3)}{1-x}}\tag{ix}$$

Proof of (6):

Let $$a_n=H_nH_n^{(2)}$$ in (v) we get

\begin{align} \sum_{n=1}^\infty H_nH_n^{(2)}x^n&=\frac1{1-x}\sum_{n=1}^\infty \left(H_nH_n^{(2)}-H_{n-1}H_{n-1}^{(2)}\right)x^n\\ &=\frac1{1-x}\sum_{n=1}^\infty\left(\frac{H_n}{n^2}+\frac{H_n^{(2)}}{n}-\frac1{n^3}\right)x^n\\ &=\frac1{1-x}\sum_{n=1}^\infty\frac{H_n}{n^2}x^n+\frac1{1-x}\sum_{n=1}^\infty\frac{H_n^{(2)}}{n}x^n-\frac{\operatorname{Li}_3(x)}{1-x} \end{align}

Substituting the results from (ii) and (iv) we get

$$\small{\sum_{n=1}^\infty H_nH_n^{(2)}x^n= \frac{\operatorname{Li}_3(x)+\operatorname{Li}_3(1-x)+\frac12\ln x\ln^2(1-x)-\zeta(2)\ln(1-x)-\zeta(3)}{1-x}}\tag{x}$$

Proof of (7):

Combine the results from (ix) and (x) along with $$\sum_{n=1}^\infty H_n^{(3)}x^n=\frac{\operatorname{Li}_3(x)}{1-x}$$ we get

$$\sum_{n=1}^\infty\left(H_n^3-3H_nH_n^{(2)}+2H_n^{(3)}\right)x^n=-\frac{\ln^3(1-x)}{1-x}$$

Different approach to prove (7):

again by using the generalization

$$\sum_{n=1}^\infty a_nx^n=\frac1{1-x}\sum_{n=1}^\infty (a_n-a_{n-1})x^n,\quad a_{0}=0$$

and setting $$a_n=H_n^3-3H_nH_n^{(2)}+2H_n^{(3)}$$ we have

$$\sum_{n=1}^\infty \left(H_n^3-3H_nH_n^{(2)}+2H_n^{(3)}\right)x^n\\=\frac1{1-x}\sum_{n=1}^\infty\left(H_n^3-3H_nH_n^{(2)}+2H_n^{(3)}-H_{n-1}^3+3H_{n-1}H_{n-1}^{(2)}-2H_{n-1}^{(3)}\right)\\ =\frac1{1-x}\sum_{n=1}^\infty\left[3\left(\frac{H_n^2-H_n^{(2)}}{n}\right)-6\frac{H_n^{(2)}}{n}+\frac6{n^3}\right]x^n\\ =\frac1{1-x}\cdot3\sum_{n=1}^\infty\left(H_n^2-H_n^{(2)}\right)\frac{x^n}{n}-\frac1{1-x}\cdot 6\sum_{n=1}^\infty\frac{H_n}{n^2}x^n+\frac{6\operatorname{Li}_3(x)}{1-x}$$

Combine the results from (ii) and (vii) we get

$$\sum_{n=1}^\infty\left(H_n^3-3H_nH_n^{(2)}+2H_n^{(3)}\right)x^n=-\frac{\ln^3(1-x)}{1-x}$$

Proof of the generalization:

\begin{align} \color{blue}{1}\sum_{n=0}^\infty a_nx^n&=\left(\color{blue}{\frac1{1-x}-\frac{x}{1-x}}\right)\sum_{n=0}^\infty a_nx^n\\ &=\frac1{1-x}\sum_{n=0}^\infty a_nx^n-\frac1{1-x}\sum_{n=0}^\infty a_nx^{n+1}\\ &=\frac1{1-x}\sum_{n=0}^\infty a_nx^n-\frac1{1-x}\sum_{n=1}^\infty a_{n-1}x^{n},\quad \text{assuming}\ \color{red}{a_{0}=0}\\ \sum_{n=\color{red}{1}}^\infty a_nx^n&=\frac1{1-x}\sum_{n=\color{red}{1}}^\infty a_nx^n-\frac1{1-x}\sum_{n=1}^\infty a_{n-1}x^{n} \end{align}

Then

$$\sum_{n=1}^\infty a_nx^n=\frac1{1-x}\sum_{n=1}^\infty (a_n-a_{n-1})x^n,\quad a_{0}=0$$

All the generating functions may be found in the book, (Almost) Impossible Integrals, Sums, and Series.

The versions $$\displaystyle \sum_{n=1}^{\infty} x^{n+1} \frac{H_n}{(n+1)^2}$$ and $$\displaystyle \sum_{n=1}^{\infty} x^{n+1} \frac{H_n^{(2)}}{n+1}$$ may be found calculated in $$(6.18)$$ and $$(6.19)$$, page $$351$$ from which we extract $$(1)$$ and $$(2)$$ in the post statement by simply readjusting.

The results in $$(3)$$ and $$(7)$$ in the post statement may be found on page $$355$$, and a generalization of $$\displaystyle \frac{\log^n(1-x)}{1-x}$$ expressed with the help of symmetric polynomials may be found on pages $$354$$-$$355$$. Also, $$(3)$$ in the post statement may be obtained by combining the generating functions in $$(4.6)$$ and $$(4.7)$$.

The result in $$(4)$$ in the post statement may be extracted by using the result in $$(4.7)$$, page $$284$$.

Also, the results in $$(5)$$ and $$(6)$$ are found on page $$284$$.

Proof of $$(8)$$:

From integrating the generating function after dividing by $$x$$

$$\sum_{n=1}^\infty H_n^{(3)}x^n=\frac{\operatorname{Li}_3(x)}{1-x}$$

it follows that

$$\sum_{n=1}^\infty \frac{H_n^{(3)}}{n}x^n=\int\frac{\operatorname{Li}_3(x)}{x(1-x)}dx=\int\frac{\operatorname{Li}_3(x)}{x}dx+\int\frac{\operatorname{Li}_3(x)}{1-x}dx$$

$$=\operatorname{Li}_4(x)-\ln(1-x)\operatorname{Li}_3(x)+\int\frac{\ln(1-x)\operatorname{Li}_2(x)}{x}dx$$

$$=\operatorname{Li}_4(x)-\ln(1-x)\operatorname{Li}_3(x)-\frac12\operatorname{Li}_2^2(x)+C$$

Set $$x=0\Longrightarrow C=0$$

then

$$\sum_{n=1}^\infty \frac{H_n^{(3)}}{n}x^n=\operatorname{Li}_4(x)-\ln(1-x)\operatorname{Li}_3(x)-\frac12\operatorname{Li}_2^2(x)$$

Proof of $$(9)$$ and $$(10)$$:

We proved above

$$\sum_{n=1}^\infty (H_n^2-H_n^{(2)})x^{n}=\frac{\ln^2(1-x)}{1-x}$$

Integrating both sides gives

$$\sum_{n=1}^\infty \frac{H_n^2-H_n^{(2)}}{n+1}x^{n+1}=\int\frac{\ln^2(1-x)}{1-x}dx=-\frac13\ln^3(1-x)+C$$

where $$C=0$$ if we set $$x=0$$

Then

$$\sum_{n=1}^\infty \frac{H_n^2}{n+1}x^{n+1}=\sum_{n=1}^\infty \frac{H_n^{(2)}}{n+1}x^{n+1}-\frac13\ln^3(1-x)\tag1$$

From

$$\sum_{n=1}^\infty H_n^{(2)}x^n=\frac{\operatorname{Li}_2(x)}{1-x}$$

It follows that

$$\sum_{n=1}^\infty\frac{ H_n^{(2)}}{n+1}x^{n+1}=\int \frac{\operatorname{Li}_2(x)}{1-x}dx=f(x)\tag2$$

now let $$1-x=y$$ then use the reflection formula $$\operatorname{Li}_2(1-y)=\zeta(2)-\ln(y)\ln(1-y)-\operatorname{Li}_2(y)$$

$$f(x)=-\zeta(2)\int\frac{dy}{y}+\int\frac{\ln(y)\ln(1-y)}{y}dy+\int\frac{\operatorname{Li}_2(y)}{y}dx\\=-\zeta(2)\ln(y)+\left[-\operatorname{Li}_2(y)\ln(y)+\int\frac{\operatorname{Li}_2(y)}{y}dy\right]+\operatorname{Li}_3(y)\\=2\operatorname{Li}_3(y)-\operatorname{Li}_2(y)\ln(y)-\zeta(2)\ln(y)\\=2\operatorname{Li}_3(1-x)-\operatorname{Li}_2(1-x)\ln(1-x)-\zeta(2)\ln(1-x)+C$$

Set $$x=0$$ $$\Longrightarrow C=-2\zeta(3)$$

Then

$$f(x)=2\operatorname{Li}_3(1-x)-\operatorname{Li}_2(1-x)\ln(1-x)-\zeta(2)\ln(1-x)-2\zeta(3)\tag3$$

Plugging $$(3)$$ in $$(2)$$ yields

$$\sum_{n=1}^\infty\frac{ H_n^{(2)}}{n+1}x^{n+1}=2\operatorname{Li}_3(1-x)-\operatorname{Li}_2(1-x)\ln(1-x)-\zeta(2)\ln(1-x)-2\zeta(3)\tag4$$

Finally plug $$(4)$$ in $$(1)$$ we get

$$\small{\sum_{n=1}^\infty\frac{ H_n^{2}}{n+1}x^{n+1}=2\operatorname{Li}_3(1-x)-\operatorname{Li}_2(1-x)\ln(1-x)-\frac13\ln^3(1-x)-\zeta(2)\ln(1-x)-2\zeta(3)}$$

Or

$$\small{\sum_{n=1}^\infty\frac{ H_n^{2}}{n+1}x^{n}=\frac{6\operatorname{Li}_3(1-x)-3\operatorname{Li}_2(1-x)\ln(1-x)-\ln^3(1-x)-3\zeta(2)\ln(1-x)-6\zeta(3)}{3x}}$$

Different way to prove

$$\sum_{n=1}^\infty (H_n^2-H_n^{(2)})x^{n}=\frac{\ln^2(1-x)}{1-x}$$

Using the proved-above identity

$$\sum_{n=1}^\infty a_nx^n=\frac1{1-x}\sum_{n=1}^\infty (a_n-a_{n-1})x^n,\quad a_{0}=0$$

Set $$a_n=H_n^2-H_n^{(2)}$$

we get

$$\sum_{n=1}^\infty (H_n^2-H_n^{(2)})x^{n}=\frac1{1-x}\sum_{n=1}^\infty (H_n^2-H_n^{(2)}-H_{n-1}^2+H_{n-1}^{(2)})x^{n}$$ $$=\frac{1}{1-x}\sum_{n=1}^\infty\left(\frac{2H_n}{n}-\frac{2}{n^2}\right)x^n=\frac{2}{1-x}\left(\color{blue}{\sum_{n=1}^\infty\frac{H_n}{n}x^n-\operatorname{Li}_2(x)}\right)$$

$$=\frac{2}{1-x}\left(\color{blue}{\frac12\ln^2(1-x)}\right)=\frac{\ln^2(1-x)}{1-x}$$

where the blue result follows from dividing both sides of $$\sum_{n=1}^\infty H_nx^n=-\frac{\ln(1-x)}{1-x}$$ by $$x$$ then integrating from $$x=0$$ to $$x=x$$

$$\sum_{n=1}^\infty \frac{H_n}{n}x^n=-\int_0^x\frac{\ln(1-x)}{x(1-x)}\ dx=\operatorname{Li}_2(x)+\frac12\ln^2(1-x)$$

or $$\sum_{n=1}^\infty\frac{H_n}{n}x^n-\operatorname{Li}_2(x)=\frac12\ln^2(1-x)$$


text:
# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary 7.3.6.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories equipped with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$, and suppose that $F_0$ admits a left Kan extension along $\delta$. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor and let $\beta : F_0 \rightarrow F \circ \delta$ be a natural transformation. The following conditions are equivalent:

$(1)$

The natural transformation $\beta$ exhibits $F$ as a left Kan extension of $F_0$ along $\delta$.

$(2)$

For every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the composite map

$\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( F \circ \delta , G \circ \delta ) \xrightarrow { \circ [\beta ] } \operatorname{Hom}_{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}( F_0, G \circ \delta )$

is a homotopy equivalence of Kan complexes.

$(3)$

For every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the composite map

$\operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})} }( F, G ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(K, \operatorname{\mathcal{D}})} }( F \circ \delta , G \circ \delta )) \xrightarrow { \circ [\beta ] } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}}( F_0, G \circ \delta )$

is a bijection of sets.

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 7.3.6.1 and the implication $(2) \Rightarrow (3)$ is immediate. We will complete the proof by showing that $(3) \Rightarrow (1)$. By assumption, there exists a functor $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\beta ': F_0 \rightarrow F' \circ \delta$ which exhibits $F'$ as a left Kan extension of $F$ along $\delta$. Applying Proposition 7.3.6.1, we deduce that there exists a natural transformation $\gamma : F' \rightarrow F$ for which $\beta$ is a composition of $\beta '$ with the induced transformation $\gamma |_{K}: (F' \circ \delta ) \rightarrow (F \circ \delta )$. For each object $G \in \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, we have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})} }( F, G ) \ar [rr]^{ \circ [\gamma ] } \ar [dr] & & \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})} }( F', G ) \ar [dl] \\ & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}}( F_0, G \circ \delta ), & }$

where the right vertical map is bijective. If condition $(3)$ is satisfied, then the left vertical map is also bijective. Allowing the functor $G$ to vary, it follows that the homotopy class $[\gamma ]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }$, so that $\gamma$ is an isomorphism in $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Invoking Remark 7.3.1.12, we conclude that $\beta$ exhibits $F$ as a left Kan extension of $F_0$ along $\delta$. $\square$


text:
## A metastable dominated convergence theorem

2012-02-01T00:00:00Z (GMT) by

The dominated convergence theorem implies that if (fn) is a sequence of functions on a probability space taking values in the interval [0, 1], and (fn) converges pointwise a.e., then (R fn) converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound on the rate of metastable convergence in the conclusion that is independent of the sequence (fn) and the underlying space. We prove a slight strengthening of Tao’s theorem which, moreover, provides an explicit description of the second bound in terms of the first. Specifically, we show that when the first bound is given by a continuous functional, the bound in the conclusion can be computed by a recursion along the tree of unsecured sequences. We also establish a quantitative version of Egorov’s theorem, and introduce a new mode of convergence related to these notions.