## Homework 3 Solution PHYS 3022 Introduction to Cosmology

### Introduction to Cosmology [book review]

1 Ryden Problem 6 UW Madison Astronomy Department. 4 1 An Introduction to Cosmology expansion presented in Section 3.9 suggests that either a fundamentally new theory of gravity is required, or that we do not know what constitutes most of the energy density of the universe (see Fig. 1.5). 1.2 The Cosmological Hypothesis The Friedmann Lema tre model of cosmic evolution is based on the cosmological, Barbara Ryden Solutions. Below are Chegg supported textbooks by Barbara Ryden. Select a textbook to see worked-out Solutions. Books by Barbara Ryden with Solutions Foundations of Astrophysics 1st Edition 191 Problems solved: Bradley M Peterson, Barbara Ryden: Introduction to Cosmology 0th Edition 0 Problems solved: Barbara Ryden.

### Homework 3 Solution PHYS 3022 Introduction to Cosmology

Wiley Introduction to Cosmology 4th Edition Matts Roos. Introduction to Cosmology Barbara Ryden Addison Wesley, San Francisco, 2003. $59.00 (244 pp.). ISBN 0-8053-8912-1 For more than 30 years, IвЂ™ve been teaching cosmology to third- and fourth-year undergraduates. For as many years, IвЂ™ve been seeking an ideal cosmology text for such students. Is Barbara RydenвЂ™s Introduction to Cos-mology that, May 27, 2019В В· Introduction to Cosmology - Kindle edition by Barbara Ryden. Download it once and read it on your Kindle device, PC, phones or tablets. Use features like bookmarks, note taking and highlighting while reading Introduction to Cosmology..

My main research interests are in galaxies and in the large scale structure of the universe. My textbook Introduction to Cosmology (1st edition В©2003, 2nd edition В©2017), won the Chambliss Astronomical Writing Award of the American Astronomical Society. Foundations of Astrophysics, written with Brad Peterson, was published in 2009. Cosmology - Introduction: When people hear the word Cosmology, they fear the worst: relativity, dark matter, galaxy clusters.While these topics are a part of the study of cosmology, they are not the mysterious subjects some make them out to be.

Sep 04, 2018В В· 'Barbara Ryden's Introduction to Cosmology is now published in a second edition, following the well-received first edition of 2002 вЂ¦ This is a course book for physics students; its approach is quantitative and the basic equations and mathematical descriptions are extensively outlined from first principles in all the areas covered. May 27, 2019В В· Introduction to Cosmology - Kindle edition by Barbara Ryden. Download it once and read it on your Kindle device, PC, phones or tablets. Use features like bookmarks, note taking and highlighting while reading Introduction to Cosmology.

Jul 24, 2017В В· The addition of new material on the baryonic component of the universe links cosmology to many modern research topics in astrophysics.' Crystal Martin, University of California, Santa Barbara 'I am delighted that a second edition of Barbara Ryden's Introduction to Cosmology is now available. Jun 06, 2016В В· ICTP Summer School on Cosmology 2016 6 June 2016 - 09:15

Introduction to Cosmology Professor Barbara Ryden Department of Astronomy Propagating wave solutions. 6/6/16 14 We describe the local curvature of spacetime with a metric. Metric = relation that gives the shortest distance Intro-Cosmo-Ryden-1 Author: Barbara Ryden My main research interests are in galaxies and in the large scale structure of the universe. My textbook Introduction to Cosmology (1st edition В©2003, 2nd edition В©2017), won the Chambliss Astronomical Writing Award of the American Astronomical Society. Foundations of Astrophysics, written with Brad Peterson, was published in 2009.

My main research interests are in galaxies and in the large scale structure of the universe. My textbook Introduction to Cosmology (1st edition В©2003, 2nd edition В©2017), won the Chambliss Astronomical Writing Award of the American Astronomical Society. Foundations of Astrophysics, written with Brad Peterson, was published in 2009. May 27, 2019В В· Introduction to Cosmology - Kindle edition by Barbara Ryden. Download it once and read it on your Kindle device, PC, phones or tablets. Use features like bookmarks, note taking and highlighting while reading Introduction to Cosmology.

Jan 13, 2006В В· View Essay - Ryden_IntroCosmo from PHYS 236 at University of California, Santa Barbara. Introduction to Cosmology Barbara Ryden Department of Astronomy The Ohio State University January 13, Introduction to Cosmology (Physics 371) Sean Carroll, Physics Department, University of Chicago Spring quarter, 2006 (the WMAP [Wilkinson Microwave Anisotropy Probe] satellite, from the WMAP Home Page) Description. Physics 371 is a graduate-level introduction to cosmology, the study of the structure and evolution of the universe.

Overview. Lectures can be downloaded as Powerpoint or pdf files from the syllabus below. The pdf files are in handout format with three slides per sheet plus space for note taking. Each lecture has an associated example exam question, with hints and answers. Introduction to Cosmology Barbara Ryden Addison Wesley, 2002 ISBN 0805389121 (hardcover) This book entails a really fabulous introduction to the basic concepts of cosmology, nearly entirely up to date. It is one of the few textbooks really fun reading, a marvel of didactic writing.

Introduction To Cosmology Ryden Solutions Manual Read/Download Introduction To Cosmology Ryden Solution Manual is designed to help you forget about your Introduction To Cosmology Ryden Solution Manual menu,. Solutions Manual - A Primer For The Mathematics Of Financial Engineering, Second Edition Stefanica, Dan Extragalactic Astronomy Jun 08, 2016В В· ICTP Summer School on Cosmology 2016 8 June 2016 - 09:15. рџ”ґ Relaxing Piano Music 24/7, Sleep Music, Piano Music, Insomnia, Meditation, Sleep, Study, Relax Yellow Brick Cinema - вЂ¦

PHYS432/532:Cosmology Winter2018 SolutionstoHomework1 Ryden4.1.(7points) Thetotalenergyofthecosmologicalconstantcontainedinasphereofradius1AUis E = 4 3 r3" 4(4870MeV Introduction to Cosmology Barbara Ryden Addison Wesley, San Francisco, 2003. $59.00 (244 pp.). ISBN 0-8053-8912-1 For more than 30 years, IвЂ™ve been teaching cosmology to third- and fourth-year undergraduates. For as many years, IвЂ™ve been seeking an ideal cosmology text for such students. Is Barbara RydenвЂ™s Introduction to Cos-mology that

Introduction to cosmology / Matt Roos. вЂ“ 3rd ed. p. cm. Includes bibliographical references and index. ISBN 0-470-84909-6 (acid-free paper) вЂ“ ISBN 0-470-84910-X (pbk. : acid-free paper) 1. Cosmology. I. Title. QB981.R653 2003 523.1 вЂ” dc22 2003020688 British Library Cataloguing in Publication Data TASI Lectures: Introduction to Cosmology Mark Trodden1 and Sean M. Carroll2 1Department of Physics Syracuse University Syracuse, NY 13244-1130, USA 2Enrico Fermi Institute, Department of Physics, and Center for Cosmological Physics University of Chicago 5640 S. Ellis Avenue, Chicago, IL 60637, USA February 2, 2008 Abstract

Introduction to Cosmology (2nd ed.) by Barbara Ryden. Read online, or download in secure PDF or secure ePub format A substantial update of this award-winning and highly regarded cosmology textbook, for advanced undergraduates in physics and astronomy. Sep 04, 2018В В· 'Barbara Ryden's Introduction to Cosmology is now published in a second edition, following the well-received first edition of 2002 вЂ¦ This is a course book for physics students; its approach is quantitative and the basic equations and mathematical descriptions are extensively outlined from first principles in all the areas covered.

Wiley Introduction to Cosmology 4th Edition Matts Roos. Jul 24, 2017В В· The addition of new material on the baryonic component of the universe links cosmology to many modern research topics in astrophysics.' Crystal Martin, University of California, Santa Barbara 'I am delighted that a second edition of Barbara Ryden's Introduction to Cosmology is now available., Cosmology - Introduction: When people hear the word Cosmology, they fear the worst: relativity, dark matter, galaxy clusters.While these topics are a part of the study of cosmology, they are not the mysterious subjects some make them out to be..

### Wiley Introduction to Cosmology 4th Edition Matts Roos

Introduction to Cosmology by Barbara Ryden Goodreads. Overview. Lectures can be downloaded as Powerpoint or pdf files from the syllabus below. The pdf files are in handout format with three slides per sheet plus space for note taking. Each lecture has an associated example exam question, with hints and answers., Introduction to Cosmology Professor Barbara Ryden Department of Astronomy Propagating wave solutions. 6/6/16 14 We describe the local curvature of spacetime with a metric. Metric = relation that gives the shortest distance Intro-Cosmo-Ryden-1 Author: Barbara Ryden.

Homework 3 Solution PHYS 3022 Introduction to Cosmology. Bookmark File PDF Barbara Ryden Solutions Manual Barbara Ryden Solutions Manual If you ally obsession such a referred barbara ryden solutions manual book that will manage to pay for you worth, acquire the categorically best seller from us currently from several preferred authors., Bookmark File PDF Barbara Ryden Solutions Manual Barbara Ryden Solutions Manual If you ally obsession such a referred barbara ryden solutions manual book that will manage to pay for you worth, acquire the categorically best seller from us currently from several preferred authors..

### Introduction to Cosmology [book review]

Introduction to Cosmology (Physics 371) вЂ“ Sean Carroll. Introduction to Cosmology (2nd ed.) by Barbara Ryden. Read online, or download in secure PDF or secure ePub format A substantial update of this award-winning and highly regarded cosmology textbook, for advanced undergraduates in physics and astronomy. Introduction to Cosmology (Physics 371) Sean Carroll, Physics Department, University of Chicago Spring quarter, 2006 (the WMAP [Wilkinson Microwave Anisotropy Probe] satellite, from the WMAP Home Page) Description. Physics 371 is a graduate-level introduction to cosmology, the study of the structure and evolution of the universe..

PHYS432/532:Cosmology Winter2018 SolutionstoHomework1 Ryden4.1.(7points) Thetotalenergyofthecosmologicalconstantcontainedinasphereofradius1AUis E = 4 3 r3" 4(4870MeV Introduction to Cosmology Barbara Ryden Addison Wesley, 2002 ISBN 0805389121 (hardcover) This book entails a really fabulous introduction to the basic concepts of cosmology, nearly entirely up to date. It is one of the few textbooks really fun reading, a marvel of didactic writing.

TASI Lectures: Introduction to Cosmology Mark Trodden1 and Sean M. Carroll2 1Department of Physics Syracuse University Syracuse, NY 13244-1130, USA 2Enrico Fermi Institute, Department of Physics, and Center for Cosmological Physics University of Chicago 5640 S. Ellis Avenue, Chicago, IL 60637, USA February 2, 2008 Abstract Jun 08, 2016В В· ICTP Summer School on Cosmology 2016 8 June 2016 - 09:15. рџ”ґ Relaxing Piano Music 24/7, Sleep Music, Piano Music, Insomnia, Meditation, Sleep, Study, Relax Yellow Brick Cinema - вЂ¦

Introduction to Cosmology, Fourth Edition includes: New theoretical approaches and in-depth material on observational astrophysics and expanded sections on astrophysical phenomena; Illustrations throughout and comprehensive references with problems at the end of вЂ¦ Introduction to cosmology / Matt Roos. вЂ“ 3rd ed. p. cm. Includes bibliographical references and index. ISBN 0-470-84909-6 (acid-free paper) вЂ“ ISBN 0-470-84910-X (pbk. : acid-free paper) 1. Cosmology. I. Title. QB981.R653 2003 523.1 вЂ” dc22 2003020688 British Library Cataloguing in Publication Data

Introduction To Cosmology Ryden Solution Manual is a bonsai collection management software for Unix systems and windows, it helps you to manage your trees, your pots Introduction To Cosmology Ryden Solutions Manual - Title: Introduction To Cosmology Ryden Solutions Manual Keywords: introduction to 4 1 An Introduction to Cosmology expansion presented in Section 3.9 suggests that either a fundamentally new theory of gravity is required, or that we do not know what constitutes most of the energy density of the universe (see Fig. 1.5). 1.2 The Cosmological Hypothesis The Friedmann Lema tre model of cosmic evolution is based on the cosmological

COVERAGE: Lecture Notes 4 and 5, and through the section on ``Dynamics of a Flat Radiation-Dominated Universe'' of Lecture Notes 6; Problem Sets 4, 5, 6, and 7; Weinberg, The First Three Minutes, Chapters 4 through the end of the book; In Ryden's Introduction to Cosmology, we have read Chapters 4, 5, and Sec. 6.1 during this period. Jun 08, 2016В В· ICTP Summer School on Cosmology 2016 8 June 2016 - 09:15. рџ”ґ Relaxing Piano Music 24/7, Sleep Music, Piano Music, Insomnia, Meditation, Sleep, Study, Relax Yellow Brick Cinema - вЂ¦

Introduction to cosmology / Matt Roos. вЂ“ 3rd ed. p. cm. Includes bibliographical references and index. ISBN 0-470-84909-6 (acid-free paper) вЂ“ ISBN 0-470-84910-X (pbk. : acid-free paper) 1. Cosmology. I. Title. QB981.R653 2003 523.1 вЂ” dc22 2003020688 British Library Cataloguing in Publication Data May 27, 2019В В· Introduction to Cosmology - Kindle edition by Barbara Ryden. Download it once and read it on your Kindle device, PC, phones or tablets. Use features like bookmarks, note taking and highlighting while reading Introduction to Cosmology.

## Introduction to Cosmology by Barbara Ryden Goodreads

Introduction to Cosmology (Physics 371) вЂ“ Sean Carroll. Introduction to cosmology / Matt Roos. вЂ“ 3rd ed. p. cm. Includes bibliographical references and index. ISBN 0-470-84909-6 (acid-free paper) вЂ“ ISBN 0-470-84910-X (pbk. : acid-free paper) 1. Cosmology. I. Title. QB981.R653 2003 523.1 вЂ” dc22 2003020688 British Library Cataloguing in Publication Data, 1 Ryden, Problem 6.3 One of the more recent speculations in cosmology is that the universe may contain a quantum п¬Ѓeld, called вЂњquintesseceвЂќ, which has a positive energy density a nd a negative value of the equationвЂ“ ofвЂ“state parameter w. Assume, for the purpose of this problem, that the universe is spatially.

### Introduction to Cosmology by Barbara Ryden Goodreads

Rien van de Weijgaert Home Page. Introduction To Cosmology Ryden Solution Manual is a bonsai collection management software for Unix systems and windows, it helps you to manage your trees, your pots Introduction To Cosmology Ryden Solutions Manual - Title: Introduction To Cosmology Ryden Solutions Manual Keywords: introduction to, Bookmark File PDF Barbara Ryden Solutions Manual Barbara Ryden Solutions Manual If you ally obsession such a referred barbara ryden solutions manual book that will manage to pay for you worth, acquire the categorically best seller from us currently from several preferred authors..

4 1 An Introduction to Cosmology expansion presented in Section 3.9 suggests that either a fundamentally new theory of gravity is required, or that we do not know what constitutes most of the energy density of the universe (see Fig. 1.5). 1.2 The Cosmological Hypothesis The Friedmann Lema tre model of cosmic evolution is based on the cosmological Jun 06, 2016В В· ICTP Summer School on Cosmology 2016 6 June 2016 - 09:15

Oct 08, 2002В В· "Introduction to Cosmology" provides a rare combination of a solid foundation of the core physical concepts of cosmology and the most recent astronomical observations. The book is designed for advanced undergraduates or beginning graduate students and assumes no prior knowledge of general relativity. Oct 08, 2002В В· "Introduction to Cosmology" provides a rare combination of a solid foundation of the core physical concepts of cosmology and the most recent astronomical observations. The book is designed for advanced undergraduates or beginning graduate students and assumes no prior knowledge of general relativity.

Introduction to Cosmology (Physics 371) Sean Carroll, Physics Department, University of Chicago Spring quarter, 2006 (the WMAP [Wilkinson Microwave Anisotropy Probe] satellite, from the WMAP Home Page) Description. Physics 371 is a graduate-level introduction to cosmology, the study of the structure and evolution of the universe. Barbara Ryden Solutions. Below are Chegg supported textbooks by Barbara Ryden. Select a textbook to see worked-out Solutions. Books by Barbara Ryden with Solutions Foundations of Astrophysics 1st Edition 191 Problems solved: Bradley M Peterson, Barbara Ryden: Introduction to Cosmology 0th Edition 0 Problems solved: Barbara Ryden

Overview. Lectures can be downloaded as Powerpoint or pdf files from the syllabus below. The pdf files are in handout format with three slides per sheet plus space for note taking. Each lecture has an associated example exam question, with hints and answers. Introduction to Cosmology, Fourth Edition includes: New theoretical approaches and in-depth material on observational astrophysics and expanded sections on astrophysical phenomena; Illustrations throughout and comprehensive references with problems at the end of вЂ¦

Barbara Ryden Solutions. Below are Chegg supported textbooks by Barbara Ryden. Select a textbook to see worked-out Solutions. Books by Barbara Ryden with Solutions Foundations of Astrophysics 1st Edition 191 Problems solved: Bradley M Peterson, Barbara Ryden: Introduction to Cosmology 0th Edition 0 Problems solved: Barbara Ryden Introduction to Cosmology Professor Barbara Ryden Department of Astronomy Propagating wave solutions. 6/6/16 14 We describe the local curvature of spacetime with a metric. Metric = relation that gives the shortest distance Intro-Cosmo-Ryden-1 Author: Barbara Ryden

Jun 08, 2016В В· ICTP Summer School on Cosmology 2016 8 June 2016 - 09:15. рџ”ґ Relaxing Piano Music 24/7, Sleep Music, Piano Music, Insomnia, Meditation, Sleep, Study, Relax Yellow Brick Cinema - вЂ¦ Bookmark File PDF Barbara Ryden Solutions Manual Barbara Ryden Solutions Manual If you ally obsession such a referred barbara ryden solutions manual book that will manage to pay for you worth, acquire the categorically best seller from us currently from several preferred authors.

Jul 24, 2017В В· The addition of new material on the baryonic component of the universe links cosmology to many modern research topics in astrophysics.' Crystal Martin, University of California, Santa Barbara 'I am delighted that a second edition of Barbara Ryden's Introduction to Cosmology is now available. Bookmark File PDF Barbara Ryden Solutions Manual Barbara Ryden Solutions Manual If you ally obsession such a referred barbara ryden solutions manual book that will manage to pay for you worth, acquire the categorically best seller from us currently from several preferred authors.

My main research interests are in galaxies and in the large scale structure of the universe. My textbook Introduction to Cosmology (1st edition В©2003, 2nd edition В©2017), won the Chambliss Astronomical Writing Award of the American Astronomical Society. Foundations of Astrophysics, written with Brad Peterson, was published in 2009. Oct 08, 2002В В· "Introduction to Cosmology" provides a rare combination of a solid foundation of the core physical concepts of cosmology and the most recent astronomical observations. The book is designed for advanced undergraduates or beginning graduate students and assumes no prior knowledge of general relativity.

Introduction to Cosmology (2nd ed.) by Barbara Ryden. Read online, or download in secure PDF or secure ePub format A substantial update of this award-winning and highly regarded cosmology textbook, for advanced undergraduates in physics and astronomy. Introduction To Cosmology Ryden Solutions Manual Read/Download Introduction To Cosmology Ryden Solution Manual is designed to help you forget about your Introduction To Cosmology Ryden Solution Manual menu,. Solutions Manual - A Primer For The Mathematics Of Financial Engineering, Second Edition Stefanica, Dan Extragalactic Astronomy

PHYS432/532:Cosmology Winter2018 SolutionstoHomework1 Ryden4.1.(7points) Thetotalenergyofthecosmologicalconstantcontainedinasphereofradius1AUis E = 4 3 r3" 4(4870MeV PHYS432/532:Cosmology Winter2018 SolutionstoHomework1 Ryden4.1.(7points) Thetotalenergyofthecosmologicalconstantcontainedinasphereofradius1AUis E = 4 3 r3" 4(4870MeV

Introduction to Cosmology (2nd ed.) by Barbara Ryden. Read online, or download in secure PDF or secure ePub format A substantial update of this award-winning and highly regarded cosmology textbook, for advanced undergraduates in physics and astronomy. 1 Ryden, Problem 6.3 One of the more recent speculations in cosmology is that the universe may contain a quantum п¬Ѓeld, called вЂњquintesseceвЂќ, which has a positive energy density a nd a negative value of the equationвЂ“ ofвЂ“state parameter w. Assume, for the purpose of this problem, that the universe is spatially

### Introduction to Cosmology by Barbara Ryden Goodreads

1 An Introduction to Cosmology Wiley-VCH. COVERAGE: Lecture Notes 4 and 5, and through the section on ``Dynamics of a Flat Radiation-Dominated Universe'' of Lecture Notes 6; Problem Sets 4, 5, 6, and 7; Weinberg, The First Three Minutes, Chapters 4 through the end of the book; In Ryden's Introduction to Cosmology, we have read Chapters 4, 5, and Sec. 6.1 during this period., barbara ryden introduction to cosmology solutions PDF may not make exciting reading, but barbara ryden introduction to cosmology solutions is packed with вЂ¦.

Homework 3 Solution PHYS 3022 Introduction to Cosmology. Introduction to Cosmology Barbara Ryden Addison Wesley, 2002 ISBN 0805389121 (hardcover) This book entails a really fabulous introduction to the basic concepts of cosmology, nearly entirely up to date. It is one of the few textbooks really fun reading, a marvel of didactic writing., Jan 13, 2006В В· View Essay - Ryden_IntroCosmo from PHYS 236 at University of California, Santa Barbara. Introduction to Cosmology Barbara Ryden Department of Astronomy The Ohio State University January 13,.

### TASI Lectures Introduction to Cosmology arXiv

PHYS 432/532 Cosmology mimas.physics.drexel.edu. Overview. Lectures can be downloaded as Powerpoint or pdf files from the syllabus below. The pdf files are in handout format with three slides per sheet plus space for note taking. Each lecture has an associated example exam question, with hints and answers. PHYS 3022 Introduction to Cosmology Spring 2014 Homework Solutions Week 3 Problem 1 - Ryden 4.2 For EinsteinвЂ™s static universe: О› = 4 ПЂGПЃ m. Assume that at some instant in time some matter density (Оґ) is converted into radiation density: ПЃ m в†’ ПЃ m-Оґ + ПЃ r + Оґ..

'Barbara Ryden's Introduction to Cosmology is now published in a second edition, following the well-received first edition of 2002 вЂ¦ This is a course book for physics students; its approach is quantitative and the basic equations and mathematical descriptions are extensively outlined from first principles in all the areas covered. Introduction to Cosmology Professor Barbara Ryden Department of Astronomy Propagating wave solutions. 6/6/16 14 We describe the local curvature of spacetime with a metric. Metric = relation that gives the shortest distance Intro-Cosmo-Ryden-1 Author: Barbara Ryden

Introduction to cosmology / Matt Roos. вЂ“ 3rd ed. p. cm. Includes bibliographical references and index. ISBN 0-470-84909-6 (acid-free paper) вЂ“ ISBN 0-470-84910-X (pbk. : acid-free paper) 1. Cosmology. I. Title. QB981.R653 2003 523.1 вЂ” dc22 2003020688 British Library Cataloguing in Publication Data Jun 08, 2016В В· ICTP Summer School on Cosmology 2016 8 June 2016 - 09:15. рџ”ґ Relaxing Piano Music 24/7, Sleep Music, Piano Music, Insomnia, Meditation, Sleep, Study, Relax Yellow Brick Cinema - вЂ¦

Jan 13, 2006В В· View Essay - Ryden_IntroCosmo from PHYS 236 at University of California, Santa Barbara. Introduction to Cosmology Barbara Ryden Department of Astronomy The Ohio State University January 13, Sep 04, 2018В В· 'Barbara Ryden's Introduction to Cosmology is now published in a second edition, following the well-received first edition of 2002 вЂ¦ This is a course book for physics students; its approach is quantitative and the basic equations and mathematical descriptions are extensively outlined from first principles in all the areas covered.

Jun 06, 2016В В· ICTP Summer School on Cosmology 2016 6 June 2016 - 09:15 Cosmology - Introduction: When people hear the word Cosmology, they fear the worst: relativity, dark matter, galaxy clusters.While these topics are a part of the study of cosmology, they are not the mysterious subjects some make them out to be.

Oct 08, 2002В В· "Introduction to Cosmology" provides a rare combination of a solid foundation of the core physical concepts of cosmology and the most recent astronomical observations. The book is designed for advanced undergraduates or beginning graduate students and assumes no prior knowledge of general relativity. Introduction to Cosmology Barbara Ryden Addison Wesley, 2002 ISBN 0805389121 (hardcover) This book entails a really fabulous introduction to the basic concepts of cosmology, nearly entirely up to date. It is one of the few textbooks really fun reading, a marvel of didactic writing.