Mastering Differential Equations: The Visual Method [TTC Video]

Mastering Differential Equations: The Visual Method [TTC Video]
Mastering Differential Equations: The Visual Method [TTC Video] by Robert L Devaney
Course No 1452 | M4V, AVC, 1500 kbps, 640x480 | AAC, 128 kbps, 2 Ch | 24x30 mins | + PDF Guidebook | 7.74GB

For centuries, differential equations have been the key to unlocking nature's deepest secrets. Over 300 years ago, Isaac Newton invented differential equations to understand the problem of motion, and he developed calculus in order to solve differential equations.

Since then, differential equations have been the essential tool for analyzing the process of change, whether in physics, engineering, biology, or any other field where it's important to predict how something behaves over time.

The pinnacle of a mathematics education, differential equations assume a basic knowledge of calculus, and they have traditionally required the rote memorization of a vast "cookbook" of formulas and specialized tricks needed to find explicit solutions. Even then, most problems involving differential equations had to be simplified, often in unrealistic ways; and a huge number of equations defied solution at all using these techniques.

But that was before computers revolutionized the field, extending the reach of differential equations into previously unexplored areas and allowing solutions to be approximated and displayed in easy-to-grasp computer graphics. For the first time, a method exists that can start a committed learner on the road to mastering this beautiful application of the ideas and techniques of calculus.

Mastering Differential Equations: The Visual Method takes you on this amazing mathematical journey in 24 intellectually stimulating and visually engaging half-hour lectures taught by a pioneer of the visual approach, Professor Robert L. Devaney of Boston University, coauthor of one of the most widely used textbooks on ordinary differential equations.

Differential Equations without Drudgery

A firm believer that there is no excuse for drudgery in a subject as fascinating as differential equations, Professor Devaney draws on the power of the computer to explore solutions visually. Throughout these graphics-intensive lectures, you investigate the geometric behavior of differential equations, seeing how the computer can calculate approximate solutions with as much precision as needed. And you may be surprised to learn how easily you can calculate and display approximate solutions yourself, even using nothing more than an ordinary spreadsheet. Best of all, the visual method means that unrealistic simplifications need not be applied to a problem.

Among those who will benefit from the exciting approach in Mastering Differential Equations are

  • college students currently enrolled in a differential equations course, who want the enriching perspective of a leader in the visual approach to the subject;
  • anyone who has completed calculus, is ready to take the next step, and is eager to see how the tools of calculus are applied to give startling insights into nature;
  • those who took differential equations in the past and would like a refresher course, especially one that shows today's revolutionary new tools for demystifying and extending the reach of the subject;
  • anyone who finds math exciting, is up for a challenge, and wants a new window into the elegantly simple structure at the heart of nature's most complex phenomena.

Beautiful Ideas plus Amazing Applications

Differential equations involve velocity, acceleration, growth rates, and other quantities that can be interpreted by derivatives, which are a fundamental concept of calculus. Often expressed with utmost simplicity and mathematical elegance, differential equations underlie some of nature's most intriguing phenomena:

  • The first and most famous differential equation is Isaac Newton's second law of motion (F = ma), which relates force, mass, and acceleration, allowing the velocity and position of an accelerating object to be determined at any point in time.
  • The Lorenz differential equation for modeling weather describes the behavior of the atmosphere in terms of a single fluid particle, showing how nature's inherent chaos can be modeled with surprisingly simple mathematics.
  • Differential equations have been used to model the catastrophic behavior of the Tacoma Narrows Bridge, which famously collapsed in 1940, and London's Millennium Bridge, which appeared headed for the same fate before it was quickly closed for modifications in 2000.
  • The precipitous drop in the North Atlantic haddock population can be understood as the bifurcation point in a differential equation, in which a slight change in one parameter—the harvesting rate—produces a drastic effect on population growth.

These and countless other applications illustrate the unrivaled ability of differential equations to stop time and sharpen our view into the past and future—a power that has grown enormously with advances in computer technology, as you explore in depth in this course.

A Taste of 21st-Century Mathematics

Each of the four sections of Mastering Differential Equations begins with a phenomenon that can be modeled with differential equations. As you probe deeper, fundamental ideas (the derivative, integral, and vector field) and other relevant tools from calculus are introduced, along with new mathematics, including four lectures on linear algebra and five lectures on chaos theory.

In the first section, you cover first-order differential equations, which involve only the first derivative of the missing function being sought as a solution. When possible, you solve the equations analytically, while making use of a wide range of visual tools, including slope fields, phase lines, and bifurcation diagrams. You also learn how computers use a simple algorithm to generate approximate solutions—and how these techniques can sometimes fail, often due to chaos.

In the second section, you turn to second-order differential equations—those that involve both the first and second derivatives. Using the mass-spring system from physics as a model, you learn that solutions are relatively straightforward as long as the mass-spring system is not forced. But when periodic forcing occurs, much more complicated behaviors arise.

In the third section, you focus on systems of differential equations, starting with linear systems and the techniques of linear algebra, which are pivotal for solving such problems. Then you shift to nonlinear systems, investigating competing species, oscillating chemical reactions, and the Lorenz system of weather modeling—which led to the famous "butterfly effect," one of the ideas that spawned chaos theory.

The final section goes deeper into chaos theory, introducing you to the cutting-edge field of dynamical systems, whose development has exploded, thanks to the rise of visual methods. Here you focus on iterated functions, also known as difference equations. Using the logistic population model from biology, you learn to analyze and understand the sudden appearance of chaos. Then you move onto the complex plane to graph the visually stunning chaos that emerges in such fractal forms as the Mandelbrot set, taking you into realms of cutting-edge mathematics.

The winner of many teaching honors, including the prestigious Deborah and Franklin Tepper Haimo Award for Distinguished University Teaching from the Mathematical Association of America, Professor Devaney is especially excited about the modern mathematics that he brings to this course. Just a few years ago, he notes, students studying differential equations seldom ventured beyond 18th-century mathematics. But Mastering Differential Equations guides you into the 21st century, showing how this deceptively simple tool—the differential equation—continues to give surprising and spectacular insights into both the world of mathematics and the workings of the universe.

Mastering Differential Equations: The Visual Method [TTC Video]

Learning to Play Guitar: Chords, Scales, and Solos [TTC Video]

Learning to Play Guitar: Chords, Scales, and Solos [TTC Video]
Learning to Play Guitar: Chords, Scales, and Solos [TTC Video] by Colin McAllister
Course No. 7776 | .MP4, AVC, 1200 kbps, 856x480 | AAC, 192 kbps, 2 Ch | 24x30 mins | + PDF Guidebook | 7.51GB

The guitar is the perfect first instrument to not only learn how to play, but to learn how to read, understand, and enjoy music. And, whether you’ve had a few years of lessons, taught yourself, or never picked a guitar up, this course is the perfect way to start.

Brilliantly designed so you can learn from the ground up, Learning to Play Guitar: Chords, Scales, and Solos is a unique and effective way to rapidly access one of the most meaningful, enjoyable, and satisfying forms of self-expression. You don’t need to know how to read a note when you start this course, but even if you do, you’ll quickly gain new techniques, knowledge, and understanding.

The highly original modular format of this course is designed to immediately get you from learning to playing. Each of the 24-lectures begins with an engaging historical narrative, or personal story, and then dives right in with five modular units of instruction:

  • Technique—Learn the correct way to hold the guitar, how to move naturally around the instrument, and all of the core skills of guitar playing, from left hand fingering and shifting to right hand strumming, finger picking, and pick playing.
  • Musicianship and Note Reading—Learn how to read three kinds of musical notation, and discover that reading music is actually quite simple if you learn it in short, easy steps; grasp how to play and read rhythms, and develop the ability to play by ear.
  • Chords—Discover how to play a wide range of basic chords--the harmonic building blocks of music and the musical foundation under a melody; study chord theory, and the fundamentals of musical structure.
  • Scales and Melodic Patterns--Study the fundamental scales for guitar—another vital building block of guitar playing—as well as guitar melodic patterns or “licks,” tools that give you the ability to play lead guitar lines and melodies.
  • Putting it All Together—From the very first lesson, put together what you’ve learned in real guitar tunes and pieces, covering a variety of different musical styles. Each lesson is capped off with a song that puts all the material to use.

Why the guitar? Like the piano, the guitar is a stand-alone solo instrument, and an endlessly adaptable “partner” for musical and vocal accompaniment. Amazingly versatile, the guitar is an instrument with an unparalleled range of expression, encompassing rock, pop, jazz, country, blues, and classical styles of music. And yet, the fundamentals of the guitar are surprisingly easy to learn—and quickly! The guitar is also one of most affordable and portable instruments. What you learn from studying the guitar easily translates to all musical instruments and disciplines, including singing, songwriting, and arranging.

Your instructor is Dr. Colin McAllister, a popular teacher and performer, whose wide musical range extends from rock bands, and the Avant garde, to classical solo guitar, chamber music. symphony orchestras, and jazz groups. Dr. McAllister studied under the “First Family” of the classical guitar—the Romero brothers—and befriended one of America’s greatest jazz guitarists, Johnny Smith. He sprinkles this course with well-told stories and deft lessons from more than 25 years of professional experience. He also shatters the myth that you must have special talent, or start at an early age, to play an instrument well. Sharing his own stories, which range from poignant to hilarious, and joined by a promising guitar student and guest vocalists, Dr. McAllister shows you that learning even four guitar chords will open up a world of possibilities, encompassing hundreds of songs, as well as styles and eras.

As you make your way through each lesson on the path from beginner to accomplished musician, you will gain the equivalent of two semesters of university-level instruction, or dozens of hours of private instruction, along with the material for hundreds of hours of practice.

This course quickly becomes an experience—one that will deepen your appreciation of everything musical that you hear.

Master the Basics of a World-Spanning Instrument

TGC’s Learning to Play Guitar: Chords, Scales, and Solos presents an engaging and adaptable learning system, developed by Dr. McAllister over decades of professional playing and teaching, which will have you playing the guitar in a remarkably short amount of time. The rapid results of these lessons are a result of Dr. McAllister’s ability to present each element of guitar playing in clear, graspable steps.

Starting from the beginning, and building one element on another, Dr. McAllister introduces you to a spectrum of basic principles and musical material, giving you a solid foundation for playing the guitar and for further study in any style. Beginning with simple chords and bass lines, you’ll learn how to play both melodies and accompaniment, as well as a range of musical styles and guitar techniques, encompassing traditions such as jazz, rock, classical, flamenco, blues, reggae, and many more.

Dr. McAllister also takes you on an odyssey into the history of the guitar, its greatest players, and the styles they created and embodied, from guitar legends to many pivotal but lesser known figures. Each lesson explores a significant topic related to the lore of the guitar, giving you a wealth of material to explore on your own.

One of the biggest benefits to this course of study is that each lesson is customizable, allowing you to study the material in a preferred sequence or focus on areas that interest you, and are easy to reference for review and practice. And the learning system is highly visual, giving you the chance to learn by watching the professor play, and using diagrams, drawings, and animations to help you incorporate the material.

Build A Solid Base for Playing in Many Styles

This course is expertly designed to give you a base knowledge of both guitar playing skills and different guitar styles. For beginners as well as more experienced players, these are techniques and materials which you can use immediately, and will give you much to explore and expand on. Some important elements you will cover include:

The Art of Improvising:

Improvising on the guitar is based on a fundamental grasp of chords and scales, as the title of the course suggests. As the lessons unfold, you’ll learn major, minor, seventh, dominant seventh, and diminished chords in different keys, and “power” chords for rock music, as well as the major and minor scales, blues scale, pentatonic (five-tone) scales, “modal” scales, and arpeggios (broken chords)—all of which will equip you to improvise in any style.

Guitar Skills:

Enriching your left and right hand technique, you’ll practice a range of key guitar skills, which will take your music-making in many exciting directions. Among these, you’ll learn “Travis picking” for folk, country, and rock playing; tremolo, which gives the illusion of a sustained tone; legato techniques (called “hammer-ons” and “pull-offs”) for shaping melodies; rasgueado, a dynamic flamenco strumming technique, harmonics (beautiful, bell-like guitar tones), barre technique (for “movable” chords), and how to accompany a vocalist.

Guitar Styles:

Throughout the lessons, you’ll learn the elements of many different musical styles. As key examples, you’ll grasp the basics of jazz harmony, making use of your seventh chords and modal scales, blues, using the blues scale and licks, and you’ll explore rhythm guitar, “piano-style” guitar, classical styles, Spanish flamenco, and others. In all cases, you’ll learn to play songs in the styles you’ve studied.

Elements of Music Theory:

In this course, for everything you learn to play, you will also learn to read it in music notation. Step by step, you’ll grasp how music is organized in terms of rhythm and harmony, both conceptually and on the page, and how different pitches, rhythms, keys, and tonalities are written. In the process, you’ll learn to read musical key signatures and time signatures, rests (musical silences), how to use a metronome, how to understand common chord progressions, and more.

Pathways to Strong Guitar Technique:

In instrumental music, “technique” is the how of playing the instrument naturally and efficiently. Proper technique is vital to successful guitar playing—as important as any other element—for achieving ease with the instrument and avoiding physical problems in playing. Throughout these lessons, you’ll learn elements of good technique such as “planting” (a practice for building technical accuracy), efficient hand position, exercises for the independence of the fingers, dexterity with a guitar pick, speed and accuracy in playing melodies, moving efficiently between chords, and how best to practice the different guitar skills you learn.

Travel into the Fascinating History and Lore of the Guitar

Throughout the lessons, Dr. McAllister’s passion for the guitar and for teaching, as well as his energetic and charismatic style, make this a highly enjoyable learning process. He enriches the lessons with inspiring and eloquent commentary on the nature of practice, the habits of successful guitar players, the value of music, and how best to study and learn.

In bringing you into the world of the guitar, he speaks vividly of the instrument’s evolution, history, and musical roles, and of the lives of guitarists—from his own professional experiences and road stories to the contributions and innovations of the instrument’s most legendary players, such as:

  • John Fahey--a master of steel-string fingerstyle guitar playing;
  • Augustin Barrios “Mangoré“--Paraguayan guitarist, composer, and poet; an outlandish genius, innovator, and one of the first classical guitarists to record;
  • Emily Remler--an astonishing talent, an improbable figure in jazz (white, middle class, woman), and one of the finest players of her time;
  • Charlie Christian--the first modern jazz guitarist, who transformed big band music, and reinvented the jazz guitar as a solo instrument; and
  • Michael Hedges--one of most innovative and influential acoustic guitarists, who pioneered ingenious alternate tunings of the instrument.

Learning to Play Guitar: Chords, Scales, and Solos offers you an engaging and highly effective way to master the fundamentals of this incredibly versatile instrument. With these 24 expertly designed lessons, the world of the guitar and the sheer joy of playing it are yours.

Learning to Play Guitar: Chords, Scales, and Solos [TTC Video]

Learning Statistics: Concepts and Applications in R [TTC Video]

Learning Statistics: Concepts and Applications in R [TTC Video]
Learning Statistics: Concepts and Applications in R [TTC Video] by Talithia Williams
Course No 1480 | MP4, AVC, 1100 kbps, 856x480 | AAC, 192 kbps, 2 Ch | 24x30 mins | + PDF Guidebook | 8.54GB

“Show me the data!” This is coin of the realm in science, medicine, business, education, journalism, and countless other fields. Of course, it’s more complicated than that, because raw data without interpretation is useless. What they mean is “Show me the statistics”—well-founded, persuasive distillations of data that support a claim under discussion.

The ability of statistics to extract insights from a random collection of facts is one of the most astonishing and useful feats of applied mathematics. That power is all the more accessible today through the statistical programming language R, a free, open-source computer language with millions of users worldwide—everyone from students and nonprofessionals to managers and researchers at the forefront of their disciplines.

In this era of big data, with a solid understanding of statistics and the tools for interpreting data, you don’t have to trust someone else’s analysis of medical treatments, financial returns, crop yields, voting trends, home prices, or any other interpretation of data. You can do it yourself.

Designed for those who appreciate math or want an introduction to an essential toolkit for thinking about the uncertainty inherent in all sorts of information, Learning Statistics: Concepts and Applications in R teaches you elementary statistical methods and how to apply them in R, which is made even more powerful when combined with the user interface of RStudio. (Both R and RStudio are free and downloadable for multiple platforms.)

In 24 challenging and in-depth half-hour lectures, award-winning Professor Talithia Williams of Harvey Mudd College walks you through major concepts of an introductory college-level statistics course, and beyond, using examples developed and presented in R. Compared with “canned” statistics packages, R brings users into a more hands-on, mind-engaging approach that is becoming the standard at top-tier statistics programs throughout the country.

An Associate Professor of Mathematics and the Associate Dean for Research and Experiential Learning at Harvey Mudd, Dr. Williams is a nationally recognized innovator in statistics education, noted for her popular TED Talk, “Own Your Body’s Data,” and she is cohost of the PBS NOVA series NOVA Wonders.

R You Ready for a Fresh Approach to Statistics?

In a course that repays multiple viewings, Professor Williams presents the most widely-used statistical measures, concepts, and techniques: how and when to use them, what they mean, and how to recognize when arguments or conclusions based on statistical data are suspect or wrong.

Learning Statistics will especially benefit those who want to go beyond a beginner level and get a deeper, fuller understanding of the discipline. And for anyone who learned statistics many years ago, this course gives an updated experience of what is going on in the field today and how user access to the R programming language is transforming the everyday practice of statistics.

The special advantages of this video-only course include:

  • Statistics concepts combined with R examples: Viewers get a two-for-one combination of thorough grounding in statistical concepts with ground-up demonstrations of how problems are solved with the R programming languge
  • A guided tour of R in action: Viewers get a gentle introduction to R in use—from how to download R and RStudio, to importing and exporting data, writing code, and generating plots. All examples in the course are conducted in R.
  • Enhanced graphics: On-screen graphics are based on outputs from RStudio, but with frequent enhancements to make the visuals even easier to read and understand.
  • Large screen or handheld: The presentation has been optimized for everything from TVs and computers to mobile devices, meaning you can watch it on a handheld device with the same comfort and clarity as on a television screen.
  • Links to the R community: When you finish these lectures, you are not on your own. Professor Williams helps you join the worldwide community of R users, who have been advising the novice and expert alike for two decades.

Professor Williams has organized the course so that it can be taken straight through, proceeding from elementary descriptive statistics to standard and advanced techniques in statistical inference. Those with a background in other statistics software may also find the progression very helpful, while students seeking help in specific areas can jump in and out at any point throughout the course.

Discover a Powerful Set of Statistical Tools

Learning Statistics begins with an overview of the field, including how to calculate and display summaries of data. Professor Williams then introduces R and discusses its advantages over other statistical analysis packages. Unlike many such products, which are costly to purchase and upgrade, R and RStudio are entirely free. Before the end of Lecture 2, you are up and running R code.

The next six lectures cover descriptive statistics and probability, in which you learn to draw conclusions from a given sample of data by using visual aids such as histograms, scatterplots, and box plots. Employing concepts such as the normal distribution, central limit theorem, and correlation, you explore a variety of probability distributions and graphical analysis techniques. You are introduced to the formulas for these operations as well as the simple R commands that run them automatically.

Starting in Lecture 8, you explore the remarkable power of statistics to make inferences about an entire population, based on a small sample. You discover how to frame a hypothesis, build a model, and deduce propositions from the resulting data. You study simple linear regression, multiple linear regression, ANOVA (analysis of variance), and other cornerstone techniques, while also using R to run simulations of many different scenarios from the R Datasets Package.

In the last third of the course, you learn how statisticians go beyond what beginners are often taught, developing branches of applied statistics that have spun off to form their own immensely productive specialties. These include:

  • Experimental design: While there are many techniques for analyzing data you already have, even more powerful is designing an experiment to decide how data is collected from the start. Consider such elements of good design as blocking, randomization, and replication to ensure that your experiment produces sound statistical results.
  • Spatial statistics: Maps have always been information-rich artifacts, but they are now more useful than ever thanks to the advent of GPS-enabled data-gathering devices and powerful computers, combined with a panoply of statistical tools for treating spatial autocorrelation as a rich new source of information.
  • Time series analysis: Just as fascinating as spatial data is information collected sequentially over time—in finance, meteorology, biology, agriculture, and other fields. One of the most important goals of time series analysis is forecasting, which extracts short- and longer-term patterns in the data.
  • Bayesian inference: Textbook statistics is often based on a “frequentist” paradigm, in which sampling is theoretically unlimited. But for many real-life situations, your information is almost always incomplete, and likely to be revised. This is the forte of Bayesian inference.

You close the course with a lecture on how to customize R to select and combine information in whatever way you want, so that R best serves your own needs.

Dr. Williams has made it her life’s work to get students, parents, educators, and the community at large excited about mathematics and especially statistics, which she describes as “a powerful framework for THINKING—for reaching insights and solving problems.” As witnessed by her TED Talk, which has been viewed over one million times, Dr. Williams has a gift for demystifying statistics and making it relevant to everyone—because whenever you hear a statistical argument that directly affects your health, livelihood, autonomy, or your firmly held beliefs, you should say, “Show me the data, so I can decide for myself.” With this course, you will be able to do exactly that.

Learning Statistics: Concepts and Applications in R [TTC Video]

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