Online courses directory (19947)

This course will focus on the wars and military conflicts that have shaped the social, political, and economic development of the United States from the colonial era through the present.  You will learn how these conflicts have led to significant changes in America social and political life during this 300-year period.  The course will be structured chronologically.  Each unit will include representative primary-source documents that illustrate important overarching themes, such as how colonial conflicts in the 18th century shaped the political organization of the United States, how regional conflicts in the 19th century culminated in the Civil War, how America cemented its status as a major world power through participation in the First and Second World Wars, how Cold War conflicts destabilized American social and political life, and how modern conflicts continue to redefine American social and political values and ideals.  By the end of the course, you will understand how three centuries of warfare have…

This course will introduce you to a comparative history of New World societies from 1400 to 1750. You will learn about European exploration and colonization as well as the cultures of native peoples of the Americas. The course will be structured geographically; each unit will focus on a particular New World society during a specific time period. Each unit will include representative primary-source documents that illustrate important overarching political, economic, and social themes, such as the fifteenth-century conceptualization of the “New World” and colonization, the indigenous peoples living in the Americas at the time of European contact, and the effect of New World societies on native peoples and Africans. By the end of the course, you will understand how the new communities in the New World evolved from fledgling settlements into profitable European colonies and how New World societieswhether French, Spanish, Portuguese, English, or indigenouswere highly varied polities.

This course will introduce you to the history of Central Eurasia and the Silk Road from 4500 B.C.E to the nineteenth century.  You will learn about the culture of the nomadic peoples of Central Eurasia as well as the development of the Silk Road.  The course will be structured chronologically; each unit will focus on one aspect of the Silk Road during a specific time period.  Each unit will include representative primary- and secondary-source documents that illustrate important overarching political, economic, and social themes, such as the discovery and production of silk in China, diplomatic relations between Han China and nomadic peoples of the Eurasian steppe, the international scope of the Silk Road trade routes, European interest in finding a “new silk route” to China, and the “Great Game” between China, Russia, and Great Britain in Central Eurasia in the nineteenth century.  By the end of the course, you will understand how the Silk Road influenced the development of nomadic societies in Ce…

This course will introduce you to the history of the Middle East from the rise of Islam to the twenty-first century.  The course will emphasize the encounters and exchanges between the Islamic world and the West.  It will be structured chronologicallyeach unit will focus on the emergence of a particular Middle Eastern society or empire during a specific time period.  Each unit will include representative primary-source documents that illustrate important overarching political, economic, and social themes, such as the emergence of Islam in the seventh century, conflicts between Islamic and Christian peoples during the Crusades, European domination of Muslim territories in the nineteenth century, independence movements and the rise of nationalism in the 1900s, and the formation of Islamic fundamentalist groups and anti-Western sentiment in the latter twentieth century.  By the end of the course, you will understand how Islam became a sophisticated and far-reaching civilization and how conflicts with the Wes…

In the 1970s, the Chinese Communist leader Zhou Enlai was asked to assess the outcomes of the French Revolution of 1789.  He supposedly answered: “It is too soon to say.”  Though this story has a somewhat apocryphal status, it captures a fundamental truth about the world in which we live: it is a world which has been shaped by revolutions, and their legacies are always difficult to evaluate. In this course, you will gain a better understanding of the modern world by studying some of the most important political revolutions that took place between the 17th century and today.  You will seek to understand the causes of each revolution, analyze the ideologies that inspired the revolutionaries, examine revolutionary uses of violence, and consider how historical revolutions still shape contemporary politics.  Close and critical readings of historical sources will be crucial in this process. The course begins with a theoretical analysis of revolutions and a careful examination of pre-revolutionary Europe…

This course will focus on the emergence and evolution of industrial societies around the world.  We will begin by comparing the legacies of industry in ancient and early modern Europe and Asia and examining the agricultural and commercial advances that laid the groundwork for the Industrial Revolution of the 18th and 19th centuries.  We will then follow the history of industrialization in different parts of the world, taking a close look at the economic, social, and environmental effects of industrialization.  The course is organized chronologically and thematically.  Each unit will focus on key developments in the history of industry as well as on representative areas of the globe, using primary-source documents, secondary readings, and multimedia resources to illustrate the dynamic nature of industrial change.  By the end of the course, you will understand how industrialization developed, spread across the globe, and shaped everyday life in the modern era.

This course will focus on the history of humankind’s relationship with the environment.  We use the word “environment” to refer to the nonhuman components of the natural world.  We will examine how environmental factors have shaped the development and growth of civilizations around the world and analyze how these civilizations have altered their environments in positive and negative ways.  The course will be structured chronologically.  Each unit will include representative primary-source documents that illustrate important overarching themes, such as how early humans adapted natural resources for new purposes, how the expansion of civilizations led to environmental changes, how the interaction between European explorers and Native Americans led to significant and unexpected environmental consequences, and how modern societies have responded to environmental problems that threaten the well-being of humans and the environment.  By the end of the course, you will better understand the reciprocal rela…

This course provides an introduction to the history of technology for the Science, Technology, and Society (STS) major.  The course surveys major technological developments from ancient to modern times with particular attention to social, political, and cultural contexts in Europe and the United States.  You will also think critically about the theory of technological determinism, the ways in which technology has defined “progress” and “civilization”, and the major ethical considerations surrounding today’s technological decisions. This course begins with discussions of the promotion of technology in centralized states of the ancient and medieval worlds: the Roman Empire, Song and Ming China, and the Islamic Abbasid Empire.  After a period of relative decline, the states of Western Europe centralized and flourished once again, having benefited from the westward transmission of key ideas and technologies from the East. The focus of the course then shifts to the West, to the technologies of the…

Precalculus II continues the in-depth study of functions addressed in Precalculus I by adding the trigonometric functions to your function toolkit. In this course, you will cover families of trigonometric functions, as well as their inverses, properties, graphs, and applications. Additionally, you will study trigonometric equations and identities, the laws of sines and cosines, polar coordinates and graphs, parametric equations and elementary vector operations. You might be curious how the study of trigonometry, or “trig,” as it is more often referred to, came about and why it is important to your studies still. Trigonometry, from the Greek for “triangle measure,” studies the relationships between the angles of a triangle and its sides and defines the trigonometric functions used to describe those relationships. Trigonometric functions are particularly useful when describing cyclical phenomena and have applications in numerous fields, including astronomy, navigation, music theory, physics, chemistry…

Calculus can be thought of as the mathematics of CHANGE. Because everything in the world is changing, calculus helps us track those changes. Algebra, by contrast, can be thought of as dealing with a large set of numbers that are inherently CONSTANT. Solving an algebra problem, like y = 2x + 5, merely produces a pairing of two predetermined numbers, although an infinite set of pairs. Algebra is even useful in rate problems, such as calculating how the money in your savings account increases because of the interest rate R, such as Y = X0+Rt, where t is elapsed time and X0 is the initial deposit. With compound interest, things get complicated for algebra, as the rate R is itself a function of time with Y = X0 + R(t)t. Now we have a rate of change which itself is changing. Calculus came to the rescue, as Isaac Newton introduced the world to mathematics specifically designed to handle those things that change. Calculus is among the most important and useful developments of human thought. Even though it is over…

This course is the second installment of Single-Variable Calculus.  In Part I (MA101) [1], we studied limits, derivatives, and basic integrals as a means to understand the behavior of functions.  In this course (Part II), we will extend our differentiation and integration abilities and apply the techniques we have learned. Additional integration techniques, in particular, are a major part of the course.  In Part I, we learned how to integrate by various formulas and by reversing the chain rule through the technique of substitution.  In Part II, we will learn some clever uses of substitution, how to reverse the product rule for differentiation through a technique called integration by parts, and how to rewrite trigonometric and rational integrands that look impossible into simpler forms.  Series, while a major topic in their own right, also serve to extend our integration reach: they culminate in an application that lets you integrate almost any function you’d like. Integration allows us to calculat…

Multivariable Calculus is an expansion of Single-Variable Calculus in that it extends single variable calculus to higher dimensions.  You may find that these courses share many of the same basic concepts, and that Multivariable Calculus will simply extend your knowledge of functions to functions of several variables.  The transition from single variable relationships to many variable relationships is not as simple as it may seem; you will find that multi-variable functions, in some cases, will yield counter-intuitive results. The structure of this course very much resembles the structure of Single-Variable Calculus I and II.  We will begin by taking a fresh look at limits and continuity.  With functions of many variables, you can approach a limit from many different directions.  We will then move on to derivatives and the process by which we generalize them to higher dimensions.  Finally, we will look at multiple integrals, or integration over regions of space as opposed to intervals. The goal of Mu…

The main purpose of this course is to bridge the gap between introductory mathematics courses in algebra, linear algebra, and calculus on one hand and advanced courses like mathematical analysis and abstract algebra, on the other hand, which typically require students to provide proofs of propositions and theorems.  Another purpose is to pose interesting problems that require you to learn how to manipulate the fundamental objects of mathematics: sets, functions, sequences, and relations.  The topics discussed in this course are the following: mathematical puzzles, propositional logic, predicate logic, elementary set theory, elementary number theory, and principles of counting.  The most important aspect of this course is that you will learn what it means to prove a mathematical proposition.  We accomplish this by putting you in an environment with mathematical objects whose structure is rich enough to have interesting propositions.  The environments we use are propositions and predicates, finite sets and…

Differential equations are, in addition to a topic of study in mathematics, the main language in which the laws and phenomena of science are expressed.  In basic terms, a differential equation is an expression that describes how a system changes from one moment of time to another, or from one point in space to another.  When working with differential equations, the ultimate goal is to move from a microscopic view of relevant physics to a macroscopic view of the behavior of a system as a whole. Let’s look at a simple differential equation.  Based on previous math and physics courses, you know that a car that is constantly accelerating in the x-direction obeys the equation d2x/dt2 = a, where a is the applied acceleration.  This equation has two derivations with respect to time, so it is a second-order differential equation; because it has derivations with respect to only one variable (in this example, time), it is known as an  ordinary differential equation, or an ODE. Let’s say that we want to sol…

The study of “abstract algebra” grew out of an interest in knowing how attributes of sets of mathematical objects behave when one or more properties we associate with real numbers are restricted.  For example, we are familiar with the notion that real numbers are closed under multiplication and division (that is, if we add or multiply a real number, we get a real number).  But if we divide one integer by another integer, we may not get an integer as a resultmeaning that integers are not closed under division.  We also know that if we take any two integers and multiply them in either order, we get the same resulta principle known as the commutative principle of multiplication for integers.  By contrast, matrix multiplication is not generally commutative.  Students of abstract algebra are interested in these sorts of properties, as they want to determine which properties hold true for any set of mathematical objects under certain operations and which types of structures result when we perform certain o…

This course is designed to introduce you to the rigorous examination of the real number system and the foundations of calculus of functions of a single real variable. Analysis lies at the heart of the trinity of higher mathematics algebra, analysis, and topology because it is where the other two fields meet. In calculus, you learned to find limits, and you used these limits to give a rigorous justification for ideas of rate of change and areas under curves. Many of the results that you learned or derived were intuitive in many cases you could draw a picture of the situation and immediately “see” whether or not the result was true. This intuition, however, can sometimes be misleading. In the first place, your ability to find limits of real-valued functions on the real line was based on certain properties of the underlying field on which undergraduate calculus is founded: the real numbers. Things may have become slightly more complicated when you began to work in other spaces. For instance, you may r…

This course will introduce students to the field of computer science and the fundamentals of computer programming.  It has been specifically designed for students with no prior programming experience, and does not require a background in Computer Science.  This course will touch upon a variety of fundamental topics within the field of Computer Science and will use Java, a high-level, portable, and well-constructed computer programming language developed by Sun Microsystems, to demonstrate those principles.  We will begin with an overview of the topics we will cover this semester and a brief history of software development.  We will then learn about Object-Oriented programming, the paradigm in which Java was constructed, before discussing Java, its fundamentals, relational operators, control statements, and Java I/0.  The course will conclude with an introduction to algorithmic design.  By the end of the course, you should have a strong understanding of the fundamentals of Computer Science and the Java p…

Linear algebra is the study of vector spaces and linear mappings between them.  In this course, we will begin by reviewing topics you learned in Linear Algebra I, starting with linear equations, followed by a review of vectors and matrices in the context of linear equations.  The review will refresh your knowledge of the fundamentals of vectors and of matrix theory, how to perform operations on matrices, and how to solve systems of equations.  After the review, you should be able to understand complex numbers from algebraic and geometric viewpoints to the fundamental theorem of algebra.  Next, we will focus on eigenvalues and eigenvectors.  Today, these have applications in such diverse fields as computer science (Google's PageRank algorithm), physics (quantum mechanics, vibration analysis, etc.), economics (equilibrium states of Markov models), and more.  We will end with the spectral theorem, which provides a decomposition of the vector space on which operators act, and singular-value decomposition, w…

Numerical analysis is the study of the methods used to solve problems involving continuous variables.  It is a highly applied branch of mathematics and computer science, wherein abstract ideas and theories become the quantities describing things we can actually touch and see.  The real number line is an abstraction where many interesting and useful ideas live, but to actually realize these ideas, we are forced to employ approximations of the real numbers.  For example, consider marking a ruler at sqrt{2}.  We know that sqrt{2} approx 1.4142, but if we put the mark there, we know we are in error for there is an infinite sequence of nonzero digits following the 2.  Even more: a number doesn’t have any width, yet any mark we make would have a width, and in that width lives an infinite number of real numbers.  You may ask yourself: isn’t it sufficient to represent sqrt{2} with 1.414?  This is the kind of question that this course will explore.  We have been trying to answer such questions for over 2,0…

This course is a continuation of Abstract Algebra I: we will revisit structures like groups, rings, and fields as well as mappings like homomorphisms and isomorphisms.  We will also take a look at ring factorization, which will lead us to a discussion of the solutions of polynomials over abstracted structures instead of numbers sets.  We will end the section on rings with a discussion of general lattices, which have both set and logical properties, and a special type of lattice known as Boolean algebra, which plays an important role in probability.  We will also visit an important topic in mathematics that you have likely encountered already: vector spaces.  Vector spaces are central to the study of linear algebra, but because they are extended groups, group theory and geometric methods can be used to study them. Later in this course, we will take a look at more advanced topics and consider several useful theorems and counting methods.  We will end the course by studying Galois theoryone of the most im…