This course is a continuation of Abstract Algebra I: the student will revisit structures like groups, rings, and fields as well as mappings like homomorphisms and isomorphisms. The student will also take a look at ring factorization, general lattices, and vector spaces. Later this course presents more advanced topics, such as Galois theory - one of the most important theories in algebra, but one that requires a thorough understanding of much of the content we will study beforehand. Upon successful completion of this course, students will be able to: Compute the sizes of finite groups when certain properties are known about those groups; Identify and manipulate solvable and nilpotent groups; Determine whether a polynomial ring is divisible or not and divide the polynomial (if it is divisible); Determine the basis of a vector space, change bases, and manipulate linear transformations; Define and use the Fundamental Theorem of Invertible Matrices; Use Galois theory to find general solutions of a polynomial over a field. (Mathematics 232)
This course is oriented toward US high school students. The course is divided into 10 units of study. The first five units build the foundation of concepts, vocabulary, knowledge, and skills for success in the remainder of the course. In the final five units, we will take the plunge into the domain of inferential statistics, where we make statistical decisions based on the data that we have collected.
This course introduces the history of the Age of Revolutions in the Atlantic World from 1776 to 1848. Running alongside and extending beyond these political revolutions is the First Industrial Revolution. The Atlantic World, dominated by European empires in 1776, was transformed through revolution into a series of independent states by 1848, experiencing profound changes through the development and consolidation of capitalism. Upon successful completion of this course, the student will be able to: think analytically about the history of the revolutionary age between 1776 and 1848; define what a revolution" means as well as describe what made 1776-1848 an "age of revolution"; define the concept of the Atlantic World and describe its importance in World History; explain the basic intellectual and technical movements associated with the Enlightenment and their relations to the revolutionary movements that follow; identify and describe the causes of the American Revolution; identify and describe the many stages of the French Revolution: the end of absolutist monarchy, the implementation of constitutional monarchy, and the rise of the Jacobin Republic; compare and contrast the Declaration of the Rights of Man and other major statements of the Revolutionary period and Enlightenment thinking; identify and describe the impact of the first successful slave rebellion in world history--the Haitian Revolution; compare and contrast the debate between Edmund Burke and Thomas Paine; analyze and interpret primary source documents that elucidate the causes and effects of the Age of Revolutions. This free course may be completed online at any time. (History 303)
This course will introduce the student to the history of the Atlantic slave trade from 1500 to 1900. The student will learn about the slave trade, its causes, and its effects on Africa, Europe, and the Americas. By the end of the course, the student will understand how the Atlantic slave trade began as a fledgling enterprise of the English, Portuguese, and Spanish in the 1500s and why, by the mid-eighteenth century, the trade dominated Atlantic societies and economies. Upon completion of this course, students will be able to: think analytically about the various meanings of 'slave' and 'slavery' during the age of the Atlantic slave trade; identify and describe the 'triangular trade' and define the Atlantic World; identify and describe the logic for enslavement of Africans by Europeans; identify and describe the African ethnic groups enslaved by Europeans and those captives' New World destinations; identify and describe the early slaving voyages of the Portuguese and Spanish. Students will also be able to describe how the Dutch and English later inserted themselves into the trade; identify and describe the expansion of the plantation complex in the New World in the 1600s and its impact on the Atlantic slave trade; identify and analyze the rise of European empires and the parallel expansion of the Atlantic slave trade; identify and analyze slavery within African societies. They will also be able to identify and describe the trans-Saharan slave trade and the Red Sea/Indian Ocean slave trade; identify and describe the nature of the African slave market and principal slaving ports in western Africa; analyze and describe New World slave societies and their impact on the Atlantic slave trade; identify and describe the 'Middle Passage' of the Atlantic slave trade; identify and describe the causes for the abolition of the Atlantic slave trade in the nineteenth century; analyze and interpret primary source documents that elucidate all aspects of the Atlantic slave trade. (History 311)
This course discusses how to use algebra for a variety of everyday tasks, such as calculate change without specifying how much money is to be spent on a purchase, analyzing relationships by graphing, and describing real-world situations in business, accounting, and science.
This course will cover American political thought from the nation's founding through the 1960s, exploring the political theories that have shaped its governance. As there is no one philosopher or idea that represents the totality of American political thought, the student will survey the writings and speeches of those who have had the greatest impact over this period of time. Much of the study required in this course is based on the original texts and speeches of those who influenced political thought throughout American history. The student will learn and understand the impact that their views and actions have had on the modern American state. Upon successful completion of this course, students will be able to: describe the religious and political origins of the American political system; explain how Enlightenment thinkers, such as John Locke, Thomas Hobbes, and Jean-Jacques Rousseau, and Baron de Montesquieu, influenced the political philosophies of American founding fathers; analyze how the colonial American experience shaped many of the core values represented in American government and expressed in the Declaration of Independence and the U.S. Constitution; compare and contrast the differing opinions on the role of the government that the founders expressed; trace the development and evolution of the concepts of 'states rights' and 'federal (national) supremacy'; connect the observations of De Tocqueville in Democracy in America to the concepts of equality, individuality, and civic engagement in American political discourse; examine the evolution of race in the American political system (from slavery to the 2008 election of Barack Obama); discuss the changes in the political role of women in America from its colonial days to the present; connect the concept of 'American Exceptionalism' to the industrial revolution, capitalism, and imperialism; analyze the roots of reform in the Progressive Era and their impact on modern political discourse; explain major principles of American foreign relations over time; assess the purpose and impact of ĺÎĺĺĺŤAmerican war rhetoricĄ_ĺĺö over time; differentiate between 'liberal' and 'conservative' political beliefs in modern American government; illustrate how the political turmoil in the 1960s greatly shaped contemporary American political discourse; evaluate the current political discourse as represented in the 2008 and 2010 elections. (Political Science 301)
In this course, the student will study the emergence of the major civilizations of the ancient world, beginning with the Paleolithic Era (about 2.5 million years ago) and finishing with the end of the Middle Ages in fifteenth century A.D. The student will pay special attention to how societies evolved across this expanse of time - from fragmented and primitive agricultural communities to more advanced and consolidated civilizations. By the end of the course, the student will possess a thorough understanding of important overarching social, political, religious, and economic themes in the ancient world, ranging from the emergence of Confucian philosophy in Asia to the fall of imperial Rome. Upon successful completion of this course, the student will be able to: Identify and define the world's earliest civilizations, including the Neolithic Revolution, and describe how it shaped the development of these early civilizations; Identify, describe, and compare/contrast the first advanced civilizations in the world - Mesopotamia and Egypt; Identify and describe the emergence of the earliest civilizations in Asia: the Harappan and Aryan societies on the Indian subcontinent and the Shang and Zhou societies in China; Identify and describe the emergence of new philosophies - Daoism and Confucianism - during the Warring States period in China. Identify and describe the subsequent rise of the Qin and Han dynasties; Identify and describe the different periods that characterized ancient Greece - Archaic Greece (or the Greek Dark Ages), classical Greece, and the Hellenistic era; Identify and describe the characteristics of the Roman Kingdom, the Roman Republic, and Imperial Rome; Analyze the emergence of the Mauryan and Gupta empires during the 'classical age' in India; Identify and analyze the Buddhist and Vedic (Hindu) faiths; Identify and describe the rise of civilizations in the Americas, particularly in Meso and South America; Analyze and describe the rise of Islam in the Middle East; Identify and describe the emergence of the Arab caliphate, the Umayyad dynasty, and Abbasid dynasty; Identify and describe the rise and fall of the Byzantine Empire; Identify and analyze key facets of medieval society in Western EuropeĺÎĺĚ_ĺÜthe Catholic Church, feudalism, and the rise of technology and commerce; Analyze and interpret primary-source documents that elucidate the exchanges and advancements made in civilizations across time and space. (History 101)
This course is an exploration of visual art forms and their cultural connections for the student with little experience in the visual arts. It includes a brief study of art history and in depth studies of the elements, media, and methods used in creative processes and thought. Upon successful completion of this course, students will be able to: interpret examples of visual art using a five-step critical process that includes description, analysis, context, meaning, and judgment; identify and describe the elements and principles of art; use analytical skills to connect formal attributes of art with their meaning and expression; explain the role and effect of the visual arts in societies, history, and other world cultures; articulate the political, social, cultural, and aesthetic themes and issues that artists examine in their work; identify the processes and materials involved in art and architectural production; utilize information to locate, evaluate, and communicate information about visual art in its various forms. Note that this course is an alternative to the Saylor FoundationĺÎĺ_ĺĚĺ_s ARTH101A and has been developed through a partnership with the Washington State Board for Community and Technical Colleges; the Saylor Foundation has modified some WSBCTC materials. This free course may be completed online at any time. (Art History 101B)
This course is also intended to provide the student with a strong foundation for intermediate algebra and beyond. Upon successful completion of this course, you will be able to: simplify and solve linear equations and expressions including problems with absolute values and applications; solve linear inequalities; find equations of lines; and solve application problems; add, subtract, multiply, and divide various types of polynomials; factor polynomials, and simplify square roots; evaluate, simplify, multiply, divide, add, and subtract rational expressions, and solve basic applications of rational expressions. This free course may be completed online at any time. It has been developed through a partnership with the Washington State Board for Community and Technical Colleges; the Saylor Foundation has modified some WSBCTC materials. (Mathematics 001)
Introductory survey of quantitative methods (QM), or the application of statistics in the workplace. Examines techniques for gathering, analyzing, and interpreting data in any number of fieldsĺÎĺ from anthropology to hedge fund management.
This course is oriented toward US high school students. Its structure and materials are aligned to the US Common Core Standards. Foci include: derivatives, integrals, limits, approximation, and applications.
This course begins with a review of algebra specifically designed to help and prepare the student for the study of calculus, and continues with discussion of functions, graphs, limits, continuity, and derivatives. The appendix provides a large collection of reference facts, geometry, and trigonometry that will assist in solving calculus problems long after the course is over. Upon successful completion of this course, the student will be able to: calculate or estimate limits of functions given by formulas, graphs, or tables by using properties of limits and LĺÎĺ_ĺĚĺ_hopitalĺÎĺ_ĺĚĺ_s Rule; state whether a function given by a graph or formula is continuous or differentiable at a given point or on a given interval and justify the answer; calculate average and instantaneous rates of change in context, and state the meaning and units of the derivative for functions given graphically; calculate derivatives of polynomial, rational, common transcendental functions, and implicitly defined functions; apply the ideas and techniques of derivatives to solve maximum and minimum problems and related rate problems, and calculate slopes and rates for function given as parametric equations; find extreme values of modeling functions given by formulas or graphs; predict, construct, and interpret the shapes of graphs; solve equations using NewtonĺÎĺ_ĺĚĺ_s Method; find linear approximations to functions using differentials; festate in words the meanings of the solutions to applied problems, attaching the appropriate units to an answer; state which parts of a mathematical statement are assumptions, such as hypotheses, and which parts are conclusions. This free course may be completed online at any time. It has been developed through a partnership with the Washington State Board for Community and Technical Colleges; the Saylor Foundation has modified some WSBCTC materials. (Mathematics 005)
In this course, the student will explore campaigns and elections, learning their purpose and significance and observing the impact that they have on the American political system. The course will focus on the history and evolution of elections and voting laws in the United States, as well as what compels individuals to run for office and how campaigns are structured. Also, the course will teach the student the role that political parties, interest groups, voters, and the media play in elections. Lastly, the student will take a closer look at electoral outcomes and the impact that elections have on public policy after votes are counted, as well as what types of proposals could be implemented to improve our electoral system. Upon successful completion of this course, the student will be able to: explain the importance of elections, voting, democracy, and citizenship in the United States; describe the various types of elections that exist within the American political system; identify the legal and constitutional bases of campaigns and elections in the United States; explain the types of individuals that run for political office and why; analyze the influence of incumbency in elections; explain how candidates develop campaigns and financing; discuss the role of money in political campaigns; discuss the influence of political parties on campaigns and elections; describe the characteristics of the U.S. party system; explain the role of interest groups in influence campaigns and election outcomes; explain the various influences and motivations of the American voter; describe the factors associated with both nonvoter and voter disenfranchisement in contemporary elections; analyze and explain the critical role of the media in campaigns and elections; explain how election outcomes impact government actions and public policy; analyze both historical and contemporary election reforms. (POLSC333)
The purpose of this course is to trace the twin paths of capitalism and democracy through American history. This course is premised on the idea that capitalism and democracy are intertwined, though they have often conflicted with one another. It provides students with a brief introduction to the history of capitalism and democracy in Europe and then to explore how they evolved in North America between 1600 and the present. Upon successful completion of this course, students will be able to: define and identify the terms 'capitalism' and 'democracy' in a variety of different modern historical eras; identify and define the historical connections between capitalism and democracy and identify periods of tension between capitalism and democracy, explaining how they both strengthen and weaken one another; identify important events, personalities, and concepts related to American democracy and capitalism; identify and describe the emergence and development of both capitalism and democracy in the United States; identify and describe the different periods of American history as they relate to the concepts of capitalism and democracy. (History 312)
This course will introduce the student to the history of Latin and South America from the year in which European explorers first discovered and began to colonize the region to the early 19th century, when many Latin and South American colonies declared their independence from European rule. The student will learn about the major political, economic, and social changes that took place throughout Latin and South America during this 400-year period. By the end of the course, the student will understand how the interaction between native peoples and European settlers created diverse and complex colonial societies throughout Latin and South America, and why the colonies of the region eventually declared their independence from European political control. Upon successful completion of this course, student will be able to: Think critically about the history of Latin and South America from the pre-colonial period though the beginning of the 19th century; Compare and contrast the political, economic, and social practices of the peoples of Iberia, Africa, and the Americas in the pre-colonial period; Analyze the political, social, and military interactions between Iberian explorers and conquerors and the indigenous peoples of the Americas in the 15th and 16th centuries; Identify how Spanish colonists settled Latin and South America in the 16th century and analyze the role played by imperial and religious institutions in colonization efforts; Assess the role of European Mercantile policies in the formation of colonial economies and trade networks; Analyze the structure of Spanish and Portuguese colonial societies and assess the role of women, indigenous peoples, and Afro-Latinos in these societies; Students will be able to assess the status of Latin and South American colonies in the Spanish and Portuguese Empires of the 17th and 18th centuries and identity how European conflicts affected political and economic life in the colonies; Identify how the Napoleonic Wars of the early 19th century led to the rise of independence movements in the colonies of Latin and South America; Assess how political revolutions and wars for independence throughout Latin and South America ended European colonial control of the region, and compare and contrast the consequences of these revolutions for ethnic European and indigenous populations; Analyze and interpret primary source documents from the pre-colonial period though the beginning of the 19th century using historical research methods. (History 221)
This course will introduce the student to a comparative history of New World societies from 1400 to 1750. The student will learn about European exploration and colonization as well as the culture of native peoples of the Americas. By the end of the course, you will understand how the New World evolved from fledgling settlements into profitable European colonies and how New World societies were highly varied polities. Upon successful completion of this course, the student will be able to: analyze what constituted the 'New World' in the fifteenth century; identify and describe the major tribes/native civilizations of North America, Central America, South America, and the Caribbean at the time of European contact; identify and describe the effects of European colonization on native peoples; identify and describe the reasons for the European Age of Discovery in the New World; identify and describe early New World exploration and initial settlements by Portugal and Spain; identify and describe how and why the consolidation of powerful European states in the 1600s resulted in New World exploration, settlement, and commerce; compare and contrast New France, French Louisiana, the French West Indies, and French Guiana; compare and contrast British North America (New England, Middle and Lower Colonies), the British West Indies, and British Central and South America; compare and contrast New Spain, the Spanish Caribbean, and Spanish South America; analyze and describe Portuguese Brazil; identify and describe the African slave trade and will also be able to compare and contrast the enslavement of Africans in New World societies; identify and describe inter-European conflicts and European-Native Indian violence in the New World; analyze and interpret primary source documents that elucidate the causes and effects of exploration and colonization in the New World. (History 321)
This course is an introduction to complex analysis, or the theory of the analytic functions of a complex variable. Put differently, complex analysis is the theory of the differentiation and integration of functions that depend on one complex variable. Because of the algebraic properties of the complex numbers and the inherently geometric flavor of complex analysis, this course will feel quite different from Real Analysis, although many of the same concepts, such as open sets, metrics, and limits will reappear. Upon successful completion of this course, the student will be able to: manipulate complex numbers in various representations, define fundamental topological concepts in the context of the complex plane, and define and calculate limits and derivatives of functions of a complex variable; represent analytic functions as power series on their domains and verify that they are well-defined; define a branch of the complex logarithm; classify singularities and find Laurent series for meromorphic functions; state and prove fundamental results, including CauchyĺÎĺ_ĺĚĺ_s Theorem and CauchyĺÎĺ_ĺĚĺ_s Integral Formula, the Fundamental Theorem of Algebra, MoreraĺÎĺ_ĺĚĺ_s Theorem and LiouvilleĺÎĺ_ĺĚĺ_s Theorem; use them to prove related results; calculate contour integrals; calculate definite integrals on the real line using the Residue Theorem; define linear fractional transformations and prove their essential characteristics; find the image of a region under a conformal mapping; state, prove, and use the Open Mapping Theorem. This free course may be completed online at any time. (Mathematics 243)
This course focuses on linear ordinary differential equations (or ODEs) and will introduce several other subclasses and their respective properties. Despite centuries of study, numerical approximation is the only practical approach to the solution of complicated ODEs that has emerged; this course will introduce you to the fundamentals behind numerical solutions. Upon successful completion of this course, students will be able to: Identify ordinary differential equations and their respective orders; Explain and demonstrate how differential equations are used to model certain situations; Solve first order differential equations as well as initial value problems; Solve linear differential equations with constant coefficients; Use power series to find solutions of linear differential equations, Solve linear systems of differential equations with constant coefficients; Use the Laplace transform to solve initial value problems; Use select methods of numerical approximation to find solutions to differential equations. (Mathematics 221; See also: Mechanical Engineering 003)
This course introduces the history of the Middle East and Southwest Asia from the pre-Islamic period to the end of World War I. Upon successful completion of this course, the student will be able to: discuss the history of East Asia from the pre-Islamic period through the beginning of the 20th century; analyze the interactions between ancient civilizations of the Middle East and Southwest Asia in the pre-Islamic period; identify the origins of Islam, and assess the political and cultural impact of the Muslim faith on the peoples of the Middle East and the Mediterranean Basin; identify the origins of the Umayyad and Abbasid Empires, and assess how these dynasties reshaped political and economic life throughout the Middle East and Southwest Asia; describe and assess the social and cultural impact of Islam on the peoples of the Middle East and the Mediterranean Basin; identify external threats to the Muslim world during the Middle Ages, and analyze how Muslim leaders responded to these threats; identify the origins of the Ottoman Empire, and assess how the Ottomans established political and economic control over the Eastern Mediterranean and the Middle East; analyze the political, economic, and military interactions between the Ottoman Empire and the nations of Europe in the 18th and 19th centuries; explain how European imperialism destabilized the Middle East and Southwest Asia in the 19th and early 20th centuries and allowed European nations to establish political control over many Middle Eastern nations; analyze the political impact of World War I on the peoples and nations of the Middle East; analyze and interpret primary source documents from the pre-Islamic period through the beginning of the 20th century using historical research methods. This free course may be completed online at any time. (History 231)
This course will focus on the history of mankind's relationship with the natural world. The student 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. By the end of the course, the student will better understand the reciprocal relationship between human beings and the natural environment and how this relationship has evolved throughout human history. Upon successful completion of this course, the student will be able to: think critically about the historical relationship between humans and the natural environment; identify how early humans modified and adapted natural resources for agricultural and commercial purposes; analyze how human settlements altered the natural environment and evaluate how environmental factors shaped the growth of early civilizations; evaluate how new agricultural and commercial practices altered the natural environment across the globe during the Middle Ages; identify how environmental factors, such as disease and pollution, shaped political and social life in Europe during the Early-Modern Era; evaluate how the Columbian Exchange resulted in significant ecological and biological changes in Europe and the Americas and dramatically altered human societies on both sides of the Atlantic Ocean; analyze the impact of industrialization on human society during the Modern Era and evaluate how governmental and nongovernmental actors have attempted to ameliorate the negative environmental consequences of industrialization; identify current environmental challenges facing humanity and analyze these challenges from a historical perspective; analyze and interpret primary and secondary source documents relating to environmental history using historical research methods. (History 364)
Foundations of Business Law and the Legal Environment is an up-to-date textbook with comprehensive coverage of legal and regulatory issues for your introductory Legal Environment or Business Law course.
The text is organized to permit instructors to tailor the materials to their particular approach.
The authors take special care to engage students by relating law to everyday events with which they are already familiar with their clear, concise and readable style.
Business Law and the Legal Environment provides students with context and essential concepts across a broad range of legal issues with which managers and business executives must grapple. The text provides the vocabulary and legal savvy necessary for business people to talk in an educated way to their customers, employees, suppliers, government officials — and to their own lawyers.
In this course, you will study the relationships between lines and angles. You will learn to calculate how much space an object covers, determine how much space is inside of a three-dimensional object, and other relationships between shapes, objects, and the mathematics that govern them.
This course is oriented toward US high school students. Its structure and materials are aligned to the US Common Core Standards. Foci include: formulas for calculating the volume of prisms, cylinders, pyramids, cones, and spheres; and assist you in using geometric modeling to solve problems involving three-dimensional figures.
This course will focus on the emergence and evolution of industrial societies around the world. The student 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. The student 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. This course ultimately examines how industrialization developed, spread across the globe, and shaped everyday life in the modern era. Upon successful completion of this course, students will be able to: identify key ideas and events in the history of industrialization; identify connections between the development of capitalism and the development of modern industry; use analytical tools to evaluate the factors contributing to industrial change in different societies; identify the consequences of industrialization in the 19th and 20th centuries in different societies; critique historical interpretations of the causes and effects of industrialization; and analyze and interpret primary source documents describing the process of industrialization and life in industrial societies. (History 363)
This course introduces the history of Africa from 300,000 BCE to the era of European imperialism in the nineteenth century. The story continues in HIST 252, which covers the last 120 years of African history. Upon successful completion of this course, the student will be able to: locate major regions, geographic features, and populations in Africa and label them on a map; identify major events and trends in the history of Africa prior to 1890 that describe change over time; demonstrate the impact of the African environment on human history in Africa and explain how humans in turn changed that environment; compare and contrast the diverse social and political structures and systems devised by Africans; summarize the connections between Africans and other peoples of the world and the ways in which those connections changed over time; demonstrate the usefulness, best practices, and limitations of different types of sources for understanding the African past; appraise various conceptions of the African past given the evidence from that past; assess the degree to which there can be said to be one, shared African history before 1890. This free course may be completed online at any time. (History 251)
A survey of major technological developments from ancient to modern times with particular attention to social, political, and cultural contexts in Europe and the United States.
This course is a continuation of MA001: Beginning Algebra, and will focus on compound inequalities, systems of linear equations, radicals, rational exponents, quadratic equations and techniques used to solve these equations, and finally, general functions and graphs with an emphasis on the exponential and logarithmic functions.
This course will survey physics concepts and their respective applications; it is intended as a basic introduction to the current physical understanding of our universe. In this course, the student will study physics from the ground up, learning the basic principles of physical law, their application to the behavior of objects, and the use of the scientific method in driving advances in this knowledge. This course focuses on Newtonian mechanics--how objects move and interact--rather than Electromagnetism or Quantum Mechanics. While mathematics is the language of physics, the student need only be familiar with high school-level algebra, geometry, and trigonometry; the small amount of additional math needed will be developed during the course. (Physics 101; See also: Biology 109, Chemistry 001, Mechanical Engineering 005)
Partial differential equations (PDEs) describe the relationships among the derivatives of an unknown function with respect to different independent variables, such as time and position. Experiment and observation provide information about the connections between rates of change of an important quantity, such as heat, with respect to different variables. Upon successful completion of this course, the student will be able to: State the heat, wave, Laplace, and Poisson equations and explain their physical origins; Define harmonic functions; State and justify the maximum principle for harmonic functions; State the mean value property for harmonic functions; Define linear operators and identify linear operations; Identify and classify linear PDEs; Identify homogeneous PDEs and evolution equations; Relate solving homogeneous linear PDEs to finding kernels of linear operators; Define boundary value problem and identify boundary conditions as periodic, Dirichlet, Neumann, or Robin (mixed); Explain physical significance of boundary conditions; Show uniqueness of solutions to the heat, wave, Laplace and Poisson equations with various boundary conditions; Define well-posedness; Define, characterize, and use inner products; Define the space of L2 functions, state its key properties, and identify L2 functions; Define orthogonality and orthonormal basis and show the orthogonality of certain trigonometric functions; Distinguish between pointwise, uniform, and L2 convergence and show convergence of Fourier series; Define Fourier series on [0,pi] and [0,L] and identify sufficient conditions for their convergence and uniqueness; Compute Fourier coefficients and construct Fourier series; Use the method of characteristics to solve linear and nonlinear first-order wave equations; Solve the one-dimensional wave equation using d'Alembert's formula; Use similarity methods to solve PDEs; Solve the heat, wave, Laplace, and Poisson equations using separation of variables and apply boundary conditions; Define the delta function and apply ideas from calculus and Fourier series to generalized functions; Derive Green's representation formula; Use Green's functions to solve the Poisson equation on the unit disk; Define the Fourier transform; Derive basic properties of the Fourier transform of a function, such as its relationship to the Fourier transform of the derivative; Show that the inverse Fourier transform of a product is a convolution; Compute Fourier transforms of functions; Use the Fourier transform to solve the heat and wave equations on unbounded domains. (Mathematics 222)
In this course, the student will learn the basic terminology and concepts of probability theory, including sample size, random experiments, outcome spaces, discrete distribution, probability density function, expected values, and conditional probability. The course also delves into the fundamental properties of several special distributions, including binomial, geometric, normal, exponential, and Poisson distributions. Upon successful completion of this course, the student will be able to: Define probability, outcome space, events, and probability functions; Use combinations to evaluate the probability of outcomes in coin-flipping experiments; Calculate the union of events and conditional probability; Apply Bayes's theorem to simple situations; Calculate the expected values of discrete and continuous distributions; Calculate the sums of random variables; Calculate cumulative distributions and marginal distributions; Evaluate random processes governed by binomial, multinomial, geometric, exponential, normal, and Poisson distributions; Define the law of large numbers and the central limit theorem. (Mathematics 252)
This course covers descriptive statistics, the foundation of statistics, probability and random distributions, and the relationships between various characteristics of data. Upon successful completion of the course, the student will be able to: Define the meaning of descriptive statistics and statistical inference; Distinguish between a population and a sample; Explain the purpose of measures of location, variability, and skewness; Calculate probabilities; Explain the difference between how probabilities are computed for discrete and continuous random variables; Recognize and understand discrete probability distribution functions, in general; Identify confidence intervals for means and proportions; Explain how the central limit theorem applies in inference; Calculate and interpret confidence intervals for one population average and one population proportion; Differentiate between Type I and Type II errors; Conduct and interpret hypothesis tests; Compute regression equations for data; Use regression equations to make predictions; Conduct and interpret ANOVA (Analysis of Variance). (Mathematics 121; See also: Biology 104, Computer Science 106, Economics 104, Psychology 201)
This course will introduce the student to United States history from the colonial period to the Civil War. The student will learn about the major political, economic, and social changes that took place in America during this 250-year period. Upon successful completion of this course, the student will be able to: Analyze the first encounters between the Native inhabitants of North America with Spanish, French, and English colonizers and determine the effect of European colonization on Native Americans; Describe and assess the creation of English/British America; Interpret the main social, political, and economic development of colonies in British North America, including the emergence of a slave economy; Analyze how and why an independent United States was created in 1776 by interpreting the ideological, political, and economic roots of American independence as it developed through the Seven YearsĺÎĺĺÎĺ War, the Imperial Crisis and the American Revolution; Analyze the myriad political and economic crises that plagued the Early American Republic in the 1780s and 1790s and identify and describe the expansion of slavery, partisan politics, economic innovation, westward expansion, and the outbreak of the War of 1812; Interpret the main developments of the Age of Jackson, the Indian Removal Act, the Nullification Crisis, the rise of the Whig Party, the Bank War; Interrogate the definition of 'democracy' in 1820s and 1830s America; Analyze the era of reform in antebellum America and identify and describe the emergence of new religious groups- Shakers, Mormons, evangelicals - as well as moral reformers who sought to curb alcoholism, improve the prison system, increase women's rights, end slavery, or modify the American education system; Analyze antebellum America and the emergence of sectionalism, and identify and describe how Northerners and Southerners apparently opposing viewpoints about labor systems, political economy, and race often obscured many similarities; Analyze the impact of the ideology of Manifest Destiny on the development of the American West as it affected Native Americans and white settlers; Identify and describe the West, the California Gold Rush, the Mexican War, and the contested boundary in the Pacific Northwest; Interpret how the question of slaveryĺÎĺĺÎĺs expansion affected American political parties, law, and created sectional conflict - both political and ideological - between 1820 and the 1850s; Analyze the American Civil War; identify and describe how and why the federal union that was created in 1776 collapsed in 1861; and assess the major facets of the war (including military engagements, the home fronts, Lincoln's presidency, and the question of slavery). (History 211)
This course will introduce the student 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. By the end of the course, the student will understand how Islam became a sophisticated and far-reaching civilization and how conflicts with the West shaped the development of the Middle East from the medieval period to the present day. Upon successful completion of this course, students will be able to: identify and describe the nature of pre-Islamic society, culture, and religion. They will also be able to describe the subsequent rise of the prophet Muhammad and his monotheistic religion, Islam; identify and describe the elements of Islamic law, religious texts and practices, and belief systems; identify and describe the rise of the Umayyad and Abbasid dynasties in the Middle East. Students will also be able to compare and contrast the two empires; identify and describe the emergence of the Umayyad dynasty in Spain. Students will also be able to analyze the conflicts between Muslims and Christians on the Iberian Peninsula; identify and describe the Crusades. They will be able to describe both Muslim and Christian perceptions of the holy wars; identify and describe the impact of the Mongol invasions on the Middle East; compare and contrast the Ottoman and Safavid empires; analyze the decline of the Ottoman Empire and the beginning of European imperialism/domination of the Middle East in the 1800s; identify and describe how and why European powers garnered increased spheres of influence after the collapse of the Ottoman Empire and the end of World War I; analyze and describe the rise of resistance and independence movements in the Middle East; identify and describe the rise of Islamic nationalism and the emergence of violent anti-Western sentiment; analyze (and synthesize) the relationship between the Middle East and the West between the 600s and the present day; analyze and interpret primary source documents that elucidate the exchanges and conflicts between the Islamic world and the West over time. (History 351)
This course introduces the student to the study of linear algebra. Practically every modern technology relies on linear algebra to simplify the computations required for internet searches, 3-D animation, coordination of safety systems, financial trading, air traffic control, and everything in between. Upon completion of this course, the student will be able to: Define and identify linear equations; Write a system of equations in matrix-vector form; Explain the geometric interpretation of a system of linear equations; Solve linear equations using a variety of methods; Define general, particular, and homogeneous solutions; Identify how many solutions a linear system has; Correctly manipulate vectors algebraically and perform matrix-vector and matrix-matrix multiplication; Define linear combination and span; Define and distinguish between singular and nonsingular matrices and calculate a matrix inverse; Define and compute LU decompositions; Relate invertibility of matrices to solvability of linear systems; Define and characterize Euclidean space; Define and compute dot and cross-products; Define and identify vector spaces and subspaces; Define spanning set and determine the span of a set of vectors; Define and verify linear independence; Define basis and dimension; Show that a set of vectors is a basis; Define and compute column space, row space, nullspace, and rank; Define and identify isomorphisms and homomorphisms; Use row and column space to solve linear systems; State the rank-nullity theorem; Define inner product, inner product space, and orthogonality; Interpret inner products geometrically; Define determinants using the permutation expansion; State the properties of determinants, such as that the determinant of the product is the product of the determinants; Compute the determinant using cofactor expansions, row reduction, and Cramer's Rule; Define and compute the characteristic polynomial of a matrix; Define and compute eigenvalues and eigenvectors; Explain the geometric significance of eigenvalues and eigenvectors; Define similarity and diagonalizability; Identify similar matrices; Identify some necessary conditions for diagonalizability. (Mathematics 211; See also: Computer Science 105)
Linear Algebra is both rich in theory and full of interesting applications; in this course the student will try to balance both. This course includes a review of topics learned in Linear Algebra I. Upon successful completion of this course, the student will be able to: Solve systems of linear equations; Define the abstract notions of vector space and inner product space; State examples of vector spaces; Diagonalize a matrix; Formulate what a system of linear equations is in terms of matrices; Give an example of a space that has the Archimedian property; Use the Euclidean algorithm to find the greatest common divisor; Understand polar form and geometric interpretation of the complex numbers; Explain what the fundamental theorem of algebra states; Determine when two matrices are row equivalent; State the Fredholm alternative; Identify matrices that are in row reduced echelon form; Find a LU factorization for a given matrix; Find a PLU factorization for a given matrix; Find a QR factorization for a given matrix; Use the simplex algorithm; Compute eigenvalues and eigenvectors; State Shur's Theorem; Define normal matrices; Explain the composition and the inversion of permutations; Define and compute the determinant; Explain when eigenvalues exist for a given operator; Normal form of a nilpotent operator; Understand the idea of Jordan blocks, Jordan matrices, and the Jordan form of a matrix; Define quadratic forms; State the second derivative test; Define eigenvectors and eigenvalues; Define a vector space and state its properties; State the notions of linear span, linear independence, and the basis of a vector space; Understand the ideas of linear independence, spanning set, basis, and dimension; Define a linear transformation; State the properties of linear transformations; Define the characteristic polynomial of a matrix; Define a Markov matrix; State what it means to have the property of being a stochastic matrix; Define a normed vector space; Apply the Cauchy Schwarz inequality; State the Riesz representation theorem; State what it means for a nxn matrix to be diagonalizable; Define Hermitian operators; Define a Hilbert space; Prove the Cayley Hamilton theorem; Define the adjoint of an operator; Define normal operators; State the spectral theorem; Understand how to find the singular-value decomposition of an operator; Define the notion of length for abstract vectors in abstract vector spaces; Define orthogonal vectors; Define orthogonal and orthonormal subsets of R^n; Use the Gram-Schmidt process; Find the eigenvalues and the eigenvectors of a given matrix numerically; Provide an explicit description of the Power Method. (Mathematics 212)
Examination of the use of mathematical logic and computation for solving some of the world's fundamental problems.
A math based economics course designed to provide the skills needed to solve fundamental problems in both macroeconomics and microeconomics, by covering concepts in precalculus and calculus.
This course will introduce the student to the history of Latin and South America from the early 19th century, when many Latin and South American colonies declared their independence from European rule, to the present day. The student will learn about the major political, economic, and social changes that took place throughout Latin and South America during this 200-year period, such as efforts by independent Latin and South American nations to create stable economies in the 19th century, political and economic conflicts between independent states and European imperial powers, the emergence of violent left-wing and right-wing political and social movements in the 20th century, and the developmental challenges that many Latin and South American nations face today. Upon successful completion of this course, students will be able to: Analyze and interpret primary source documents from the 19th and 20th centuries using historical research methods; Think critically about the history of Latin and South America from the 19th century to the present; Analyze how the peoples of Latin and South America attempted to organize viable nation-states following independence from Spanish and Portuguese colonial rule; Assess how the United States used economic imperialism to control the economic and political development of the nations of Latin and South America; Identify the origins of the 1910 Mexican Revolution and assess the political, economic, and social impact of the revolution for the people of Mexico; Assess the role that Latin and South American nations played in the global economy in the 19th and 20th centuries; Analyze the role that cultural agents such as the Catholic Church played in the development of Latin and South American nations; Identify the role played by women, indigenous peoples, and Afro-Latinos in the social and political development of Latin and South America; Identify the political and economic factors that led to the emergence of political dictatorships in many Latin and South American nations in the early 20th century; Assess how Cold War struggles between capitalist and Communist ideologies influenced political life in the nations of Latin and South America and led to the rise of repressive, authoritarian regimes in the 1970s and 1980s; Identify important contemporary political, economic, and social trends in Latin and South America based on an analysis of the region's history; Analyze and interpret primary source documents from the 19th and 20th centuries using historical research methods. (History 222)
This course will introduce the student to the history of the nations and peoples of the Middle East and Southwest Asia from 1919 to the present. The course covers the major political, economic, and social changes that took place throughout the region during this 100-year period. Upon successful completion of this course, the student will be able to: Identify and explain major political, social and economic trends, events, and people in history of the Middle East and Southwest Asia from the beginning of the 20th century to the present; Explain how the countries of the region have overcome significant social, economic, and political problems as they have grown from weak former colonies into modern nation-states; Identify and explain the emergence of nationalist movements following World War I, European political and economic imperialism during the first half of the 20th century, the creation of the nation of Israel, regional economic development, and the impact of secular and religious trends on Middle Eastern society and culture during the second half of the 20th century; Identify and explain the important economic, political, and social developments in the Middle East and Southwest Asia during the twentieth and twenty-first centuries; Analyze and interpret primary source documents from the 20th and 21st centuries that illustrate important overarching political, economic, and social themes. (History 232)