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Welcome!
Welcome!
I'm Oscar Fernandez, professor of mathematics at Wellesley College. I originally set up this site to house the digital versions of the courses I teach at Wellesley. But now that they're on here, there's no reason not to make these courses and their resources available to everyone. That's the ultimate goal of this website, and of my teaching: to make mathematics as accessible, engaging, and easy to learn as possible.
Below are the courses currently on the site. These are complete courses: they come with lecture notes, practice problems, solutions to those problems, and short videos explaining the content. I will be adding more courses over time, so feel free to bookmark this site or check in periodically .
One final thing to know. While Google Sites makes creating new sites easy, it is fairly restrictive in what one can do. (Example: I cannot change the color of this font!) Some time ago I maintained another website, surroundedbymath.com, which ran on Wordpress and so was very customizable. I no longer have that site up. But if you see "surroundedbymath" pop up anywhere in the materials for the courses below, now you know what that is. (I moved all that site's content to this site.)
I hope you enjoy the courses below. If you find any errors or would like to leave other feedback, feel free to email me (ofernand@wellesley.edu).
Available Courses
Available Courses
This course develops the core concepts and techniques of single-variable calculus, starting from a review of precalculus topics (e.g., functions) and continuing on to limits, differentiation, and integration. The course is based on the book I wrote, Calculus Simplified (whose cover is pictured above, in part). As such, it follows the approach I took in that book: learn calculus in the context of algebraic functions first and only after all the aforementioned calculus topics have been covered go back and learn about the calculus of transcendental functions. The course includes many applications to the physical, biological, and social sciences. Content is taught across 27 lessons.
Prerequisites: Precalculus (ideally), but if not, a strong College Algebra background
This course picks up where Calculus 1 left off and continues the calculus adventure in three new directions. The first direction explores infinite sequences and series and how we can tell when they "converge" (loosely speaking: approach or add up to a limiting value). The second direction generalizes the linear approximation theory from Calculus 1 to develop a powerful method to approximate a function with a polynomial, leading to what are called power series and Taylor series. The final direction explores advanced integration techniques and the applications of integration to calculating volumes. The course's 33 lessons discuss many, many applications of real-world phenomena, including in the sciences and social sciences.
Prerequisites: Calculus 1
This course explores the world around us--three-dimensional space--and develops concepts and tools to describe how it changes over time. The course investigates this question mathematically by studying the differentiation and integration of functions of more than one variable ("multivariable functions"), and later expanding that to vector calculus, which describes the dynamics of vectors and vector fields (quantities that describe the magnitude and direction of phenomena at various points in space, like a wind field or magnetic field). The course includes many applications to the physical, biological, and social sciences. Content is taught in 18 lessons.
Prerequisites: Calculus 2
This course explores one of the most fundamental questions humans have pondered: How is the future determined by the present and the past? The course investigates this question mathematically by studying various types of equations involving an unknown function and its derivatives. These differential equations help us understand how quantities change over time, and how that change depends on current and prior phenomena. The course's 19 lessons discuss many, many applications of differential equations to real-world phenomena, including in the sciences and social sciences.
Prerequisites: Calculus 2
This course seeks to "rigorize" calculus. While there isn't a unique way to do that, this course follows the approach now called "standard analysis," which rids calculus of the infinitesimals you likely learned about in your calculus course and grounds calculus instead on topological concepts like accumulation points. (So-called "non-standard analysis" retains the usage of infinitesimals to rigorize calculus.) As a result, this course -- and real analysis courses in general -- introduce topology and topological concepts. The course is divided into 27 lessons covering the real numbers, basic topology, limits, differentiation, and Riemannian integration. For most of the course, the setting is R^n (n-dimensional Euclidean space).
Prerequisites: Multivariable calculus, linear algebra, and some training in proofs and proof-writing
This course explores what happens to real analysis when we allow for the square root of negative one to be a number. The resulting set of "complex numbers," denoted C, turns out how have a lot in common with two-dimensional Euclidean space. But the similarities soon give way to stark differences when ones studies how to represent complex numbers in polar form, how to find roots of complex numbers, and how to visualize mappings of complex numbers. The results of pursuing these differences, complex analysis, turn out to provide a rich and beautiful set of theorems, methods, and applications. Furthermore, perhaps the most famous unsolved problem in mathematics -- the Riemann hypothesis -- is squarely a complex analysis problem. We'll end the course working up to the statement of that conjecture. The course is divided into 34 lessons, a few of which include applications of complex analysis to fluid dynamics.
Prerequisites: Real Analysis
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