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Course Criteria
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3.00 Credits
No course description available.
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3.00 Credits
Applications of calculus in mathematics, science, economics, psychology, the social sciences, involve several variables. This course extends calculus to several variables: vectors, partial derivatives and multiple integrals. The goal of the course is Stokes Theorem, a deep and profound generalization of the Fundamental Theorem of Calculus. The difference between this course and Mathematics 105 is that Mathematics 105 covers infinite series instead of Stokes Theorem. Students with the equivalent of BC 3 or higher should enroll in Mathematics 106, as well as students who have taken the equivalent of an integral calculus and who have already been exposed to infinite series. For further clarification as to whether or not Mathematics 105 or Mathematics 106 is appropriate, please consult a member of the math/stat department.
Prerequisite:
BC 3 or higher or integral calculus with infinite series
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3.00 Credits
Have you ever wondered what keeps your credit card information safe everytime you buy something online? Number theory! Number Theory is one of the oldest branches of mathematics. In this course, we will discover the beauty and usefulness of numbers, from ancient Greece to modern cryptography. We will look for patterns, make conjectures, and learn how to prove these conjectures. Starting with nothing more than basic high school algebra, we will develop the logic and critical thinking skills required to realize and prove mathematical results. Topics to be covered include the meaning and content of proof, prime numbers, divisibility, rationality, modular arithmetic, Fermat's Last Theorem, the Golden ratio, Fibonacci numbers, coding theory, and unique factorization.
Prerequisite:
Mathematics 100/101/102 (or demonstrated proficiency on a diagnostic test) or permission of instructor
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3.00 Credits
Historically, much beautiful mathematics has arisen from attempts to explain physical, chemical, biological and economic processes. A few ingenious techniques solve a surprisingly large fraction of the associated ordinary and partial differential equations, and geometric methods give insight to many more. The mystical Pythagorean fascination with ratios and harmonics is vindicated and applied in Fourier series and integrals. We will explore the methods, abstract structures, and modeling applications of ordinary and partial differential equations and Fourier analysis.
Prerequisite:
Mathematics 105; students may not normally get credit for both Mathematics 209 and Mathematics/Physics 210
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3.00 Credits
This course covers a variety of mathematical methods used in the sciences, focusing particularly on the solution of ordinary and partial differential equations. In addition to calling attention to certain special equations that arise frequently in the study of waves and diffusion, we develop general techniques such as looking for series solutions and, in the case of nonlinear equations, using phase portraits and linearizing around fixed points. We study some simple numerical techniques for solving differential equations. A series of optional sessions in Mathematica will be offered for students who are not already familiar with this computational tool.
Prerequisite:
Mathematics 105 or 106 and familiarity with Newtonian mechanics at the level of Physics 131
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3.00 Credits
Many social, political, economic, biological, and physical phenomena can be described, at least approximately, by linear relations. In the study of systems of linear equations one may ask: When does a solution exist? When is it unique? How does one find it? How can one interpret it geometrically? This course develops the theoretical structure underlying answers to these and other questions and includes the study of matrices, vector spaces, linear independence and bases, linear transformations, determinants and inner products. Course work is balanced between theoretical and computational, with attention to improving mathematical style and sophistication.
Prerequisite:
Mathematics 105 or 209 or 210 or 251, or Statistics 201
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3.00 Credits
As a complement to calculus, which is the study of continuous processes, this course focuses on the discrete, including finite sets and structures, their properties and applications. Topics will include basic set theory, infinity, graph theory, logic, counting, recursion, and functions. The course serves as an introduction not only to these and other topics but also to the methods and styles of mathematical proof.
Prerequisite:
Mathematics 104 or Mathematics 103 with Computer Science 134 or one year of high school calculus with permission of instructor
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3.00 Credits
Under faculty supervision, student-teachers will prepare and conduct scheduled weekly extra sessions for Mathematics 103, for smaller, assigned groups of students. For these sessions they will prepare presentations, assign and grade homework, and answer questions on the course material and on the homework. They will be available to their students outside of class, attend and assist at Mathematics 103 lectures (3 hours a week), and visit and evaluate each other's sessions. There is a weekly meeting, for an hour or two, including organizational matters, deeper study of the mathematics discussed, and practical teaching skills. In addition, there will be other special meetings as needed. There will be assigned readings, discussion, drills, and weekly homework or papers. This is a seminar whose focus is both on education and transforming lives, as well as on mathematics and the mechanics of teaching it.
Prerequisite:
Permission of instructor, preferably early in the previous Spring
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3.00 Credits
Real analysis is the theory behind calculus. It is based on a precise understanding of the real numbers, elementary topology, and limits. Topologically, nice sets are either closed (contain their limit points) or open (complement closed). You also need limits to define continuity, derivatives, integrals, and to understand sequences of functions.
Prerequisite:
Mathematics 105 and 211, or permission of instructor
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3.00 Credits
The calculus of complex-valued functions turns out to have unexpected simplicity and power. As an example of simplicity, every complex-differentiable function is automatically infinitely differentiable. As examples of power, the so-called "residue calculus" permits the computation of "impossible" integrals, and "conformal mapping" reduces physical problems on very general domains to problems on the round disc. The easiest proof of the Fundamental Theorem of Algebra, not to mention the first proof of the Prime Number Theorem, used complex analysis.
Prerequisite:
Mathematics 301 or 305
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