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Course Criteria
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4.00 Credits
Elements of calculus and linear algebra with examples and motivation drawn from important topics in economics. Topics include derivatives of functions of one and several variables; interpretations of the derivatives; convexity; constrained and unconstrained optimization; series, including geometric and Taylor series; ordinary differential equations; matrix algebra; eigenvalues; and (if time permits) dynamic optimization and multivariable integration.
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2.00 Credits
Builds on students' intuition, informal logical argumentation, and mathematical concepts with which Department of Mathematics they are familiar. Students study a series of case problems and test the validity of mathematical statements.
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5.00 Credits
First semester of a yearlong sequence that covers the content of Calculus II and III (MATH-UA 122, 123) as well as Linear Algebra (MATH-UA 140). Sequences and series; Taylor's theorem; power series; linear systems of equations; matrices and LU decomposition; determinants; vector spaces; eigenvalues and eigenvectors; functions of several variables; vector-valued functions; partial derivatives; various applications including maxima and minima.
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5.00 Credits
Second semester of a yearlong sequence that covers the content of Calculus II and III (MATH-UA 122, 123) as well as Linear Algebra (MATH-UA 140). Multidimensional differentiation (e.g. differentials, gradients, Taylor expansions, applications); multidimensional integration (e.g. double and triple integrals, Green's theorem, divergence theorem, applications); differential equations (e.g. first-order linear equations, second-order linear equations, applications); and additional topics in linear algebra (e.g. inner products, orthogonality, applications).
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4.00 Credits
Brief review of multivariate calculus: partial derivatives, chain rule, Riemann integral, change of variables, line integrals. Lagrange multipliers. Inverse and implicit function theorems and their applications. Introduction to calculus on manifolds: definition and examples of manifolds, tangent vectors and vector fields, differential forms, exterior derivative, line integrals and integration of forms. Gauss's and Stokes's theorems on manifolds.
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4.00 Credits
Introduction to dynamical processes that drive the circulation of the atmosphere and ocean, and their interaction. Goal of the lectures is to develop an understanding of the unifying principles of planetary fluid dynamics. Topics include the global energy balance, convection and radiation (the greenhouse effect), effects of planetary rotation (the Coriolis force), structure of the atmospheric circulation (the Hadley cell and wind patterns), structure of the oceanic circulation (wind-driven currents and the thermohaline circulation), and climate and climate variability (including El Niño and anthropogenic warming).
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4.00 Credits
Introduction to the mathematical techniques of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, Markov chains, applications.
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4.00 Credits
Introduction to the mathematical foundations and techniques of modern statistical analysis used in the interpretation of data in quantitative sciences. Mathematical theory of sampling; normal populations and distributions; chi-square, t, and F distributions; hypothesis testing; estimation; confidence intervals; sequential analysis; correlation, regression, and analysis of variance. Applications to the sciences.
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4.00 Credits
Combination of MATH-UA 233 and MATH-UA 234 at a more elementary level to acquaint students with both probability and statistics in a single term. In probability: mathematical treatment of chance; Department of Mathematics combinatorics; binomial, Poisson, and Gaussian distributions; law of large numbers and the normal distribution; application to coin-tossing; radioactive decay. In statistics: sampling; normal and other useful distributions; testing of hypotheses; confidence intervals; correlation and regression; applications to scientific, industrial, and financial data.
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4.00 Credits
Techniques for counting and enumeration, including generating functions, the principle of inclusion and exclusion, and Polya counting. Graph theory. Modern algorithms and data structures for graph theoretic problems.
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