Upon completion of this course the student is able to:
- give and work with precise definitions of some important concepts from real analysis and write coherent and understandable mathematical proof related to these concepts. In particular, the student is able to
- explain and use the concepts of Riemann integral of single and multiple variable functions, including improper integrals,
- explain and use the concepts of convergence/divergence of a series of real numbers and the same for a sequence or series of real-valued functions,
- explain and work with the concepts related to continuity and partial/directional derivatives of real-valued functions of multiple variables,
- properly apply the definitions and important theorems to perform explicit calculations. In particular, the student is able to
- apply the fundamental theorem of calculus,
- apply various properties of and techniques for integration of functions of one variable, such as integration by parts, substitution of variable, direct calculation, and apply them to modeling problems,
- work with and manipulate power series and Taylor series,
- apply the concepts of limit and continuity of functions of several variables,
- work with and apply to modeling problems the concepts such as partial derivative, gradient, directional derivative, and Lagrange multipliers,
- work with the concepts, properties, and techniques for integration of functions of several variables, such as iterated integrals and change of variable to polar, spherical and cylindrical coordinates, and apply them to modeling problems.
Within the Analysis-line of Applied Mathematics, one learns the fundamental concepts in Real Analysis that are indispensable for further study. Another integral part of the Analysis-line is the proper and rigorous manner of proving results. In this second course, the following topics will be treated.
- Integrability on ℝ
The Riemann integral, Riemann sums, The fundamental theorem of calculus and integration techniques, Improper Riemann integration, Applications to solids of revolution, arc length, volume and surface area of a solid of revolution, mass, moments, center of mass, centroids.
- Infinite series of real numbers
Notations and definitions; geometric, harmonic, telescopic series, Series with nonnegative terms, Absolute convergence, Alternating series, Estimation of series.
- Infinite series of functions
Convergence of sequences and series of functions, Uniform convergence, Power series, Analytic functions.
- Partial differentiation and multiple integration
Functions of several variables, partial derivatives, Gradients and directional derivatives, Double integrals, Change of variable in double integral, triple integrals, Change of variables in triple integrals.
|Bachelor Applied Mathematics||Required materials|
Recommended materials-Instructional modes
|An Introduction to Analysis, William Wade, 4th edition. ISBN:9781292357874|
|Calculus: A Complete Course, R.A. Adams & C. Essex, 10th edition. ISBN: 9780135732588|