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Real Analysis-II

Course Code: MA 356; MM 451  |  Credits: 4  |  Prerequisite: Real Analysis-I, Linear Algebra

Syllabus Details

  • Unit I: (Functions of several variables) Functions of several variables, directional derivative, partial derivative, total derivative, Jacobian, chain rule and mean value theorems, higher derivatives, interchange of the order of differentiation, Taylor’s theorem.
  • Unit II: (Inverse and implicit mapping theorem) Inverse mapping theorem, implicit function theorem.
  • Unit III: (Extremum problems) Extremum problems, extremum problems with constraints, Lagrange’s multipliers method.
  • Unit IV: (Multiple integrals) Multiple integrals, properties of integrals, existence of integrals, iterated integrals, change of variables.
  • Unit V: (Theorems of Green, Gauss and Stokes) Curl, gradient, div, Laplacian in cylindrical and spherical coordinates, line integrals, surface integrals, theorems of Green, Gauss and Stokes.

Course Learning Objective

  1. Calculate existence or nonexistence of limits, derivatives of scalar fields, and vector fields.
  2. Calculate derivatives using the chain rule.
  3. Optimize non-linear functions using the Lagrange multipliers to compute multiple integrals using the Fubini’s theorem, and by the change of variables.
  4. Parametrize curves and surfaces to find line integrals, surface integrals.
  5. Apply the Green’s, Gauss’, and Stokes’ theorems to find line, surface and volume integrals.
  1. Apostol, Tom M., Mathematical Analysis, second edition, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1974. (Chapter 6,7,10 and 11.2).
  2. Apostol, Tom M., Calculus Vol. II: Multi-variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability, second edition, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1969.
  3. Munkres, James R., Analysis on Manifolds, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1991.
  4. Spiegel, Murray R., Schaum’s Outline of Vector Analysis, Schaum’s Outline Series, 1959.
  5. Spivak, Michael, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965.
  6. Moskowitz, M., and Paliogiannis F., Functions of Several Real Variables, World Scientific, 2011.
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