Course Details:

Limits and continuity of functions defined on open subsets of Rn. Extreme value theorem for real valued continuous functions on closed, bounded subsets of Rn

Partial derivatives of real-valued functions of several variables; total derivative and directional derivative. Level sets, gradient and its geometric interpretation.

Derivatives of functions from intervals to Rn: differentiable curves; tangent vector. Curvature of curves.

Differentiability of functions from open sets in Rn to Rm; derivative as a linear approximation of a map and the linear map associated to the derivative. Chain rule.

Taylor’s theorem for functions of several variables; tests for local maxima/minima. Lagrange multiplier method for constrained optimization problems.

Double and triple Integrals. Change-of-variables formula. Applications of integration to computations of area and volume.

Curves: parametrization, arc-length and its invariance. Line integrals of scalar and vector fields along curves. Curl, divergence; conservative vector fields. Green’s theorem. Surface integrals & Stokes’ theorem, Divergence theorem and their applications.

Text Books:

  1. S. J. Colley, Vector Calculus, Pearson (2012)
  2.  G. B. Thomas Jr., M.D. Weir and J.R. Hass, Thomas Calculus, Pearson Education (2009)
  3. Jerrold E. Marsden, Anthony Tromba, Alan Weinstein, Basic Multivariable Calculus, (Springer India Private Ltd)

Reference Books:

  1. E. Kreyszig, Advanced engineering mathematics, 10th Edition, John Wiley & sons (2011)
  2. S. R. Ghorpade, B. V. Limaye, A Course in Multivariable Calculus and Analysis, Springer (2010)
  3. Tom M. Apostol, Calculus, Volume II: Multi-variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability, 2nd Ed , John Wiley & Sons (2007)