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.
- S. J. Colley, Vector Calculus, Pearson (2012)
- G. B. Thomas Jr., M.D. Weir and J.R. Hass, Thomas Calculus, Pearson Education (2009)
- Jerrold E. Marsden, Anthony Tromba, Alan Weinstein, Basic Multivariable Calculus, (Springer India Private Ltd)
- E. Kreyszig, Advanced engineering mathematics, 10th Edition, John Wiley & sons (2011)
- S. R. Ghorpade, B. V. Limaye, A Course in Multivariable Calculus and Analysis, Springer (2010)
- 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)