Calculus and Vector Analysis

Unit 1 · Indian Knowledge System & Vedic Calculus

Contributions of Indian mathematicians to calculus — Aryabhata, Madhava and Jyeshthadeva. Vedic calculus — differentiation by the Dhvaja-Ghata sutra; successive differentiation and the derivative of a quotient of polynomials by the Urdhva-Triyagbhyam sutra; integration by the Ekadhikena Purvena and Paravartya Yojayet sutras; integration of a product of two functions; and the Vedic approach to areas under curves.

Unit 2 · Differential Calculus – I

Successive differentiation, Leibnitz theorem, Maclaurin's and Taylor's series expansions; basic concepts of partial derivatives of functions of two and three variables; asymptotes of algebraic and polar curves and the condition for their existence, including parallel asymptotes.

Unit 3 · Differential Calculus – II: Curvature & Curve Tracing

Curvature — radius of curvature, curvature at the origin, centre of curvature; concavity and convexity, points of inflexion, singular and multiple points; tracing of curves given by Cartesian and polar equations.

Unit 4 · Integral Calculus

Integration of transcendental functions; double and triple integrals; reduction formulae; quadrature and rectification (Cartesian and polar coordinates); volumes and surfaces of solids of revolution.

Unit 5 · Vector Analysis – I

Vector differentiation and rules of differentiation, derivatives of triple products; gradient, divergence and curl; directional derivatives; vector identities.

Unit 6 · Vector Analysis – II

Vector integration; Gauss, Green and Stokes theorems and problems based on them; applications to geometry — curves in space, curvature and torsion, and the Frenet–Serret formulae.

Unit 7 · Industrial Applications (Case Study)

Applications of calculus and vector calculus to problems in industry, business and the real world.