BSc Nautical Sciences APPLIED MATHEMATICS I Syllabus - Mumbai University 2026

Module I: Complex Variables & Vector Algebra and

• Calculus Definition, Cartesian, Polar & exponential form. De- Moivre’s Theorem. Power & Roots of Exponential and Trigonometric Functions. Hyperbolic & Logarithmic Functions. Inverse Hyperbolic & Inverse Trigonometric Functions. Separation into real and imaginary parts of all types of functions. Vector Algebra and Calculus Scalar and Vector Triple Products. Differentiation of a vector functions of a single scalar variable. Derivative of a unit vector.

Module II: Vector analysis

• Gradient, divergence and curl, Divergence theorem, Stoke’s theorem, expressions for gradient, divergence and curl in orthogonal curvilinear co- ordinates, Gauss theorem, Equation of heat flow, equations of hydrodynamics. Differential Calculus Successive differentiation. Standard form to find the nth derivative. Rolle’s theorem (with proof), Lagrange’s and Cauchy’s mean value theorem (with proof).

Module III

• Spherical trigonometry & Simpson’s Rules Properties of a spherical triangle and oblique spherical triangle. Cosine formula, Haversine formula, Sine formula and four part formula and their application to Navigational problems. Polar triangle and application of their properties. Right angle and quadrantal triangles. Napier’s Rules and their application to Navigational problems. Area of spherical triangles. Inequalities, Derivation of formula by supplemental theorem, ‘Half angle’ formula, ‘Half side’ formula, Identities. Delambre’s Analogies, Napier’s Analogies, Legendre’s theorem. Derivation of Simpson’s first, second and five-eight rules and their use in the computation of areas, volumes and centroids.

Reference Books

This paper runs across both semesters of the year and the University prints one list of reference books for it, at the end of the second half. Her list is repeated here so that this half is not read as a paper she recommends no reading for.

• 1 An introduction to Spherical Trigonometry Clough-Smith J.H
• 2 Spherical Trigonometry Capt. H. Subramaniam
• 3 Higher engineering Mathematics Dr. B.S. Grewal
• 4 A Text book of applied mathematics Wartikar P.N. & Wartikar J.N.
• 5 Integral Calculus Shanti Narayan