UNIVERSITY OF MADRAS
B.Sc. DEGREE COURSE IN Nautical Science
SEMESTER SYSTEM WITH CREDITS
(Effective from the Academic Year 2003-2004)

SYLLABUS

Semester I - Paper I - Nautical Mathematics I

Duration of Examination: 3 hrs
Maximum Marks: 100
Credits: 4

1. Complex Variables

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

2. Vector Algebra and Calculus

Scalar and Vector Triple Products. Differentiation of a vector function of a single scalar variable. Derivative of a unit vector, application to curves in space, principal triad, Sennet-Frenet form.

3. Vector Analysis

Line integral, Green's theorem for the plane, properties of line integrals, lien integrals in space and their properties, application to work and to the flow of liquid, scalar and vector fields, conservative fields, potentials, gradient, divergence and curl, Divergence theorem, Stoke's theorem, expressions for gradient, divergence and curl in orthogonal curvilinear coordinates, Gauss theorem, equation of heat flow, equations of hydrodynamics.

4. Differential Calculus

Successive differentiation. Standard form to find the nth derivative. Leibnitz theorem, Rolle's theorem (with proof). Lagrange's and Cauchy's mean value theorem (with proof), Taylor's theorem, Taylor's and Maclaurin's series (without proof). Indeterminate forms, L' Hospital's rule, Expansion of function in power series (all types), Partial derivatives of first and higher orders. Total differential, Concept of communicativity of partial derivatives (without proof) Euler's theorem on homogeneous functions. Deduction from Euler theorems. Errors & Approximations. Maxima & Minima of the functions of two variables.

5. Differential Equations

(a) Exact differential equations and those which can be made exact use of integrating factors by inspection. (i) Linear Equations and reducible to linear (Bernoulli) equations, (ii) Method of substitution to reduce the equation to one of the above forms.

(b) Linear Differential Equations of the nth order with constant coefficients. Complimentary function and Particular integral when the function of the independent variable on R.H.S. is C^ax, X^m, C^ax V(x), sin (ax+b). Cauchy's Linear equation (homogeneous). Legendre's Linear equation. Variation of parameters and method of undetermine coefficients.

(c) Elementary applications of above differential equations in solving engineering problems such as Electrical Engg., Mech. Engg.

Books Recommended For Reference:

1. Elements of applied mathematics Vol. I Wartikar P.N. & Wartikar J.N
2. Textbook of applied mathematics Vol. II Wartikar P.N. & Wartikar J.N
3. Vector Algebra Shanti Narayan
4, Vector Calculus Shanti Narayan
5. Differential Calculus Shanti Narayan
6. Engineering Mathematics Bali, Saxena, Iyengar
7 Plane Trigonornetiy (Part II) Loney S.L.
8. Higher Engineering Mathematics Grewal. B.S
9. Differential Equations Raisingania
10. Engineering Mathematics Bhatia M.L.
11. Engineering Mathematics Baphana R.M.
12. Vector Methods and Vector Calculus Vaishista.