Syllabus of Zoology Optional Subject

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# Syllabus of Mathematics Optional Subject

01-01-1970

05:30:AM

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__PAPER-I__

**1. Linear Algebra :**

- Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions, Linear transformations, rank and nullity, matrix of a linear transformation.
- Algebra of Matrices; Row and column reduction, Echelon form, congruence’s and similarity; Rankof a matrix; Inverse of a matrix; Solution of system of linear equations; Eigenvalues and eigenvectors,

**2. Calculus :**

- Real numbers, functions of a real variable, limits, continuity, differentiability, mean-value theorem,
- Riemann’s definition of definite integrals; Indefinite integrals; Infinite and improper integral; Double and triple integrals (evaluation techniques only); Areas, surface and volumes.

**3. Analytic Geometry :**

- Cartesian and polar coordinates in three dimensions, second degree equations in three variables,

**4. Ordinary Differential Equations :**

- Formulation of differential equations; Equations of first order and first degree, integrating factor;
- Second and higher order liner equations with constant coefficients, complementary function, particular integral and general solution.
- Section order linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using method of variation of parameters.
- Laplace and Inverse Laplace transforms and their properties, Laplace transforms of elementary functions.
- Application to initial value problems for 2nd order linear equations with constant coefficients.

**5. Dynamics and Statics :**

- Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; Constrained motion; Work and
- Equilibrium of a system of particles; Work and potential energy, friction, Common catenary; Principle of virtual work; Stability of equilibrium, equilibrium of forces in three dimensions.

**6. Vector Analysis :**

- Scalar and vector fields, differentiation of vector field of a scalar variable; Gradient, divergence and curl in
- Application to geometry : Curves in space, curvature and torsion; Serret-Furenet's formulae.
- Gauss and Stokes’ theorems, Green's indentities.

__ ____PAPER-II__

**1. Algebra :**

- Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups,
- Rings, subrings and ideals, homomorphisms of rings; Integral domains, principal ideal domains,

**2. Real Analysis :**

- Real number system as an ordered field with least upper bound property; Sequences, limit of a sequence,
- Riemann integral, improper integrals; Fundamental theorems of integral calculus.
- Uniform convergence, continuity, differentiability and integrability for sequences and series of functions;

**3. Complex Analysis :**

- Analytic function, Cauchy-Riemann equations, Cauchy's theorem, Cauchy's integral formula, power

**4. Linear Programming :**

- Linear programming problems, basic solution, basic feasible solution and optimal solution; Graphical
- Transportation and assignment problems.

**5. Partial Differential Equations :**

- Family of surfaces in three dimensions and formulation of partial differential equations; Solution of

**6. Numerical Analysis and Computer Programming :**

- Numerical methods: Solution of algebraic and transcendental equations of one variable by bisection,
- Numerical integration: Trapezoidal rule, Simpson’s rule, Gaussian quadrature formula.
- Numerical solution of ordinary differential equations : Eular and Runga Kutta methods.
- Computer Programming : Binary system; Arithmetic and logical operations on numbers; Octal and
- Elements of computer systems and concept of memory; Basic logic gates and truth tables, Boolean algebra, normal forms.
- Representation of unsigned integers, signed integers and reals, double precision reals and long integers.
- Algorithms and flow charts for solving numerical analysis problems.

**7. Mechanics and Fluid Dynamics :**

- Generalised coordinates; D’Alembert’s principle and Lagrange’s equations; Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions.
- Equation of continuity; Euler’s equation of motion for inviscid flow; Stream-lines, path of a particle;

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