B.A. MathematicsMahatma Gandhi Kashi Vidyapith
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The Mahatma Gandhi Kashi Vidyapeeth offers B.A. Mathematics course to its students. The course structure for it is as follows:
Matrices: Types of Matrices. Hermitian and Skew-Hermitian Matrices. Elementary row and column operation on a matrix. Row and Column equivalances. Echelon form of a matrix, Rank of matrix. Inverse of
Matrix and methods of their computations (e.g. by using elementary row/column operations). Orthogonal and unitary matrices. Caley-Hamilton theorem and its use in finding matrix inverse.
Solutions of the system of linear equations. Characteristic roots and Characteristic vectors of a matrix.
Theory of Equations- Relations between roots and coefficients of a general polynomial equation in one
variable. Transformation of equations. Descartes rule of signs. Solution of cubic equations (Cardon method),
Trigonometry – DeMauvre’s Theorem and its applications. Direct and inverse circular and hyperbolic
functions. Gregory series. Logarithm of a complex quantity. Summation of series.
Coordinate Geometry-General equation of second degree. Tracing of conics. System of conics, confocal
Differential Calculus-The definition of the limit of a function( , definition), Basic properties of limits.
Continuous functions. Classification of discontinuities, differentiability. Successive differention. Leibnitz
theorem. Maclaurin’s and Tayler series expansions. Asymptotes.
Curvature. Multiple points. Curve tracing. Partial differentiations. Change of variables. Euler’s theorem on
homogeneous functions. Taylor’s theorem for functions of two variables (without proof). Envelope,
maxima, minima and saddle points of functions of two variables. Lagrange’s multiplier method.
Integral Calculus- Reduction formulae, definite integrals.
Integral Calculus-Quadrature, Rectification, Volumes and surface areas of solid of revolution
Algebra and calculus of vectors: Scalar and vector products , triple products. Quadruple products.
Reciprocal vectors. Differentiation of vectors. Gradient, divergence, and curl.
Dynamics: Velocities and accelerations along radial and transverse directions and along tangential and
normal directions. Simple harmonic motion, elastic string.
Dynamics: Motion on smooth and rough plane curves. Resisting media. Central orbits. Kepler’s laws of
Statics: Analytical conditions of equations of equilibrium of coplanar forces. Virtual work, Catenary.
Analysis: Definition and properties of continuous functions. Differentiability, Mean Value theorems with
their geometrical interpretations. Darbous’s theorem for derivatives. Taylor’s theorem with various forms of
Definition and properties of beta and gamma functions. Double and triple integrals. Dirichelt’s integrals.
Change of order of integration in double integrals.
Ordinary Differential Equations- Degree and order of a differential equation. Equations of first order and
first degree. Equations of first order but not of first degree. Clairaut’s form and singular solutions. Linear
differential equations with constant coefficients.
Differential Equations: Orthogonal trajectories. Homogeneous linear ordinary differential equations.
Linear ordinary differential equations with variable coefficients. Solution when an integral including CF is known, normal form (removal of first order derivative), change of independent variable and variation of parameters.
Sequences: Limit of sequences. Supremum and Infimum of a sequence. Monotonic sequences. Cauchy’s
general principle of convergence, series of positive terms. Comparison tests, Cauchy’s nth root and ratio tests. Series of monotonically decreasing terms. Raabe’s logarithmic, and De Morgan’s test. Leibnitz’s
theorem for alternating series.
Statement of logical Connectives: Truth tables. Tautologies. Tautologies implication and equivalences.
Quantifiers, Rules of inference. Equivalence relations and partitions. Groups, Their examples, subgroups.
Generation of groups. Cyclic groups.
Closet decompositions. Lagrange’s theorem. Homomorphisms and isomorphisms. Normal subgroups.
Quotient groups. The Fundamental theorem of homomorphism.
Groups: Permutaion groups. Even and odd permutations. The alternating groups, An . Cayley’s theorem.
Rings: Rings, subrings, integral domains, and field-basic ideas and simple result only. Characteristics of a ring.
Vector Space: Vector spaces, Subspaces, generation of subspaces. Quotient spaces, Linear dependence and
independence. Basis and Dimensions. Direct sum of vector subspaces.
Vector space: Linear transformation and their representation as matrices. Dual of a vector space. Dual basis. Rank and nullity of a linear transformation. The rank nullity theorem for linear transformations.
Solid Geometry: Plane, straight line, sphere, cone and cylinder.
General conicoids, tangent plane at a point of a central conicoid, normal , Polar planes , polar lines.
Vector Integration: Gauss, Stokes and Green’s Theorems and problems based on these.
Tensors: Contravariant and covariant vectors, Transformation formulae.
Dynamics of Rigid Body: D’Alembert’s principle, Equations of motion under finite and impulsive forces,
motion about a fixed axis. Compound pendulum. Reaction on the axis of rotation. Centre of percussion.
Dynamics of Rigid Body: Moment and product of Intertia, Momental ellipsoid, equimomental systems,
Principal axes, Equation of motion in two dimension, kinetic energy and angular momentum in two
dimensional fixed plane.