PGAT 2023 Mathematics: Geometry, Algebra, Analysis

• Syllabus for PGAT 2023 Mathematics (22)

Geometry

• Topics include straight lines and planes
• Also covers spheres, cones, and cylinders
• Includes central conicoids and generating lines
• Conics in polar coordinates are part of the syllabus

Algebra

• Focuses on sets, relations, and maps
• Groups and cyclic groups are examined
• Covers normal subgroups and quotient groups
• Lagrange's theorem is included
• Symmetric and alternating groups are studied
• Isomorphism theorems are important
• Rings, integral domains, and fields are part of the curriculum
• Ideals and quotient rings are covered
• Polynomial rings are included
• Linear and quadratic congruences are studied
• Legendre symbol is examined
• Arithmetic functions are included

Analysis

• Includes the real number system
• Convergence of sequences and infinite series of real numbers is covered
• Absolute and conditional convergence is examined
• Limits, continuity, and differentiability of functions of one real variable are studied
• Mean value theorems and their applications are a focus
• Riemann integrals are included
• Convergence of improper integrals is also covered

Limit, continuity

• Partial and directional derivatives are examined
• Differentiability of functions of several real variables is studied
• Mean value theorems and Taylor's theorem for functions of several real variables are included
• Jacobians are part of the curriculum
• Inverse and Implicit function theorem are covered
• Convergence and uniform convergence of sequences and series of functions is studied
• Basic concepts in metric spaces are included
• Compactness in metric spaces is examined
• Complex analytic functions are also covered

Differential Equations

• Encompasses differential equations of first order and their applications
• Equations of higher degree are part of the syllabus
• Singular solutions are examined
• Linear differential equations of higher order are covered
• Variation of parameters is studied
• Linear systems of first order are included
• Existence and uniqueness theorems for solutions of differential equations are examined

Vector Calculus

• Scalar and vector fields are included
• Gradient, divergence, and curl are examined
• Line integrals, double integrals, surface integrals, and volume integrals are studied
• Gauss', Stokes', and Green's theorems and their applications are covered

Linear Algebra

• Vector spaces, subspaces, linear independence, linear span, bases, and dimension are included
• Linear transformations and matrices are examined
• Rank-Nullity Theorem is covered
• Systems of linear equations are studied
• Gauss elimination is included
• Normal and Echelon form of a matrix are examined
• Determinant and Cramer's Rule are part of the syllabus
• Eigenvalues and eigenvectors are studied
• Cayley-Hamilton Theorem is included
• Diagonalization is examined
• Inner product spaces, bilinear and quadratic forms are covered

Mechanics

• Virtual work and catenary are included
• Simple harmonic motion and motion in a plane are examined
• Constrained motion and central orbits are studied
• Forces in three dimensions are included
• Moments and products of inertia of rigid bodies are examined
• D'Alembert's principle is also covered

Hydrodynamics

• Kinematics of fluid motion is examined
• Velocity potential and stream lines are included
• Euler's equation of motion is studied
• Fluid motion in two dimensions is examined
• Complex potential is part of the syllabus
• Sources, sinks, and doublets are included
• Image system with regard to a line and a circle is studied

Numerical Methods

• Numerical techniques for roots of general equations are covered
• Interpolation is examined
• Numerical differentiation and integration are included
• Numerical solution of first and second order ordinary differential equations are studied
• Matrix factorization and iterative methods for systems for linear equations are included
• Estimation of eigenvalues and eigenvectors is examined
• Least square curve fitting is covered