Abstract Algebra B.Sc. Mathematics 1st Semester University of Kota | Lecture - 1 | Binary Operation

Описание: In this Video series, we will discuss about the syllabus of BSc mathematics first semester

Binary operation. Addition and multiplication modulo operations. Definition of a group with examples and simple properties, Permutation group, cycle, transpositions, even and odd permutations, and alternating group. Order of an element of a group and its properties. Subgroups of a group with its properties, Cyclic groups, and their properties, Cosets. Index of a subgroup, Lagrange’s theorem, and its applications.
Normal subgroups with properties. Simple groups, Quotient groups.
Polar coordinates, the angle between radius vector and tangent, polar subtangent and subnormal. Perpendicular from pole on tangent. Pedal equation of a curve. Derivative of length of an arc in cartesian and polar coordinates. Curvature, Radius of curvature and its formula in various forms. Centre of curvature, chord of curvature.
Vector differentiation and integration, Problems based on Gradient, divergence and curl. Vector identities, Line and surface integrals. Theorems of Gauss, Green, Stokes(without proof) and problems based on these.
Ellipse : standard equation, auxiliary circle, eccentric angle, tangent, normal, two tangents from point, chord of contact, pole, polar, chord whose mid point given, diameter, conjugate diameters and four normals from a point.
Hyperbola : Standard equation, parametric co-ordinates, asymptotes, equation referred to asymptotes as axes, conjugate diameters and rectangular hyperbola
Cone : Homogeneous equation in x, y, z, cone with a given vertex and given base, enveloping cone, condition for the general equation to represent a cone, tangent plane,
reciprocal cone, angle between the two lines, in which a plane cuts a cone, three mutually perpendicular generators and right circular cone.
Cylinder : Right circular cylinder and enveloping cylinder.