# Partial Differential Equations Tuition Class

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Definition and examples of groups, examples of abelian and non-abelian groups, the group Zn of integers under addition modulo n and the group U(n) of units under multiplication modulo n. Hence, Join Math Tuition Class In Noida.

Cyclic groups from number systems, complex roots of unity, circle group, the general linear group GLn (n,R), groups of symmetries of (i) an isosceles triangle, (ii) an equilateral triangle, (iii) a rectangle, and (iv) a square, the permutation group Sym (n), Group of quaternions. Hence, Top Math Tutor Is Available To Help You In BSc Math.

Subgroups, cyclic subgroups, the concept of a subgroup generated by a subset and the commutator subgroup of group, examples of subgroups including the center of a group. Cosets, Index of subgroup, Lagrange’s theorem, order of an element, Normal subgroups: their definition, examples, and characterizations, Quotient groups. Hence, Join Tuition Class Of Complex Analysis In Noida.

Definition and examples of rings, examples of commutative and non-commutative rings: rings from number systems, Zn the ring of integers modulo n, ring of real quaternions, rings of matrices, polynomial rings, and rings of continuous functions. Subrings and ideals, Integral domains and fields, examples of fields: Zp, Q, R, and C. Field of rational functions. Hence, Join Tuition Class Of Differential Claculus In Noida.

Mathematical induction. Principle of inclusion and exclusion. Pigeon hole principle. Finite combinatorics. Generating functions. Partitions. Recurrence relations. Linear difference equations with constant coefficients. Partial and linear orderings. Chains and antichains. Lattices. Distributive lattices. Complementation. Graphs and Planar graphs. Paths and circuits. Hamiltonian paths. Shortest paths. Eulerian paths. Traveling salesman problem. Trees. Spanning trees. Truth functional logic and prepositional connectives. Switching circuits. Boolean algebras. Duality. Boolean functions. Normal forms. Karnaugh maps. Hence, Join Tuition Class Of Discrete Mathematics In Noida.

Applications of differential equations: the vibrations of a mass on a spring, mixture problem, free damped motion, forced motion, resonance phenomena, electric circuit problem, mechanics of simultaneous differential equations. Applications to Traffic Flow. Vibrating string, vibrating membrane, conduction of heat in solids, gravitational potential, conservation laws. Hence, Join Tuition Class Of Mathematical Methods In Noida.

Sample space, probability axioms, real random variables (discrete and continuous), cumulative distribution function, probability mass/density functions, mathematical expectation, moments, moment generating function, characteristic function, discrete distributions: uniform, binomial, Poisson, continuous distributions: uniform, normal, exponential. Joint cumulative distribution function and its properties, joint probability density functions, marginal and conditional distributions, expectation of function of two random variables, conditional expectations, independent random variables. Hence, Join Tuition Class Of Probability Theory In Noida.

The Unear programming prob lem. Problem formulation. Types of solutions. Linear programming in matrix notation. Graphical solution of linear programming problems. Some basic proper ties of convex sets, convex functions, and concave functions. Theory and applica tion of the simplex method of solution of a linear programming problem, Charne’s M-technique. The two phase method, principle of duality in linear programming problem, fundamental duality theorem, simple problems, the transportation and assignment problems. Hence, Join Tuition Class Of Linear Programming Optimization In Noida.

Finite and infinite sets, examples of countable and uncountable sets. Real line, bounded sets, suprema and infima, completeness property of R, Archimedean property of R, intervals. Concept of cluster points and statement of Bolzano-Weierstrass theorem. Real Sequence, Bounded sequence, Cauchy convergence criterion for sequences. Cauchy’s theorem on limits, order preservation and squeeze theorem, monotone sequences and their convergence (monotone convergence theorem without proof). Infinite series. Cauchy convergence criterion for series, positive term series, geometric series, comparison test, convergence of p-series, Root test, Ratio test, alternating series, Leibnitz’s test (Tests of Convergence without proof). Definition and examples of absolute and conditional convergence. Sequences and series of functions, Pointwise and uniform convergence. Mn-test, M-test, Statements of the results about uniform convergence and integrability and differentiability of functions, Power series and radius of convergence. Hence, Join Tuition Class Of Real Vector Analysis In Noida.

First order exact differential equations. Integrating factors, rules to find an integrating factor. First order higher degree equations solvable for x, y, p. Methods for solving higher-order differential equations. Basic theory of linear differential equations, Wronskian, and its properties. Solving a differential equation by reducing its order. Linear homogenous equations with constant coefficients, Linear non-homogenous equations, The method of variation of parameters, The Cauchy-Euler equation, Simultaneous differential equations, Total differential equations. Order and degree of partial differential equations, Concept of linear and non-linear partial differential equations, Formation of first order partial differential equations, Linear partial differential equation of first order, Lagrange’s method, Charpit’s method. Classification of second order partial differential equations into elliptic, parabolic and hyperbolic through illustrations only. Hence, Join Tuition Class Of Differential Equations In Noida.

Algorithms, Convergence, Bisection method, False position method, Fixed point iteration method, Newton’s method, Secant method, LU decomposition, Gauss-Jacobi, Gauss-Siedel and SOR iterative methods. Lagrange and Newton interpolation: linear and higher order, finite difference operators. Numerical differentiation: forward difference, backward difference and central Difference. Integration: trapezoidal rule, Simpson’s rule, Euler’s method. Hence, Join Tuition Class Of Numerical Methods In Noida.

Topics Covered For Differential Calculus Limit and Continuity (ε and δ definition), Types of discontinuities, Differentiability of functions, Successive differentiation, Leibnitz’s theorem, Partial differentiation, Euler’s theorem on homogeneous functions.
Tangents and normals, Curvature, Asymptotes, Singular points, Tracing of curves. Parametric
representation of curves and tracing of parametric curves, Polar coordinates and tracing of curves
in polar coordinates. Rolle’s theorem, Mean Value theorems, Taylor’s theorem with Lagrange’s and Cauchy’s forms of remainder, Taylor’s series, Maclaurin’s series of sin x, cos x, ex , log(l+x), (l+x)m, Maxima and Minima, Indeterminate forms. Hence, Join Tuition Class Of Partial Differential Equations In Noida.

Vector spaces, subspaces, algebra of subspaces, quotient spaces, linear combination of vectors, linear span, linear independence, basis and dimension, dimension of subspaces. Linear transformations, null space, range, rank and nullity of a linear transformation, matrix representation of a linear transformation, algebra of linear transformations. Dual Space, Dual Basis, Double Dual, Eigen values and Eigen vectors, Characteristic Polynomial. Isomorphisms, Isomorphism theorems, invertibility and isomorphisms, change of coordinate matrix. Hence, Join Tuition Class Of Linear Algebra In Noida.

Limits, Limits involving the point at infinity, continuity. Properties of complex numbers, regions in the complex plane, functions of complex variable, mappings. Derivatives, differentiation formulas, Cauchy-Riemann equations, sufficient conditions for differentiability. Analytic functions, examples of analytic functions, exponential function, Logarithmic function, trigonometric function, derivatives of functions, definite integrals of functions. Contours, Contour integrals and its examples, upper bounds for moduli of contour integrals. Cauchy Goursat theorem, Cauchy integral formula. Liouville’s theorem and the fundamental theorem of algebra. Convergence of sequences and series, Taylor series and its examples. Laurent series and its examples, absolute and uniform convergence of power series. Hence, Join Tuition Class Of Complex Analysis In Noida.