General revision for sets and inequalities .Relations and functions : Limits : Continuity : Definitions and theorems .Differentiation .Applications .

Cartesian and polar coordinates in plane -Order pairs or a vectors and pointsAlgebraic operation an vectors- Vector equations of st lines parametric representation and types of line. Transformation of axis, Polar coordinates :Some equation in polar form-general equation of second degree of two variables.

Matrices and its defined operations – algebraic properties of operations on Matrices- special Matrices-Transpose of a matrix –symmetric and skew symmetric Matrices-Elementary raw transformations-Equivalent Matrices- rank of a matrix –Inverse of a matrix and its properties- calculations of inverse matrices by elementary raw transformations. Determinants(Definitions and concepts)-properties of Determinants in the calculations of inverse matrix (which is non singular) -Linear equations -Vectors spaces- partialvectors spaces-Linear dependency and independency- basis and dimension.

statistics since definition (basic concepts and definitions)- presentation statistical data-measures of central tendency: ( The mean – the geometric mean – the harmonic mean – the median – the mode ) - measures of dispersion: ( the range – the mean deviation – the quartile deviation – the variance – the standard deviation – coefficient of variance ) - the correlation and regression.

Definite integratio .Indefinite Integration .Inverse trigonometric functions – logarithmic– exponential– Hyper geometric functions and their Derivatives Methodsof integration .Applications of Integration .Improper integration .

Sets: concept of a set, its elements and ways of expressing them Operations on sets including: Union, intersection, Difference, symmetric differences . Basic logical concepts: constant, variables, diagrams and statements. Operations and logical connectives: Negation, compound statements, disjunction, conjunction, quantifiers . The set of integers, Heighst common factor (HCF), prime relatively, Division algorithm, unique factorization theorem.

(basic concepts and definitions)- sample space – the event-random experiments –Set theory – axioms of probability - sample space counting methods - general laws in probability theory – the conditional probability and the independence – bayes theory- The random variables – some Special probability distributions .

Force, Resultant of concarent forest in space and in a plane Moment of a force about a point and about an axis. Equilibrium of nonconcarent forces - (in a plane and in space). Center of gravity: several point masses, linear rigid bodies, surface rigid bodies.

Multi – variable functions .Partial derivative .Double integral .Evaluation of Double integrals , applications of triple integrals , sequence of real numbers – convergence and divergence – Cauchy's sequence .Absolute and conditional convergence , Applications of the power series .

revision for the one dimensional random variables – Special discrete probability distributions ( its proprets – its moments – its moment generating function ) - Special continuous probability distributions ( its proprets – its moments– its moment generating function ).

Kinematics of particles, motion on straight line, Motion in a plane using Cartesian. Kinematics of rigid body, Velocity of a point with respect to another. Kinematics of particles, Newton's Law's and applications for all types of motions mentioned Linear momentum, angular momentum and applications in collision and coefficient of restitution. Work, power and energy in linear and rotational motions. Motion in a plane and in the three dimensional space for rigid bodies.

Linear transformations -Inner product space(Definitions –examples-and fundamental properties)- Orthogonal vector- norm-Gram- Schmidt process. Eigen values and eigen vectors for a matrix , Eigen values and eigen vectors oflinear transformation- Matrices that can be diagonalised- Diagonalisation of symmetric Matrices- Hermitian Matrices–Cayiey –Hamiltion Theorem and its applications- the characteristic polynomial.

Definitions and Examples of ordinary differential equations, first Ordered differential Equations, solution of the First Ordinary differential Equations, linear Ordinary Differential equations of second order and higher, linear Ordinary Differential equations with constant coefficients, non homogenous ordinary linear Differential equations, Laplace Transform .

Vectors: Definition Of a vector and its Magnitude - Vector algebra - unit vector - unit vectors in rectangular Differential Geometry: Concept of a curve and classifications - ordinary and normal parametric Concept of a surface, parametric representation.

Properties of real number - greatest lower bound - least upper bounds - Archimedes property .N the Euclidean space (Rn) Sequences and series in Rn .Continuous functions : continuity and compactness , (connection) - regular continuity .

Introduction of simple linear regression - Simple linear regression - Assumption of simple linear regression model - Least squares method - Alternative formula for simple linear regression - Variance estimate -Inferences concerning regression coefficient - The prediction - Regression analysis styel. - Coefficient of determination .

Complex numbers:Definition,Algebraic operations,Geometrical representation .Triangle inequality.Functions of complex variabie:Limit.Continuity.Differentiability. Cauchy-Riemann equatios.Elementary functions of a complex variable: Their properties and the mappings defined by them.Complex Integrations: Properties of the line integral,Cauchy's theorem.series: Taylor and Laurent serie.Singularities of complex function, Residues: formulae for calculating residues,theResidue theorem and its applications.

Binary operations and its properties- The Group andbasic properties –Sub group Cyclic groups and their properties – permutation group and its properties- Commutators Group And its properties –Langrahg i s theorem and its applications- Normal subgroups and its basis properties simple group group-quotient group-Homomorphism in a Group (Examples-and elementary properties)-Effect of homomorphism on the subgroup and the normal subgroup-kernel of homomorphism at its properties-First fundamental theorem of isomorphism

Derivative of real functions - L'Hospital's derivatives - Thylor's Theorem .Integrable functions,Lebesgue theorem - Fundamental theorem of Calculus .Sequences and series of functions :Regular convergence and continuity and Differentiability– (Integrability)

Systems of ordinary first order differential equations, Solution of a system of differential equations use of Marries in solving a system of homogeneous differential equations and solution of in case of some special differential equations.

Virtual work and D'lembert principal , Momentum and energy of a mechanical system, Euler Argles , General equations of Motion for a rigid body , Virtual displacement and Degrees of freedom ,Types of Mechanical Systems,Canonical Transformations,Hamilto – Jacobi Equations

Complex Integrations,Cauchy is Integral relations.Properties of Analytic function according to results of Integration.Series of functions and convergence Tests.Taylor series, Laurent Secries and Taylor and Laurent Theorem.

Definitions of a P.D.E, origin of the P.D.E, solution of the P.D.E, P.D.E of the first order in two variables, P.D.E of the second order in two variables, P.D.E of the second order in n variables, P.D.E in Physics (head equation ,wave equation, Laplace equation ).

Ring: Definitions- Examples – Basic concepts and its elementary consequences. Special types of rings and ring elements –Subrings and their properties – Integral domain and there property – Field,Ideals and its basic concepts. Principle ideals in a commutative ring - Ring homomrphisms and theirs properties ,prime and Maximal ideal.

Taylors polynomial ( calculation of error when using Taylors polynomial . Errors ( origin of error – mathematical pattern and numbers representation according to systems, use of calculators and approximations. Types of errors – calculations of errors with properties – roots of linear and non linear equations.

Bivariate distributions- multivariate distributions –marginal and conditional distributions – independence – joint mathematical expectation – probability and moment generating functions of the bivariate random variables – covariance and correlation – distributions of functions of random variables - distributions of sum of random variables – the central limit theorem – low of large numbers – normal approximation to some distributions.

Completion and rooting ( completion from the inside , completion from the outside) – numerical differentiation – numerical integration – the numerical solution of normal differential equations – the numerical solution of partial differential equations .

Topological.space:definition.of.Topology-Topological spaces-open sets –closed set – interior of asset –closure of a set- boundary of a set and a accumulation points(closter).

General definitions: Solid, liquid and gaseous bodies; density; pressure; Velocity Integral equations for motion of liquids. Equations of motion for a liquid (Viscous and non-Viscous).  Continuity equation.  Equation of reduction of Momentum.

Definition – Fredholm Equations – Volterra Equations – Singular equations, relation between Integral Equations, solution of Integral equations, solution of nonhomogeneous Integral equations, nonlinear Integral equations – Singular Integral equations.

Systems of equations:Canonical form, change baser - pivoting The simpler method :The objective function and the tableau. Computational Refinements: The revised simpler method - products form, Rein version. Duality and the dual simpler method the dual problem, the complementary slackner condition, the dual simpler, The duality theorem end some consequences, an interpretation of dual variables, exercises. Sensitivity Analysis: Discrete changes, parametric programming. Bounded variables: Implicit constraints, Definitions sensitivity analyses, Integer programming.

Metric Spaces, Separable Spaces,Normed Spaces(Definition and properties, Convergence and completion, Linear effects),Banach Spaces( Hahn – BanachTheorem, The weak convergence),Banach Algebra , Hilbert Spaces(Inner beating space and Hilbert space, OrthonormalSets, Space attached to the Hilbert space, Linear effects on Hilbert space) ,Theorems and basic Inequalities.

Fourier Series, Fourier transform, integral transformations, special functions ( Gamma function , Beta function , Bessel functions), main principal methods of Partial differential equations.