Advanced Engineering Mathematics

FUNCTIONS OF A REAL VARIABLEIntroductionLimits, Continuity, and DifferentiabilityApplication of Derivatives and Taylor's SeriesIntegration and its ApplicationsImproper IntegralsFUNCTIONS OF SEVERAL REAL VARIABLESIntroductionFunctions of Two VaraiblesPartial DerivativesHigher Order Partial DerivativesMaximum and Minimum Values of FunctionsMultiple IntegralsMATRICES AND EIGENVALUE PROBLEMSIntroductionMatricesVector SpacesSolution of General Linear System of EquationsEigenvalue ProblemsAnswers and HintsORDINARY DIFFERENTIAL EQUATIONS OF FIRST ORDERIntroductionFormation of Differential EquationsSolution of a Differential EquationInitial and Boundary Value ProblemsSolution of Equations in Separable FormExact First Order Differential EquationsLinear First Order EquationsSome Special First Order EquationsOrthogonal Trajectories of a Given Family of CurvesExistence and Uniqueness of SolutionsAnswers and HintsLINEAR DIFFERENTIAL EQUATIONSIntroductionSolution of Linear Differential EquationsMethods for Solution of Linear EquationsSolution of Non-Homogenous Linear EquationsOperator Methods for Finding Particular IntegralsSimultaneous Linear EquationsAnswers and HintsSERIES SOLUTION OF DIFFERENTIAL EQUATIONSIntroductionOrdinary and Singular Points of an EquationPower Series SolutionSeries Solution about a Regular Singular Point: Frobenius MethodAnswers and HintsLEGENDRE POLYNOMIALS, BESSEL FUNCTIONS AND STURM-LIOUVILLE PROBLEMIntroductionLegendre Differential Equation and Legendre PolynomialsBessel's Differential Equation and Bessel's FunctionsSturm-Liouville ProblemAnswers and HintsLAPLACE TRANSFORMATIONIntroductionBasic Theory of Laplace TransformsLaplace Transform Solution of Initial Value ProblemsTranslation Theorems (Shifting Theorems)Laplace Transform of Dirac-delta Function and More Properties of Laplace TransformsLaplace Transform of Periodic FunctionsLaplace Transform Method for the Solution of Some Partial Differential EquationsAnswers and HintsFOURIER SERIES, FOURIER INTEGRALS AND FOURIER TRANSFORMSIntroductionFourier SeriesFourier Half-Range SeriesFourier IntegralsApplication of Fourier Series: Separation of Variables Solution of Linear Partial Differential EquationsFourier TransformsAnswers and HintsFUNCTIONS OF A COMPLEX VARIABLE: ANALYTIC FUNCTIONSIntroductionComplex Number SystemSets of Points in the Complex PlaneFunctions of a Complex VariableElementary FunctionsLimits and ContinuityDifferentiability and AnalyticityHarmonic FunctionsAnswers and HintsINTEGRATION OF COMPLEX FUNCTIONSIntroductionDefinite IntegralsCauchy Integral TheoremChauchy Integral FormulaCauchy Integral Formula for DerivativesAnswers and HintsPOWER SERIES, TAYLOR AND LAURENT SERIESIntroductionInfinite SequencesInfinite SeriesPower SeriesTaylor SeriesLaurent SeriesAnswers and HintsZEROS, SINGULARITIES AND RESIDUESIntroductionZeros and Singularities of Complex FunctionsResiduesEvaluation of Contour Integrals Using ResiduesEvaluation of Real Integrals Using ResiduesAnswers and HintsBILINEAR TRANSFORMATION AND CONFORMAL MAPPINGIntroductionLinear and Inverse TransformationsBilinear TransformationsConformal MappingAnswers and HintsVECTOR DIFFERENTIAL AND INTEGRAL CALCULUSIntroductionParametric Representatives, Continuity and Differentiability of Vector FunctionsGradient of a Scalar Field and Directional DerivativeDivergence and Curl of a Vector FieldLine Integrals and Green's TheoremSurface Area and Surface IntegralsDivergence Theorem of Gauss and Stoke's TheoremAnswers and Hints