Advanced Engineering Mathematics, SI Edition

Rn.Vectors in the Plane and 3 Space. The Dot Product. The Cross Product. n-Vectors and the Algebraic Structure of Rn. Orthogonal Sets and Orthogonalization. Orthogonal Complements and Projections.11. Matrices, Determinants and Linear Systems.Matrices and Matrix Algebra. Row Operations and Reduced Matrices. Solution of Homogeneous Linear Systems. Solution of Nonhomogeneous Linear Systems. Matrix Inverses. Determinants, Cramer s Rule. The Matrix Tree Theorem.12. Eigenvalues, Diagonalization and Special Matrices.Eigenvalues and Eigenvectors. Diagonalization. Special Matrices and Their Eigenvalues and Eigenvectors. Quadratic Forms.PART IV: SYSTEMS OF DIFFERENTIAL EQUATIONS.13. Systems of Linear Differential Equations.Linear Systems. Solution of X = AX When A Is Constant. Exponential Matrix Solutions. Solution of X = AX + G for Constant A.14. Nonlinear Systems and Qualitative Analysis.Nonlinear Systems and Phase Portraits. Critical Points and Stability. Almost Linear Systems, Linearization.Part V: VECTOR ANALYSIS. 15. Vector Differential Calculus.Vector Functions of One Variable. Velocity, Acceleration, and Curvature. The Gradient Field. Divergence and Curl. Streamlines of a Vector Field.16. Vector Integral Calculus.Line Integrals. Green s Theorem. Independence of Path and Potential Theory. Surface Integrals. Applications of Surface Integrals. Gauss s Divergence Theorem. Stokes s Theorem.PART VI: FOURIER ANALYSIS.17. Fourier Series.Fourier Series On [-L, L]. Fourier Sine and Cosine Series. Integration and Differentiation of Fourier Series. Properties of Fourier Coefficients. Phase Angle Form. Complex Fourier Series, Filtering of Signals.18. Fourier Transforms.The Fourier Transform. Fourier Sine and Cosine Transforms.PART VII: COMPLEX FUNCTIONS.19. Complex Numbers and Functions.Geometry and Arithmetic of Complex Numbers. Complex Functions, Limits. The Exponential and Trigonometric Functions. The Complex Logarithm. Powers.20. Integration.The Integral of a Complex Function. Cauchy s Theorem. Consequences of Cauchy s Theorem.21. Series Representations of Functions.Power Series. The Laurent Expansion.22. Singularities and the Residue Theorem.Classification of Singularities. The Residue Theorem. Evaluation of Real Integrals.23. Conformal Mappings.The Idea of a Conformal Mapping. Construction of Conformal Mappings.Notation.ANSWERS TO SELECTED PROBLEMS.