Essentials of mathematical methods in science and engineering / Ş. Selçuk Bayın.

Includes bibliographical references (pages 787-792) and index.

1. Functional Analysis -- 1.1 Concept of Function -- 1.2 Continuity and Limits -- 1.3 Partial Differentiation -- 1.4 Total Differential -- 1.5 Taylor Series -- 1.6 Maxima and Minima of Functions -- 1.7 Extrema of Functions with Conditions -- 1.8 Derivatives and Differentials of Composite Functions -- 1.9 Implicit Function Theorem -- 1.10 Inverse Functions -- 1.11 Integral Calculus and the Definite Integral -- 1.12 Riemann Integral -- 1.13 Improper Integrals -- 1.14 Cauchy Principal Value Integrals -- 1.15 Integrals Involving a Parameter -- 1.16 Limits of Integration Depending on a Parameter -- 1.17 Double Integrals -- 1.18 Properties of Double Integrals -- 1.19 Triple and Multiple Integrals -- Problems -- 2. Vector Analysis -- 2.1 Vector Algebra: Geometric Method -- 2.1.1 Multiplication of Vectors -- 2.2 Vector Algebra: Coordinate Representation -- 2.3 Lines and Planes -- 2.4 Vector Differential Calculus -- 2.4.1 Scalar Fields and Vector Fields -- 2.4.2 Vector Differentiation -- 2.5 Gradient Operator -- 2.5.1 Meaning of the Gradient -- 2.5.2 Directional Derivative -- 2.6 Divergence and Curl Operators -- 2.6.1 Meaning of Divergence and the Divergence Theorem -- 2.7 Vector Integral Calculus in Two Dimensions -- 2.7.1 Arc Length and Line Integrals -- 2.7.2 Surface Area and Surface Integrals -- 2.7.3 An Alternate Way to Write Line Integrals -- 2.7.4 Green's Theorem -- 2.7.5 Interpretations of Green's Theorem -- 2.7.6 Extension to Multiply Connected Domains -- 2.8 Curl Operator and Stokes's Theorem -- 2.8.1 On the Plane -- 2.8.2 In Space -- 2.8.3 Geometric Interpretation of Curl -- 2.9 Mixed Operations with the Del Operator -- 2.10 Potential Theory -- 2.10.1 Gravitational Field of a Spherically Symmetric Star -- 2.10.2 Work Done by Gravitational Force -- 2.10.3 Path Independence and Exact Differentials -- 2.10.4 Gravity and Conservative Forces -- 2.10.5 Gravitational Potential -- 2.10.6 Gravitational Potential Energy of a System -- 2.10.7 Helmholtz Theorem -- 2.10.8 Applications of the Helmholtz Theorem -- 2.10.9 Examples from Physics -- Problems -- 3. Generalized Coordinates and Tensors -- 3.1 Transformations Between Cartesian Coordinates -- 3.1.1 Basis Vectors and Direction Cosines -- 3.1.2 Transformation Matrix and the Orthogonality Relation -- 3.1.3 Inverse Transformation Matrix -- 3.2 Cartesian Tensors -- 3.2.1 Algebraic Properties of Tensors -- 3.2.2 Kronecker Delta and the Permutation Symbol -- 3.3 Generalized Coordinates -- 3.3.1 Coordinate Curves and Surfaces -- 3.3.2 Why Upper and Lower Indices -- 3.4 General Tensors -- 3.4.1 Einstein Summation Convention -- 3.4.2 Line Element -- 3.4.3 Metric Tensor -- 3.4.4 How to Raise and Lower Indices -- 3.4.5 Metric Tensor and the Basis Vectors -- 3.4.6 Displacement Vector -- 3.4.7 Transformation of Scalar Functions and Line Integrals -- 3.4.8 Area Element in Generalized Coordinates -- 3.4.9 Area of a Surface -- 3.4.10 Volume Element in Generalized Coordinates -- 3.4.11 Invariance and Covariance -- 3.5 Differential Operators in Generalized Coordinates -- 3.5.1 Gradient -- 3.5.2 Divergence -- 3.5.3 Curl -- 3.5.4 Laplacian.

Print version record.

This book addresses the need for mathematical techniques to be introduced early on in the undergraduate program. Topics that are unique to the undergraduate curriculum, i.e. series, complex analysis, variational calculus, and integral transforms, do overl.