# Description

Appropriate for one- or two-semester Advanced courses in departments of and . This clear, pedagogically rich develops a strong understanding of the mathematical principles and practices that today’s and scientists need to know. Equally as effective as either a textbook or manual, it approaches mathematical concepts from an perspective, making applications more vivid and substantial.

Its comprehensive instructional framework supports a conversational, down-to-earth narrative style, offering easy accessibility and frequent opportunities for application and reinforcement.

# Table of Content

I. ORDINARY DIFFERENTIAL EQUATIONS.
1. Introduction to Differential Equations.
2. Equations of First Order.
3. Linear Differential Equations of Second Order and Higher.
4. Power Series Solutions.
5. Laplace Transform.
6. Quantitative Methods: Numerical Solution of Differential Equations.
7. Qualitative Methods: Phase Plane and Nonlinear Differential Equations.

II. LINEAR ALGEBRA.
8. Systems of Linear Algebraic Equations; Gauss Elimination.
9. Vector Space.
10. Matrices and Linear Equations.
11. The Eigenvalue Problem.
12. Extension to Complex Case (Optional).

III. SCALAR and VECTOR FIELD THEORY.
13. Differential Calculus of Functions of Several Variables.
14. Vectors in 3-Space.
15.Curves, Surfaces, and Volumes.
16. Scalar and Vector Field Theory.

IV. FOURIER SERIES AND PARTIAL DIFFERENTIAL EQUATIONS.
17. Fourier Series, Fourier Integral, Fourier Transform.
18. Diffusion Equation.
19. Wave Equation.
20. Laplace Equation.

V. COMPLEX VARIABLE THEORY.
21. Functions of a Complex Variable.
22. Conformal Mapping.
23. The Complex Integral Calculus.
24. Taylor Series, Laurent Series, and the Residue Theorem.

Appendix A: Review of Partial Fraction Expansions.
Appendix B: Existence and Uniqueness of Solutions of Systems of Linear Algebraic Equations.
Appendix C: Table of Laplace Transforms.
Appendix D: Table of Fourier Transforms.
Appendix E: Table of Fourier Cosine and Sine Transforms.
Appendix F: Table of Conformal Maps.