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

**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.

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