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Comprehensive Overview of Complex Numbers, Vectors, and Differential Equations

This resource provides a detailed exploration of complex numbers, including their conjugates, wave representation, and applications in vector operations such as addition, subtraction, and products. The content further delves into determinants, eigenvalues, and eigenvectors, illustrating their role in solving systems of equations and matrix rotations. Additionally, it covers key calculus concepts like derivatives, Taylor series, and integration techniques. Aimed at enhancing understanding in mathematical and physical contexts, this material is essential for students and professionals alike.

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Comprehensive Overview of Complex Numbers, Vectors, and Differential Equations

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  1. Complex Numbers i AppxA_01fig_PChem.jpg

  2. Complex Conjugate AppxA_02fig_PChem.jpg

  3. Complex Numbers Representing Waves

  4. Vectors AppxA_11fig_PChem.jpg

  5. Vector Addition and Subtraction AppxA_12fig_PChem.jpg

  6. Vector Products Dot Product Cross Product c – Unit vector perpendicular to a & b AppxA_13fig_PChem.jpg

  7. Determinants and Cross Products

  8. Systems of Equations

  9. Solving Systems of Equations

  10. Eigenvalues and Eigenvectors

  11. Eigenvalues

  12. Eigenvalues and Eigenvectors

  13. Matrices and Rotations AppxA_14fig_PChem.jpg

  14. AppxA_14fig_PChem.jpg Matrices and Rotations Rz(120o) = Rz(180o) =

  15. Eigen Representation of ANXN For a general ANXN:

  16. Eigen Representation of ANXN

  17. Eigen Representation Real Symmetric Matrix For a symmetric ANXN:

  18. Eigen Representation Real Symmetric Matrix

  19. Eigen Representation Hermitian Matrix For a Hermitian ANXN:

  20. Eigen Representation Hermitian Matrix

  21. Differentiation AppxA_03fig_PChem.jpg

  22. Derivatives of Some Important Functions AppxA_04fig_PChem.jpg

  23. Some Basic Rules Linearity Product Rule Quotient Rule Chain Rule

  24. Higher Order Derivatives And Optimization AppxA_05fig_PChem.jpg

  25. AppxA_05fig_PChem.jpg Higher Order Derivatives And Optimization

  26. AppxA_08fig_PChem.jpg Integration

  27. Integration Linearity Power Rule AppxA_08fig_PChem.jpg

  28. Even and Odd Functions Even Functions Unless f(x) is an even periodic function, Symmetric about x-axis, and a=2p Odd Functions

  29. Power Series Approximation of Functions an = n! Diverges an = 1/n! Converges AppxA_07fig_PChem.jpg

  30. Taylor Series Expansions f(x) = exp(x) about x = 0 f(x) = ln(1+x) about x = 0 AppxA_06fig_PChem.jpg

  31. AppxA_06fig_PChem.jpg Fourier Series

  32. First Order Differential Equations

  33. Second Order Linear Differential Equations Trial function

  34. Second Order Linear Differential Equations Initial Conditions:

  35. Series Solutions to D.E.’s

  36. Series Solutions to D.E.’s

  37. Series Solutions to D.E.’s

  38. Series Solutions to D.E.’s

  39. Series Solutions to D.E.’s

  40. Series Solutions to D.E.’s

  41. Operators

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