1. AppxA_01fig_PChem.jpg Complex Numbers i

2. AppxA_02fig_PChem.jpg Complex Conjugate * - z* =(a, -b)

3. Complex Numbers Representing Waves

4. AppxA_11fig_PChem.jpg Vectors

5. AppxA_12fig_PChem.jpg Vector Addition and Subtraction

6. AppxA_13fig_PChem.jpg Vector Products Dot Product Cross Product c – Unit vector perpendicular to a & b c i i j j k k + -

7. Determinants and Cross Products Cross Product 3 by 3 Major Minor

8. Systems of Equations

9. Solving Systems of Equations

10. Eigenvalues and Eigenvectors

11. Eigenvalues

12. Eigenvalues and Eigenvectors

13. AppxA_14fig_PChem.jpg Matrices and Rotations Initial and final vector components Where Recall that The final component can be re-expressed in terms of  and  :

14. AppxA_14fig_PChem.jpg Matrices and Rotations R z (180 o ) = R z (120 o ) = v 2 = R z v 1 Where Rotation of Matrices : A 2 = R z A 1 R z -1 Rotation of Vectors

15. Eigen Representation of A NXN For a general A NXN : Thus changing to the eigen-representation is like a rotation of the coordinate system to another one where the new axes give rise to equation that are diagonal.

16. Eigen Representation of A NXN

17. Eigen Representation Real Symmetric Matrix For a symmetric A NXN :

18. Eigen Representation Real Symmetric Matrix

19. Eigen Representation Hermitian Matrix For a Hermitian A NXN :

20. Eigen Representation Hermitian Matrix

21. AppxA_03fig_PChem.jpg Differentiation

22. AppxA_04fig_PChem.jpg Derivatives of Some Important Functions

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

24. AppxA_05fig_PChem.jpg Higher Order Derivatives And Optimization

25. AppxA_05fig_PChem.jpg Higher Order Derivatives And Optimization

26. AppxA_08fig_PChem.jpg Integration

27. AppxA_08fig_PChem.jpg Integration Linearity Power Rule

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

29. AppxA_07fig_PChem.jpg a n = n! a n = 1/n! Power Series Approximation of Functions Diverges Converges Add some notes on the process of fittng power series

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

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: One other known point:

35. Series Solutions to D.E.’s

37. Recursion Formula

38. Series Solutions to D.E.’s

40. Operators