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AppxA_01fig_PChem.jpg Complex Numbers i. AppxA_02fig_PChem.jpg Complex Conjugate.

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Presentation on theme: "AppxA_01fig_PChem.jpg Complex Numbers i. AppxA_02fig_PChem.jpg Complex Conjugate."— Presentation transcript:

1 AppxA_01fig_PChem.jpg Complex Numbers i

2 AppxA_02fig_PChem.jpg Complex Conjugate

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

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 AppxA_14fig_PChem.jpg Matrices and Rotations

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

15 Eigen Representation of A NXN For a general A NXN :

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

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:

35 Series Solutions to D.E.’s

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41 Operators


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