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

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Presentation transcript:

AppxA_01fig_PChem.jpg Complex Numbers i

AppxA_02fig_PChem.jpg Complex Conjugate

Complex Numbers Representing Waves

AppxA_11fig_PChem.jpg Vectors

AppxA_12fig_PChem.jpg Vector Addition and Subtraction

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

Determinants and Cross Products

Systems of Equations

Solving Systems of Equations

Eigenvalues and Eigenvectors

Eigenvalues

Eigenvalues and Eigenvectors

AppxA_14fig_PChem.jpg Matrices and Rotations

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

Eigen Representation of A NXN For a general A NXN :

Eigen Representation of A NXN

Eigen Representation Real Symmetric Matrix For a symmetric A NXN :

Eigen Representation Real Symmetric Matrix

Eigen Representation Hermitian Matrix For a Hermitian A NXN :

Eigen Representation Hermitian Matrix

AppxA_03fig_PChem.jpg Differentiation

AppxA_04fig_PChem.jpg Derivatives of Some Important Functions

Some Basic Rules Linearity Product Rule Quotient Rule Chain Rule

AppxA_05fig_PChem.jpg Higher Order Derivatives And Optimization

AppxA_05fig_PChem.jpg Higher Order Derivatives And Optimization

AppxA_08fig_PChem.jpg Integration

AppxA_08fig_PChem.jpg Integration Linearity Power Rule

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

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

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

AppxA_06fig_PChem.jpg Fourier Series

First Order Differential Equations

Second Order Linear Differential Equations Trial function

Second Order Linear Differential Equations Initial Conditions:

Series Solutions to D.E.’s

Operators