Fourier series: vector dot-product analogy; orthogonality and closure; expansions (even, odd, neither); Fourier transforms. Generalized Functions / Distributions: delta function, theta step Heavyside function, and their derivatives and Laplace transforms; applications to electric charge distributions. Ordinary Differential Equations: order, linearity, homogeneity; examples simple harmonic oscillator, damped SHO, damped driven SHO, resonance, Green functions. Coordinate Systems; Scale functions; Divergence, Gradient, Curl, Laplacian. Partial Differential Equations: Separation of variables, Cartesian coordinates in 1, 2, and 3 dimensions, Cylindrical Polar coordinates, Spherical Polar coordinates, example Laplace's Equation, Green functions. Group Theory: Discrete groups: cyclic groups Z_{n}; dihedral groups D_{n}; symmetric groups S_{n}; alternating groups A_{n}. Group multiplication tables. Representations. Sporadic groups. Lie groups: SO(2); SO(3); SO(n); SU(n); Sp(2n). Complex Analysis: Roots of Unity, Cauchy-Riemann equations, analyticity, contour integrals, residues, Laurent expansion Chaos: Sensitive Dependence on Initial Conditions Nonlinearity, Fractals, Hausdorf Dimension, Lyapunov Exponents, Hennon Map, Logistic Map, Cantor Dust, Sierpinski Gasket, Menger Sponge, Mandelbrot Set, Julia Set, Chua's Circuit, Period 3 Implies Chaos