School of Mathematics, Computer Science, and Physics

Lecturer: Samy Zafrany

## Course Whatsapp Group

Password should be given at the first lecture or can be obtained from me (Samy)

## Course Program

Course Program: sylabus, grading policy, bibliography, etc.

# Lecture Notes

Full lecture notes can be downloaded from the link above or by parts from the links below. Corrections and improvement suggestions are most welcome (send to email above).
[Infinite series chapters have been moved to Appendix A-C at the end of the booklet]

## Course Promlem Set

A list of exercises for the whole semester.
Make sure to solve as many problems as you can before the final exam (best way to prepare).

## Course Summary (sikum)

A summary of the main course theorems, definitions, and formulas.
This documnet can be used in the final exam (in printed form), but please make sure you do not add any hand written notes on it!

## Quizes

We will have 3 short quizes during the course. Here is an example (with detailed explanation) on how it looks

## Final Exam 1 - Solution

Solution of final exam given on February 07, 2018
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## Future Topics in Calculus 2 - Parametric 3D surfaces

This is an extended Spherical Harmonics 3D surface with 12 varying parameters

$r = \sin^{m_1}(m_0\phi) + \cos^{m_3}(m_2\phi) + \sin^{m_5}(m_4\theta) + \cos^{m_7}(m_6\theta) + \sin^{m_9}(m_8\theta) + \cos^{m_{11}}(m_{10}\theta) \\$
The parametric equations are: $\begin{cases} x &= r \sin(\phi) \cos(\theta) \\[0.5em] y &= r \cos(\phi) \\[0.5em] z &= r \sin(\phi) \sin(\theta) \end{cases}$ Initial values of the twelve parameters are: $\newcommand\T{\Rule{0pt}{1em}{.3em}}$ $\renewcommand{\arraystretch}{0.9} \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline m_0 & m_1 & m_2 & m_3 & m_4 & m_5 & m_6 & m_7 & m_8 & m_9 & m_{10} & m_{11} \\ \hline 4.0 & 3.0 & 2.0 & 3.0 & 6.0 & 2.0 & 6.0 & 4.0 & 8.0 & 4.0 & 6.0 & 8.0 \\ \hline \end{array}$

The Python code for generating this movie can be obtained from here.

## Classic Spherical Harmonic 3D surface (moving perspectives)

This is the classical Spherical Harmonics 3D surface with 8 static parameters

$r = \sin^{m_1}(m_0\phi) + \cos^{m_3}(m_2\phi) + \sin^{m_5}(m_4\theta) + \cos^{m_7}(m_6\theta) \\$
The parametric equations are: $\begin{cases} x &= r \sin(\phi) \cos(\theta) \\[0.5em] y &= r \cos(\phi) \\[0.5em] z &= r \sin(\phi) \sin(\theta) \end{cases}$ Static values of the eight parameters are: $\newcommand\T{\Rule{0pt}{1em}{.3em}}$ $\renewcommand{\arraystretch}{0.9} \begin{array}{|c|c|c|c|c|c|c|c|} \hline m_0 & m_1 & m_2 & m_3 & m_4 & m_5 & m_6 & m_7 \\ \hline 4.0 & 3.0 & 2.0 & 3.0 & 6.0 & 2.0 & 6.0 & 4.0 \\ \hline \end{array}$

The Python code for generating this movie can be obtained from here.

## Conic Section 3D Animation

The Cartesian equation of the Conic Section surface is given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$

## 3D Two-Sheeted Hyperboloid with a tangent plane

The Cartesian equation of the Hyperoloid surface is given by $x^2 - y^2 - z^2 = 4$ The tangent plane is $3x - 2y - z - 4 = 0$ The tangent point is $(3,2,1)$.

And of course, let's not forget all the great people who made this possible ...
(page 339 from a soon to be published book ... stay tuned :-)