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Differential and Integral Calculus 2T (104013)

Department of Mathematics

Lecturer: Samy Zafrany
Email: samyz@technion.ac.il

Moodle Course Page

Please visit this page regularly for all administrative information: homework assignments, course adminstrator, books and course materials, exams and grading policy, faculty and teaching assistants lists, office hours, study units, etc.

Course Program

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).

Recommended Software Tools

  1. GeoGebra Graphing Calculator
    Neat Examples
  2. GeoGebra 3D Calculator - Android App
  3. Desmos Graphing Calculator
  4. Desmos - Android App
  5. 3D Plot Grapher - Android App
  6. CalcPlot3D
  7. 3D Surface Plotter
  8. 3D Surface Plotter - Android App
  9. WolframAlpha Computational Intelligence
  10. Matlab Online
  11. Sympy Live (Python Programming)
  12. Plotly Matplotlib Library

Solutions to Homework Assignments

  1. Homework #1
  2. Homework #2
  3. Homework #3
  4. Homework #4
  5. Homework #5
  6. Homework #6

Solutions to Final Exams

  1. Final Exam #1
  2. Final Exam #2
  3. Final Exam #3

Youtube videos for Course Topics

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 \((3,2,1)\).

Divergence and curl | Fluid flow with complex functions, part 1

Professor Doyle