Many of you do not have access to Matlab from home. Alternatively, you may use Octave; very similar to Matlab and it is an open source! Here it is:
https://www.gnu.org/software/octave/
Also, you can use Python or any other language..
We will have an office hour every Tuesday at 5 pm. You can ask your questions or discuss any thing related. It will be held via Zoom.
We will have one quiz every Monday starting 13-4-2020. The quiz material will be the previous week material. For 13-4-2020, the material will be the Fourier series topics. The quiz will be held at 6 pm on e-learning platform.
Starting Monday 27th of April, our quizzes will be held at 10 pm as we have agreed in class. This change is due to Ramadan timings.
For our quiz on Monday 11-5-2020, the material will be "Separation of variables". This of course includes the wave equation, the heat equation and any related problem. Make sure to understand the solution, nit to memorize it. You need to understand why we had three cases for k, and when to reject and when to accept. These are the important things you need to know about separation of variables.
You can view our class notes by clicking the link. Please notice that these notes are those I write while I record my videos. So, to fully understand the scratches, the arrows, the shading and all these stuff in the pdf file, you are encouraged to watch the videos, from the youtube channel.
Second Quiz will be held on Monday 13-4-2020 Time: 6-7 pm on e-learning. Material included in the quiz: Periodic functions, Fourier series (real and complex), Parseval's identity, Error formula in Fourier series approximations.
To Schedule your project discussion, please use the form:
https://forms.gle/YF9w5uAY41XNFkWm7
All instructions are there!
Please submit your project via the link:
https://forms.gle/qiijuzCQHpeCJurB8
Please use the link to fill your project information. This is required to work on the project.
Here is the link: https://forms.gle/RcMBR3qKNm8g5iAf8
What do you need to submit for the project, and when?
The dedline for submitting your project work is Thursday 21st of May, 2020.
You need to submit:
1- A document that summarizes your project, like a word or a pdf file. You are expected to discuss the main ideas of your project and to answer all question in this document.
2- A powerpoint presentation.
3- The code
4- Any material you feel will help.
Eng. Math II is a Math course, full of new ideas and computations. However, a good portion of this course can be visualized in a way that simplifies our understanding for the topic, and hence, implies a better comprehension of the course.
Therefore, in this course, I will announce multiple ideas as "projects", where students are asked to present the idea in a different way: Poster, slides, movie, etc. These projects will be evaluated based on their novelty, accuracy and "beauty".
The project will be worth 10-20% of the final grade of the course. Therefore, students are asked to take this seriously.
Here are the guidelines that you should follow:
1- Group work of up to 3 students is allowed. In fact, having a group work is better than individuals; it is one goal of the projects to work together.
2- The questions that I pose on each project are not the only thing you can do. These are kind of the main ideas. However, you will have your other good ideas which you can strengthen your project by.
3- When you prepare your project, make sure to answer the questions that are asked clearly and in a good way that makes it easy for the reader to follow.
4- You will need to submit a report and a presentation for your project. The report is a word (pdf) file that addresses all questions; as if you are writing a "book" about the project. The presentation should be prepared using power point tool, for example, in a way that summarizes your project. In presentations, make sure to have minimal writing. Graphics and animations are way better in presentations than text.
5- After finishing your report and presentation and submitting them, your report and presentation will be graded, and you will be given partial credit.
6- The final grade will depend on your presentation.
7- The presentation will be face-to-face in university or on zoom, depending on our return to the campus.
8- In the presentation, every member of the group should present his contribution and defend it.
9- I will invite people from outside the class to attend your presentations and to participate in grading the projects.
10- You need to remember some stuff: We really trust you all. I will not ask anyone if he/she has copied from someone else, or if someone else has done the project for you. PSUT students are trustworthy, and I will not question your loyalty and honesty.
These projects may take some effort and time fro you, but it will help you better understand the topic and will qualify you for future courses. In engineering, project based courses are trending these days.
11- Of course, you will have to work on one project only.
12- Projects must be finished at least 2 weeks before the end of the semester so that we have sufficient time to grade and present them.
13- These are general guidelines that you will need to follow. I will add further guidelines if needed later.
We have learned curve parametrization in Calculus III already. However, in this course, we will need this important topic in a different way than that in Calculus III. Therefore, we spent at least one class reviewing this term and trying to understand the logic behind it. In the end of our class on Sunday 16-2-2020, I announced that you can have your first project on this idea. Although I would like to see you "invent" your ideas about this project, I will tell you a summary of what I expect from this project.
1- We spent some time explaining the geometric meaning of a curve parametrization, not to forget the physical meaning. Can you summarize these ideas in a visual way?
2- Can you summarize, in a simple way, the difference between different parametrizations of the same curve?
3- Can you explain the main differences between a curve parametrization and an (x,y) equation of the curve?
4- You, what do you have in mind about "making curve parametrization easily comprehended"!
5- Related to curves and their parameterizations, we see in the literature the terms "simple, closed, smooth, piecewise smooth" curves. Explain these terms with graphical interpretation.
6- Related to this, we see in the literature the terms "simply connected and multiply connected regions". Explain these terms with graphical interpretations.
7- As you may have already seen, graphical interpretations of curve parameterizations include our understanding of a moving particle, where the position at time t is given by r(t). Write a Matlab code that sketches a given parametric equation, and make this graph animated to show that user how the particle moves over the curve as the parameter changes.
8- Parametric equations can be used to find the length of a curve. Explain the formula, and give an idea of why this formula is valid, then implement it in your Matlab code; so that the code itself is able to find the length of the curve.
9- Parametric equations are not particularly designed for curves in the plane. The idea works in any dimension. Elaborate on curves and their parameterizations in the space R^3.
We agreed that we will have a project for this course. Each student is to participate in one project. Maximum of 3 students can have a joint project, and the expectation of 3 students is not as 1.
Our second project is about the Curl and the divergence and their applications in this course, and others!
1- We defined the curl in class and gave its formula. What is the geometric or the physical meaning of the curl?
2- We said in class that if the curl of F is zero, then F is conservative. What is the geometry or physics behind this?
3- Make a video, a presentation or any kind of "animation" that better explains the curl.
4- The curl is usually understood by imagining a vector field, representing a fluid velocity for example. Make graphical interpretation of the curl using this direction.
5- We defined the divergence of a vector field in this course. Explain the geometric meaning of the divergence. What does it mean when the divergence is zero!
6- Make sure you have a Matlab code that is able to show some graphics showing a possible meaning for the curl and divergence, given a vector field.
7- It is well known that Curl(Div F)=0. Show the truth of this identity and explain a possible physical meaning of it.
8- One big application of the curl and the divergence is their applications in surface integrals. Explain these applications and try to have physical interpretations.
9- People usually look at Green's theorem as being Stoke's theorem in R^2. explain the relation, and make some graphics to explain the relation. What does this have to do with the above mentioned stuff?!
We started Fourier series and we did some examples and graphical interpretations. This project is about "Visualization of Fourier series".
So, this project should be as follows:
1- of course, an introduction to what the Fourier series.
2- Given a function, write a Matlab or other code to compute the first "n" terms of the Fourier series. These computations could be numerical approximations.
3- This code must be able to plot the function and the partial sums of its Fourier series.
4- This code must be able to plot the function and successive partial sums to show the user how the partial sums approach the function.
5- How do communication engineers use Fourier series? Why is it so important for them?
6- In your discussion of Fourier series, what are the frequencies? How is that somehow related to signals?
7- Using the same code you implemented earlier, what is the power of the signal?
8- Can your code substitute certain values for x to obtain some infinite sums using Dirichlet-Dini result??
What else can you do!
These are the guidlines for a project on Fourier series. Any other contribution is welcomed! It is your chance to present your thought here in the project.
This project is about the "Fourier transform".
1- Begin by defining the Fourier transform. Give examples of functions that do not have Fourier transform, and explain why not. At the same time, give examples of functions that do have Fourier transform, and fund these transforms.
2- When we did Fourier series in class, we explained thoroughly the geometric meaning of the Fourier series; whether the complex or real series. However, when we started Fourier transform, we did not talk about the geometric meaning. Explain the geometric meaning, if there is any, of the Fourier transform.
3- If Fourier series is used to approximate periodic phenomenon, what is the purpose of the Fourier transform?
4- We did in details how "we obtained the Fourier series formulas". Explain how the Fourier transform definition is obtained starting from the Fourier series.
5- When the function is even or odd, the Fourier transform gives new stuff. Explain.
6- Explain fully and carefully the importance of Fourier transform in communications engineering. What is the difference between it and the Fourier series?
7- Write a code (on Matlab or anything else) that finds the Fourier transform of a function and plots it. Also, this code must be able to show the user the geometric relation between the function and the Fourier transform.
8- Remember: graphs, videos, animations and such ideas are the best tool to explain such concepts. Do your best and explain what you can using these tools!
For this project, coding is essetial. If you cannot do the coding, you will not qualify!
Here we go with the most interesting topic ever! This project is about "epicircles and Fourier series". A hidden beauty of Fourier series has not been seen in class, due to time limitations. Fourier series can be used somehow to draw any shape in a very strange way: Epicircles.
1- Visit the link: https://brettcvz.github.io/epicycles/ to better know the meaning of epicircles. As a term, epicircles mean that circles are moving over other circles, with the center of the next circle lying on the previous one.
2- Define, in your way, the meaning of "epicircles".
3- Describe fully and clearly the relation between "Fourier series" and "Epicircles". This is a very nice interpretation~
4- Write a Matlab code or Python (or anything that works like Mathematica) that does the following job: Given a function, let the program find the first n terms of the complex Fourier series, then let the program draw those n epicircles.
5- Given a shape (2D), something called SVG can be used to find coordinates of this shape, and hence can be looked as a function describing this shape. Look into this, and describe how this SVG works. Then explain how we can use Fourier series to draw any 2D shape. For example, a rabbit in 2D, your face in 2D, PSUT logo...
6- That is it!
This project is about the so called "Discrete Fourier transform". This topic has not been covered in class, due to time limitations, yet it is a very important topic for engineers, but I don't know why!
1- Define what is meant by Discrete Fourier transform "DFT".
2- Give few examples to show the computations involved with the DFT.
3- Having defined DFT, write a Matlab (or any other language) code to find the DFT of a given complex vector.
4- In understanding the DFT, the term "orthogonality" must have shown itself somehow. Explain this concept in this context, and show this orthogonality when needed.
5- How is DFT used in electrical engineering?
6- How does an engineer distinguish whether to use the Fourier transform or the DFT in his/her problem.
7- Write explicitly in full details two engineering problems where the DFT is used. Use it and solve this engineering problem using DFT.
8- What else can you do??