Engineering Math II

This is one of the most exciting Math courses that PSUT students will ever see. In this mixed course, you will learn concepts related to vector calculus, where big theorems like Green's, Stokes' and Gauss will be discussed. Then, independently, you will learn Fourier analysis, where beauty becomes clear upon seeing the mystery behind the concept. Although the course is well known to students as a partial course, partial differential equations will be covered only in the last chapter! You will see how Fourier analysis plays a major role in understanding these equations.

Syllabus

Text Book: Advanced Engineering Mathematics, by Kreyzig, 9th edition.
Course Content:

9.1 Vectors in 2-space and 3-space.

9.2 Inner (dot) product.

9.3 Vector (cross) product.

9.4 Vector and Scalar Fields .

9.5 Curves. Arc Length. Torsion.

9.6 Calculus Review.

9.7 Gradient of a Scalar Field.

9.8 Divergence of a Vector Field.

9.9 Curl of a Vector Field.

10.1 Line Integrals.

10.2 Path Independence of Line Integral.

10.3 Calculus Review. Double Integrals.

10.4 Green’s Theorem in the Plane.

10.5 Surfaces for Surface Integrals.

10.6 Surface Integrals.

10.7 Triple Integrals, Divergence Theorem for Gauss.

10.9 Stokes’s Theorem.

11.1 Fourier Series.

11.2 Functions of Any Period.

11.3 Even and Odd Functions.

11.7 Fourier Integral.

11.8 Fourier Cosine and Sine Transforms.

11.9 Fourier Transform.

11.10 Tables of Transforms.

12.1 Basic Concepts.

12.2 Modelling: Vibrating String, Wave Equation.

12.3 Separation of Variables.

12.5 Heat Equation.

12.6 Heat Equation: Solution by Fourier Integrals.

12.9 Lapacian in Polar Coordinates.

12.10 Laplace’s Equation in Cylindrical and Spherical Coordinates.

12.11 Solution by Laplace Transform.

Class Videos

Chapter 1: Vector Calculus

The first 26 minutes are about how the formula of the curve integral was obtained as a work integral. This is extremely useful in understanding how things work.

The first numerical example: From 26:20-34:25, The second numerical example: 34:30-40:15

Introduction to path independence: 41:00-43:54, An Important example about the introduction to path independence 44:00-56:55

Path Independence I:

The first 50 minutes explain the idea and related issues about path independence. This is so important to comprehend the concept.

From minute 50:00 to the end, the video discusses how to tell if the integral is path independent, using the gradient idea.

Path Independence II

The first 3 minutes revise the previous video.

3:00-22:24 A detailed example on using the gradient idea to solve path independent integrals.

22:30-33 A remark about how we test quickly for path independence, when we deal with R^2.

38:35-41 An example about the use of the formula f(B)-f(A).

Path Independence III

00:00-02:30 A quick test for path independence in R^3.

03:00-14:00 The use of Curl to test for path independence, with an example.

14:00-End A detailed example on using the Curl and the gradient in path independence.

Part II: An example on the circle

Part III: An example on the triangle

Part IV: An example on a non closed curve

Part V: Finding the area using line integrals

Surface Parametrization: Part I: Basic concepts, and an example on the disk

Part II : Parametrization of the cylinder

Part III: Parametrization of the cone

Part IV: Parametrization of the sphere, and spherical coordinates

Normal Vectors- Surface Integrals
Surface Integrals: Part I: Basic ideas and an example on the plane

Part II: An example on the cylinder

Part III: An example on the sphere

Part IV: An example on the cone

Part V: A sphere example, as an introduction to the Divergence theorem

Divergence Theorem in further details: Part I: Introduction and an example on the unit sphere

Part II: An example on the cylinder

Part III: An example on the upper hemi sphere; the surface is not closed

Part IV: Some integrals computations

Part V: An example on the cone

Part VI: On the meaning of the divergence. The infinitesimal flux

Stokes' Theorem in details: Part I: Introduction and an example on the disk

Part II: Further comments, and an example on the upper hemisphere

Part III: An example on the open cylinder

Part IV: On the relation between Stokes theorem and Green's theorem

Chapter 2: Fourier Analysis

Fourier series: Part I: Introduction to Fourier series through Taylor series

Part II: Fourier series- 1-coefficients formula

Part III: Fourier series- 2- Some examples

Minutes 00:00-26:00: The derivation of the Fourier coefficients

Minutes 26-36:38: A summary and an example

Minutes 36:40-45:53: An example. Finding the Fourier series of the function f(x)=x.

Minutes 45:55-end: The geometric meaning of the Fourier series.

Part IV: Fourier series-3- Rectangular wave

Part V: Fourier series-4 Functions of any period

Part VI: Fourier series-5 From real to complex Fourier series

Part VII: Fourier series-6 Basic examples complex Fourier series

Part VIII: Fourier series-7 Parseval’s identity

Part IX: Fourier series-8 Linear algebra and Fourier series

Part X: Fourier series-9 Error in Fourier series approximation

Fourier transform: Part I: Fourier transform-1 Definition More examples on Fourier transforms

Part II: Inversion formula

Part III: Fourier cosine and sine transforms

Part IV: Convolution 1

Part V: Convolution 2

Part VI: Convolution 3

Part VII: Fourier integrals More examples on Fourier integrals

Partial differential equations: Introduction and a basic example on separation of variables

The wave equation The heat equation

00:00 17:43 minutes explain the geometric meaning of the wave equation.

17:44-21:54 minutes explain how we deal with the boundary condition.

21:55-39:02 minutes explain in details how we got the function F.

39:03-46:30 minutes explain how we find G and then how to write u.

46:31-51:07 minutes explain how we satisfy the boundary conditions, and how the Fourier series come in!

51:08-end minutes give a summary with an explicit example.

Exercises

Past Exams

YouTube Channel

This is my youtube channel. I started uploading videos in this channel starting Spring 2020. In this semester, I taught only Eng Math II, which justifies the fact that my videos are only about this course until the moment of writing this paragraph!

https://www.youtube.com/user/sababheh1

Spring 2019-2020

Matlab alternative:

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..

Office hours:

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.

Quiz policy:

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.

Class Notes:

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.

Announcements:

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.

Project Policy:

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.

First Project:

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.

Second Project:

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?!

Third Project:

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.

Fourth 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!

Fifth Project:

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!

Sixth Project:

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??

Summer 2019-2020

Course Syllabus:

Click here to download.

Class Notes:

Recorded classes:

Projects

In this course, you are asked to do one project. Our projects cover variety of topics. Here are they. The list will be updated when we cover more topics.

Use the form: https://forms.gle/toSuocukDZzpK7vZ9 to pick your project.

Use the form: https://forms.gle/UF35wKfTwryUMAn5A to submit your project.

How does the project work?

One week before the end of the semester, you will need to submit a report about the project.
The report must include: a cover page, including students' names, course, project title, university logo, instructor name,...
An abstract; a paragraph describing in general what you are doing.
An introduction that describes in more details what you are doing and why it is important.
Sections that explains your contribution, answering all listed questions and more.
The project is not limited to the listed questions below. These are guidelines, that you can use as basics. Then it is left to you what you want to add.

Grading policy

Submitted reports will be checked against plagiarism. Any such action will result in a "zero".
In grading the projects, we have three items: 1- The oral presentation and how you present. (4 marks) 2- The slides you prepare fro your oral presentation. (3 marks) 3- The report. (4 marks)

First project

Title: Curve parametrization

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). Elaborate on this and give as much effort to make this easily understood, mathematically and graphically. Animated graphics is much appreciated here.

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, with some examples.

10- The derivative of a curve parameterization has a certain geometric/physical meaning. Explain.

Second Project

Title: Surface parameterizations

1- Explain the reason we need two parameters to define a surface parameterization.

2- Our understanding for a curve parameterization involved a moving particle with position r(t) at time t. Explain a geometric/physical meaning of a surface parameterization.

3- We had some formulae from Calculus III about finding a normal vector of a surface. List these formulae and relate them to how we find normal vectors when dealing with surface parameterizations.

4- Surface parameterizations are usually used to evaluate surface integrals. Explain the physical meaning of the surface integrals. Recall that curve integrals present the work done by a force. What is the meaning of a surface integral?

5- Explain the differences between different parameterizations of a surface.

6- Make an animation that facilitates visualizing surface parameterizations.

Third Project:

Title: Curl and Divergence

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 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- It is well known that Curl(Div F)=0. Show the truth of this identity and explain a possible physical meaning of it.

7- 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?!

Fourth Project:

Title: Fourier series

1- What is the Fourier series?

2- Explain the relation between the function and its Fourier series.

3- How do communications engineers use Fourier series? Why is it so important for them?

4- In your discussion of Fourier series, what are the frequencies? How is that somehow related to signals?

5- Explain how the Fourier series can be used to obtain certain infinite sums.

6- Why do we use Fourier series to approximate function while we have Macaurin series; a much easier form?

7- In dealing with Fourier series, we encounter two forms: Real and Complex forms. Explain the relation between the two forms and discuss the usage and easiness of each one.

8- Fourier series are used to deal with periodic functions. Explain how they can be used with non-periodic functions.

Spring 2022-2023

Course Syllabus:

Click here to download.

These class notes must be studied together with the ppt slides we discuss in class. Those written comments are just complementary notes that aim to get the students' attention to certain stuff that I, as an instructor, see important.

Class Notes: Here you find our comments

Previous notes

Recorded classes:

You can check the Spring 19-20 page for this course. There, you find all classes recorded.

Fall 2023-2024

Course syllabus

Thursdays' Video classes- Blended

Thursday 12-10-2023: Curve Parametrization
Thursday 19-10-2023: Line integral and work
Thursday 26-10-2023: Path Independence I
Thursday 26-10-2023: The curl
Thursday 2-11-2023: Green's theorem in details
Thursday 9-11-2023: Please check the 4 clips under the [Class Videos Tab- Surface Parametrization] in the course page.
Thursday 16-11-2023: Please check the clip entitled (Surface Integrals) in the [Class Videos Tab]
Thursday 23-11-2023: Please check the 6 clips under (Divergence theorem) in the [Class Videos Tab]. These are separated examples in details.
Thursday 30-11-2023: Our Midterm exam.
Thursday 7-12-2023: These clips are extremely important. They give more examples on what we did in class about the Fourier series. They also discuss the idea of finding certain sums using the fourier series. For this, you MUST watch the following clips, which are also posted in the [class videos tab].

Fourier series-2 : All is important, but pay more attention to the discussion after Minute: 16.

Fourier series-3: All is important, but pay more attention to the discussion after Minute: 12.

Thursday 21-12-2023: Please make sure to understand the video entitled (Part X: Fourier series-9 Error in Fourier series approximation ), in the Class Videos tab.
Thursday 4-1-2024: Watch the video Convolution 1. Start at Minute 8. By L1, we mean an absolutely integrable function.
Thursday 11-1-2024: Watch the video Convolution 2 from the class video tab.

Some Class notes- Supplementary

Course Projects

Project guidelines

1- Group work of up to 4 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.

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.

The projects

**** Detailed Guidelines about the project.

Spring 2023-2024

The blended classes videos

Thursday 22-02-2024: Please watch the video entitled Curve Integrals from the "Class Videos" tab. To make it easier for you, follow these instructions:
Thursday 29-02-2024: Please watch the video entitled Path Independence III. Detailed moments are mentioned in the "Class Videos" tab.
Thursday 07-03-2024: You must watch the Green's theorem videos from the "Class Videos" tab above. Here, you have 5 separated videos with less than 45 minutes in total. These are enumerated as (Part I, II, III, IV, V). Also, if you press the title "Green's theorem", you will have another video of 55 minutes. The 5 separated parts give a precise and clear idea about Green's theore. If you have free time, then it is good to watch the other long video.
Thursday 14-03-2024: Parametrization of the sphere, and spherical coordinates. The summary of the video starts at minutes: 25.
Thursday 28-03-2024: There are 4 separate videos on Stokes' theorem in the "Class Videos" tab. Make sure you watch them as a summary of Stokes' theorem.
Thursday 18-04-2024: Part VIII: Fourier series-7 Parseval’s identity
Thursday 25-04-2024: Part X: Fourier series-9 Error in Fourier series approximation

Fall 2024-2025

Course outline

The blended classes videos

Thursday 10-10-2023: Curve Parametrization and Line integral and work

For the first week, you have two videos to watch. The first video is a revision from Calculus III, where curve parametrization is explained. This is assumed to be known, but this video will make it easier for you. The other important video is basically about: How is work related to what we are doing in this course?