Engineering Mathematics: YouTube Workbook

How to use this workbook

This workbook is designed to be used in conjunction with the author's free online video tutorials. Inside this workbook each chapter is divided into learning modules (subsections), each having its own dedicated video tutorial.

View the online video via the hyperlink located at the top of the page of each learning module, with workbook and paper / tablet at the ready. Or click on the Engineering Mathematics YouTube Workbook playlist where all the videos for the workbook are located in chronological order:

Engineering Mathematics YouTube Workbook Playlist

While watching each video, fill in the spaces provided after each example in the workbook and annotate to the associated text.

You can also access the above via the author's YouTube channel Dr Chris Tisdell's YouTube Channel

The delivery method for each learning module in the workbook is as follows:

1. Briefly motivate the topic under consideration;
2. Carefully discuss a concrete example;
3. Mention how the ideas generalize;
4. Provide a few exercises (with answers) for the reader to try.

Incorporating YouTube as an educational tool means enhanced eLearning benefits, for example, the student can easily control the delivery of learning by pausing, rewinding (or fast-forwarding) the video as needed.

The subject material is based on the author's lectures to engineering students at UNSW, Sydney. The style is informal. It is anticipated that most readers will use this workbook as a revision tool and have their own set of problems to solve -- this is one reason why the number of exercises herein are limited.

Two semesters of calculus is an essential prerequisite for anyone using this workbook.

• How to use this workbook
• About the author
• Acknowledgments

1. Partial derivatives & applications
   1. Partial derivatives & partial differential equations
   2. Partial derivatives & chain rule
   3. Taylor polynomial approximations: two variables
   4. Error estimation
   5. Differentiate under integral signs: Leibniz rule
2. Some max/min problems for multivariable functions
   1. How to determine & classify critical points
   2. More on determining & classifying critical points
   3. The method of Lagrange multipliers
   4. Another example on Lagrange multipliers
   5. More on Lagrange multipliers: 2 constraints
3. A glimpse at vector calculus
   1. Vector functions of one variable
   2. The gradient field of a function
   3. The divergence of a vector field
   4. The curl of a vector field
   5. Introduction to line integrals
   6. More on line integrals
   7. Fundamental theorem of line integrals
   8. Flux in the plane + line integrals
4. Double integrals and applications
   1. How to integrate over rectangles
   2. Double integrals over general regions
   3. How to reverse the order of integration
   4. How to determine area of 2D shapes
   5. Double integrals in polar co-ordinates
   6. More on integration & polar co-ordinates
5. Ordinary differential equations
   1. Separable differential equations
   2. Linear, first–order differential equations
   3. Homogeneous, first–order ODEs
   4. 2nd–order linear ordinary differential equations
   5. Nonhomogeneous differential equations
   6. Variation of constants / parameters
6. Laplace transforms and applications
   1. Introduction to the Laplace transform
   2. Laplace transforms + the first shifting theorem
   3. Laplace transforms + the 2nd shifting theorem
   4. Laplace transforms + differential equations
7. Fourier series
   1. Introduction to Fourier series
   2. Odd + even functions + Fourier series
   3. More on Fourier series
   4. Applications of Fourier series to ODEs
8. PDEs & separation of variables
   1. Deriving the heat equation
   2. Heat equation & separation of variables
   3. Heat equation & Fourier series
   4. Wave equation and Fourier series
9. Bibliography