Wing Fai Lee

I have been teaching in the Higher Education sector for decades. My lecture and tutorial notes are thorough, coherent, and well organized, there are lot of coursework assignments and exercises complied with my lectures and tutorials I had prepared for my students, all these teaching materials could be a valuable learning support to you as well.
Some of the mathematics courses I taught before are outlined and sorted out by topics as follows:
Probability & Statistics (Theories and applications)
Linear Algebra (Theories and applications)
Calculus (Theories and its applications)
Multi-variable Functions, Advanced Calculus, and its related topics such as Multiple Integrals, Vector Calculus/ Analysis
Ordinary and Partial Differential Equations and its related topics such as Mathematical modelling, Complex Analysis, Conformal Mapping, Numerical Analysis (such as Runge-Kutta Method)
Practical examples:
In the course Linear Algebra, for the topic of Eigenvalues and Eigenvectors, I had introduced an application to solving the system of ordinary differential equations, which provided students with another technique obtained in linear algebra to solve differential equations, which could be solved by some specific methods taught in other course. That’s good for students to have a crossover between different topics in different mathematics courses.
For the advanced calculus course, there is quite a common issue I would like to share:
most of the year one students understand the partial derivatives of multivariable functions, because the concepts are quite straight forward, just directly extending the concept of derivative of single variable functions to that of multivariable functions, nevertheless, it’s a bit challenging that some of the students could not understand the concept of the total differential, they could not ’image’ how (it’s rather abstract compared to partial derivatives) and could not conceptually link up the partial derivative to the total differential, so I tried to give them a graphical illustration to visualize the mathematical meaning of the total differential, while the illustration cannot be attached here, I have reserved it for your reference upon request. In that figure, for dz (the so-called total differential of z, a function of two variables), I tried to enlarge it, link up the partial derivatives and the dz such that the differential contributed by the variable x (while the variable y is kept constant), and the differential contributed by the variable y (while the variable x is kept constant), so we have
The total differential = the differential contributed by x + the differential contributed by y
This concept could be simply and directly extended to the functions of three or more variables (despite we could not show it graphically as the function of two variables as above), those students got a clear understanding of the total differential for multivariable functions.
Furthermore, to let those students have a bit more ‘feeling’ about it, I gave them a practical example for the application of the total differential as below:
A cylindrical steel storage tanks is to be built that is 10m tall and 4m across in diameter. It is known that the steel will expand or contract with temperature changes; is the overall volume of the tank more sensitive to changes in the diameter or in the height of the tank? Answer given to my students: A 3-D cylinder with height h and radius r has volume V = pr2h.
We can view V as a function of two variables, h and r. We can compute partial derivatives of V and formulate the dV (the total differential of V) such that we could figure out the volume of the tank, after substituting h = 10 and r = 4, is more sensitive to changes in radius r than in height h. After attending my extra tutorial, those students just well understood the concept of the total differential, and most important outcome is that they learned the practical application as well as the concept of the total differential of multivariable functions.