Arick Shao 邵崇哲

Below are the courses that I have taught at the undergraduate level.

## MTH5113: Introduction to Differential Geometry

This is a second-year undergraduate module which provides an introduction to the differential geometry of curves and surfaces. The main focus will be on connecting geometric questions with ideas from calculus and linear algebra, and on using these connections to gain a better understanding of all three subjects.

Approximate list of topics covered:

1. Revisions: vector calculus with a geometric perspective
2. The geometry of curves
3. The geometry of surfaces
4. Applications: Lagrange multipliers, vector integral (Green's, Stokes', divergence) theorems

## MTH5109: Geometry II: Knots and Surfaces

This is a second-year undergraduate course on the differential geometry of curves and surfaces, along with a brief excursion to knot theory.

Approximate (combined) list of topics covered:

1. Knots, knot diagrams, and the Reidemeister theorem
2. Knot invariants: crossing number, chirality, tricolorability, Jones polynomial
3. Curves and their parametrizations
4. Curves: tangent lines, orientation
5. Curves: arc length, path/line integrals
6. Curves: curvature
7. Plane curves: signed curvature, winding number
8. Space curves: torsion, Frenet–Serret formulas
9. Surfaces and their parametrizations
10. Surfaces: tangent planes, first fundamental forms, unit normals
11. Surfaces: orientability and orientation
12. Surfaces: surface area, surface integrals
13. Surfaces: first fundamental form, second fundamental forms
14. Surfaces: principal, mean, and Gauss curvatures
15. Curves on surfaces: normal and geodesic curvatures, Euler's theorem
16. Geodesics: covariant derivatives, geodesic equations
17. Some landmark results (distance-minimizing curves, theorema egregium, Gauss–Bonnet theorem)

## MAT336: Elements of Analysis

• Position: Instructor, Course coordinator
• Location: University of Toronto
• Semester: Spring 2014
• Textbook: Real Analysis and Applications: Theory in Practice, by Kenneth. R. Davidson and Allan P. Donsig

This is a introductory course in real analysis. The course covers the basic properties of real numbers, as well as a rigorous development of various calculus concepts. Throughout, we also discuss some generalizations of these concepts to other useful settings. The course is geared toward students without experience in fully rigorous mathematics courses.

Approximate list of topics covered, by week:

1. Motivation (why real numbers are necessary), construction of real numbers (Dedekind cuts), defining properties of real numbers (least upper bound principle)
2. Cardinality, limits (real numbers), monotone sequences
3. Limits (metric spaces), Cauchy sequences, completeness
4. Series, convergence tests, absolute and conditional convergence
5. Discrete dynamical systems, fixed points, contraction mapping theorem
6. Fractals (iterated function systems)
7. Topology of metric spaces, limit and interior points, open and closed sets, continuity
8. Compactness and connectedness, Heine-Borel theorem, extreme and intermediate value theorems
9. Differentiation (1-d), mean value theorem, Taylor's theorem
10. Inverse function theorem, implicit function theorem
11. Riemann integration, fundamental theorem of calculus
12. Pointwise and uniform convergence, spaces of continuous functions, existence and uniqueness for ordinary differential equations

Some exam questions and solutions from the course:

## MAT244: Ordinary Differential Equations

• Position: Instructor
• Location: University of Toronto
• Semester: Spring 2014
• Textbook: Elementary Differential Equations and Boundary Value Problems (10th ed.), by William E. Boyce and Richard C. DiPrima

This is an introductory course on ordinary differential equations.

Approximate list of topics covered, by week:

1. First-order equations: separation of variables, linear equations
2. More first-order equations: exact equations, general integrating factors
3. Existence and uniqueness theory: linear equations and systems, nonlinear equations
4. Second-order homogeneous linear equations: equations with constant coefficients
5. Second-order nonhomogeneous linear equations: method of undetermined coefficients, variation of parameters
6. Higher-order linear equations: equations with constant coefficients, methods of undetermined coefficients and variations of parameters
7. First-order linear systems: basic theory, existence and uniqueness, connections with linear algebra
8. Homogeneous first-order linear systems: equations with constant coefficients, eigenvalue analysis
9. More homogeneous first-order linear systems: qualitative analysis, phase diagrams, stability
10. Nonhomogeneous linear systems: methods of diagonalization, undetermined coefficients, and variation of parameters
11. Nonlinear first-order systems: autonomous systems, local linearizations, and stability analysis
12. Series solutions of linear equations: power series, solutions near ordinary points

## MAT334: Complex Variables

• Position: Instructor, Course coordinator
• Location: University of Toronto
• Semesters: Spring 2013, Fall 2012
• Textbook: Complex Variables (2nd ed.), by S. D. Fisher

This is an introductory course on the theory of functions of one complex variable.

Topics (approximate):

1. Basics of complex numbers
2. Functions and limits on the complex plane
3. Line/contour integration
4. Holomorphic functions, the Cauchy-Riemann equations
5. Power series
6. Cauchy's theorem, Cauchy's formula
7. Analyticity, Morera's theorem, Liouville theorem
8. Zeroes, isolated singularities, and Laurent series
9. Residue theorem, and applications
10. Argument principle, Rouche's theorem, and the maximum modulus property
11. Linear fractional transformations
12. Conformal transformations, Riemann mapping theorem

I am slowly compiling my pile of notes from this course into a coherent set of lecture notes. The most recent version of this effort is below:

Here are some exam questions and solutions from previous versions of the course.

## MAT235: Calculus for Life Sciences II

• Position: Instructor
• Location: University of Toronto
• Semesters: Fall 2011, Spring 2012
• Textbook: Calculus (7th ed.), by J. Stewart

This is a year-long course on both differential and integral multivariable calculus.

Fall semester topics:

1. Parametrizations and parametric curves
2. Polar coordinates and curves
3. Polar curves (continued), conic sections
4. Vectors, dot products, cross products
5. Lines and planes in 3-D space
6. (More) calculus of parametric curves
7. Multivariable functions, multivariable limits
8. Partial and directional derivatives
9. Differentiability and differentials, linear approximations of functions
10. Multivariable chain rules, gradients, calculus on level sets
11. Extrema of functions, first and second derivative tests
12. Lagrange multipliers

Spring semester topics:

1. Double integrals, Fubini's theorem
2. Double integrals in polar coordinates
3. Applications of integrals, surface area of graphs
4. Triple integrals, cylindrical coordinates
5. Spherical coordinates, change of variables formula
6. Vector fields, line integrals
7. Line integrals and conservative fields
8. Green's theorem
9. Divergence and curl, parametrized surfaces
10. Surface area, surface integrals
11. Stokes' theorem
12. Divergence theorem