# Teaching

### 2013/2014 Semester 2

• Lecturer for 2D Topics in Linear Algebra and Calculus (section 2A2)
• Lecturer for 3H Mechanics of Rigid and Deformable Bodies (weeks 26-32)
Office Hours: Mondays 2-3, Tuesdays 2-3, Wednesdays 2-3, and by appointment.

### 2013/2014 Semester 1

• Lecturer for 2A Multivariable Calculus (sections 2A1, 2A3)
• Tutor for 2M Financial Modelling
Office Hours: Mondays 1-2, Tuesdays 1-2, Wednesdays 1-2, and by appointment.

### Links for Maths 2A Multivariable Calculus

• An example of a function f(x,y) for which fxy ≠ fyx. Can you think of any others?

• What are partial derivatives good for? One application is solving optimisation problems. You'll learn about this in 2D. Another important application is partial differential equations (PDEs). Pretty much everything in the world around us can be modelled by PDEs; at the heart of every theory in physics lies a PDE: Newtonian mechanics, quantum mechanics, general relativity, electromagnetism, thermodynamics, fluid mechanics, .... But PDEs are not just for applied mathematicians. In 2003 the Poincaré conjecture, a famous 100-year-old open problem in pure mathematics, was solved using PDEs!

• You learnt about how to solve some 1st order PDEs by using a change of variables, but we didn't tell you how to choose the change of variables. This is done using the Method of Characteristics. If you really can't wait until the 3H course Mathematical Methods to find out about this, see Section 2.1.1. of these lecture notes. (Warning: this is hard stuff. The notation is also different; instead of using (u,v) for the change of variables they use (ξ,η).)

• Sometimes order matters: An example of a function where the order of integration makes a difference.

• In class we saw that Beta and Gamma functions can be used to derive a formula for the integral of powers of sines and cosines. Gamma functions are also important because they are related to the Riemann zeta function. And why should you care about the Riemann zeta function? Well, it could make you rich! There is a \$1,000,000 reward for solving the Riemann Hypothesis, which is about locating the roots of the Riemann zeta function. This is one of the seven (now six) \$1,000,000 Millennium Prize Problems. (The Poincaré conjecture was also one of these problems.)

• We haven't spent a lot of time in class during the first two years on the definition of the Riemann integral, rather we have been focusing on how to compute integrals. A rigorous treatment of the Riemann integral will be given in the 3rd year course Analysis of Differentiation and Integration. If you can't wait until the third year, then have a look at the textbook Understanding Analysis by S. Abbott, Chapter 7. (Note, however, that you haven't seen yet rigorous definitions of limit and continuity. These will be given next semester in 2E.)

• A very brief history of the integral: While calculus was invented by Newton and Leibniz at the end of the 1600s, it wasn't until the 1850s that there was a rigorous definition of the definite integral. This was introduced by Bernhard Riemann and we now call it the Riemann integral. Actually the story is more complicated than this. It turns out that there are a few technical problems with the definition of the Riemann integral and a better definition of integral is the Lebesgue integral. This was introduced by Henri Lebesgue around 1904 and is covered in our 4th year course Functional Analysis. But don't worry if you are not going to do Honours Mathematics: For continuous functions the Riemann integral and the Lebesgue integral are just the same thing, and a lot of you will never encounter the technical problems with the Riemann integral (these arise if you try taking limits of sequences of integrals).

• How to plot vector fields in Wolfram Alpha: For example, to plot the vector field f(x,y)=(y,-x), type "Plot vector field (y,-x)". PDF file of the result.

• Applications of vector fields: Brilliantly demonstrated in the short film by Vi Hart using green beans: The Green Bean Matherole.

• Differential operators: In class we studied the differential operators div, grad, curl and Laplacian. These are important because they arise in lots of PDEs, e.g., the heat equation, the wave equation, the Schrödinger equation and the Navier-Stokes equations. The Navier-Stokes equations model the motion of fluids such as water, oil and whisky and are widely used in science and engineering. And yet from the mathematical point of view they are not very well understood. Solving the Navier-Stokes equations by hand is out of the question (you have to solve them on a computer), and it is not even known whether they have a "nice" solution, which means the following: If the velocity of the fluid is smooth (= nice) at time 0, does there exist a smooth (nice) solution of the Navier-Stokes equations for all time? This is another one of the Millenium Prize problems and so if you can solve it you will win \$1 million (in this case, remember that I was the person to tell you about it)!

• How to remember Green's Theorem: You might find it easier to remember the right-hand side of Green's Theorem if you notice that ∂Q/∂x - ∂P/∂y = (curl F) · k, where F(x,y) = (P(x,y),Q(x,y),0). This way you can also see that Green's Theorem is just a special case of the Stokes Theorem.

• Another application of Line Integrals: Line integrals aren't only good for computing work done. They are also important in fluid dynamics. This circulation of a fluid around a closed curve is a line integral. Read about it in this handout. Circulation appears in classical aerofoil theory in the Kutta-Joukowski Theorem, which roughly says that the lift force on an aircraft wing is proportional to the circulation of air around it. The handout also shows how Green's Theorem can be used to prove a special case of the Kelvin Circulation Theorem: The circulation of fluid around any curve in a steady irrotational flow is zero.

• General definition of the surface integral: We only defined the surface integral for surfaces that are the graphs of functions, or surfaces that can be written as the union of graphs of functions. In general, the surface integral is defined using the notion of parametrised surfaces. See here.

• An application of the Divergence Theorem: The Divergence Theorem can be used to prove Archimedes' Principle! See here.

### Other interesting links for students

A great online tool for plotting surfaces (and doing many other things), no programming experience required: What you should be aiming to solve: