The Exponential and Logarithmic Functions of a real variable.
Starting from the existence of real numbers, and Dedekind's theorem concerning sections of real numbers, state, without proof, the chain of theorems leading to the proposition that any continuous function is integrable. Establish the principal properties of integrals. Show that if \(f(x)=0\) for irrational values of \(x\), and \(f(x)=1/q\) when \(x\) is a rational \(p/q\), where \(p/q\) is in its lowest terms, then \(f(x)\) has in any finite interval a Riemann integral, whose value is zero.
Curvature.
Green's Theorem and its applications to Electrostatics.
The potentials, charges, and energy of a system of conductors.
Lines and tubes of electrostatic force, and equipotential surfaces.
The parabolic motion of a particle under gravity.
The conservation of momentum and energy; illustrate your account by considering the direct impact of spheres.
The refraction of light, with applications to prisms and simple lenses.