Covered so far
PART I Vector Calculus
Lecture 1 Introductary remarks, scalar fields, the Riemann integral for fxns and for scalar fields.
Lecture 2 Iterated integrals, an example.
Lecture 3 Iterated integrals, another example. Change of coordinates, Jacobians, polar coordinates example.
Lecture 4 More on Jacobians, three-dimensional integration.
Lecture 5 Cylindrical and spherical polar coordinates.
Lecture 6 Tutorial for PS1. A three-d cylindrical integration example.
Lecture 7 Vector fields, gradient, directional derivatives.
Lecture 8 Divergence, continuity equation, start of curl.
Lecture 9 PS2 tutorial.
Lecture 10 Halloween bank holiday.
Lecture 11 More curl, rotating field example. Vector identities.
Lecture 12 More vector identies, start of line integrals.
Lecture 13 PS3 tutorial q1, q2 and q5. Hyperbolic functions.
Lecture 14 PS3 tutorial q3 and q4, calculating line integrals.
Lecture 15 PS4 tutorial.
Lecture 16 Conservative and path independent vector fields. Conservative implies path independent. Integrating conservative fields around a closed loop.
Lecture 17 Path independent implies conservative, simply connected spaces. Finish early for 't Hooft.
Lecture 18 PS5 tutorial, q5b not covered.
Lecture 19 Simply connected spaces, surface integration.
Lecture 20 More surface integration, the parameteric form.
Lecture 21 PS6 tutorial, q4 not covered.
Lecture 22 Stokes's Theorem and Green's Theorem. Start of proof of Green's Theorem.
Lecture 23 Proof of Green's Theorem, start of proof of Stokes's Theorem.
Lecture 24 Proof of Stokes's Theorem.
Lecture 25 Recap end of Stokes's Theorem, PS6 q4: the vortex field question.
Lecture 26 Stokes's Theorem and curlF=0 => conservative for simply connected. PS7 Tutorial.
Lecture 27 Christmas Quiz.
Lecture 28 Gauss Theorem, examples.
Lecture 29 Line and surface integrals of scalars.
Lecture 30 PS8 Tutorial
PART II Fourier series and transforms
Lecture 31 Start of Fourier series.
Lecture 32 Fourier series, with example.
Lecture 33 PS9 Tutorial
Lecture 34 (Sergey Cherkis) More on the example: conditions for existence, complex.
Lecture 35 (Sergey Cherkis) Complex series, Parceval's theorem.
Lecture 36 (Jessica Stanley) PS9/10 Tutorial
Lecture 37 Countable, uncountable, Fourier integrals.
Lecture 38 More on Fourier integrals, an example.
Lecture 39 Finish PS10. Dual vector spaces and distributions.
Lecture 40 Distributions.
Lecture 41 Distributions and \delta(h(x)).
Lecture 42 PS11.
Lecture 43 Distributions and Fourier integrals.
Lecture 44 Parceval's Theorem. ODEs, first order and integrating factors.
PART III Ordinary Differential Equations
Lecture 44 Parceval's Theorem. ODEs, first order and integrating factors.
Lecture 45 ODEs: first order again, second order with constant coefficients.
Lecture 46 PS12 q1-3.
Lecture 47 More second order with constant coefficients.
Lecture 48 PS12 q4, PS13 q1.
Lecture 49 PS13 q2. Finish homogeneous constant coefficients, start constant coefficients homogeneous.
Lecture 50 Finish inhomogeneous constant coefficients. Quick guide to general inhomogeneous and Green's function.
Lecture 51 PS14 tutorial.
Lecture 52 The equations of maths physics. Euler's equation. Start of series solutions.
Lecture 53 More series solutions.
Lecture 54 PS15 tutorial.
Lecture 55 The method of Froebenius.
Lecture 56 Bessel's equation.
Lecture 57 PS16 tutorial.
Lecture 58 Fuchs theorem. Hermitian matrices.
Lecture 59 Hermetian matrices, properties. Hermetian operators, the Laplace operator is Hermetian.
Lecture 60 PS17 tutorial. q1,2 and 3(a)
Lecture 61 PS17 Q3(b)/(c) The Laplace operator and Fourier transforms.
Lecture 62 PS17 Q4. The Legendre equation as an operator equation and Hermiticity
Lecture 63 PS18 tutorial. q1,2 and 3
Lecture 64 PS18 q4. Legendre polynomials and the Legendre equation as an operator eqn. A list of PDEs.
PART IV Partial Differential Equations
Lecture 64 PS18 q4. Legendre polynomials and the Legendre equation as an operator eqn. A list of PDEs.
Lecture 65 The heat equation. Different boundary conditions. Uniqueness of solns to the Laplace Eqn.