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Covered so far
1.1 Vector calculus
Lecture 1: Introduction. Vector calculus, the scalar field. Inegration, the Riemann integral recalled, the Riemann integral for scalar fields over two-dimensions.
Lecture 2: The Riemann integral over two dimensions, the iterated integral. Examples, including centre of mass. Motivating changes of variable, polar coordinates.
Lecture 3: More on changes of variable, the area element for polar coordinates, the example again, Jacobians.
Lecture 4/5: The situation in three dimensions, spherical and cylindrical polars. A sphere with a cylindrical core removed. Scalar fields and vector fields, the gradient of a scalar field, the directional derivative, the divergence and the curl.
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Lecture 6: Problem sheet 1 tutorial class. More discussion of the Riemann integral
Lecture 7/8: More about the curl, the vector identities. Begining of the line integral. An example.
Lecture 9: Problem sheet 2 tutorial class.
1.2 Multivariate integration
Lecture 10: The line integral, parameterization independence, the example again, conservative fields and path independence.
Lecture 11: More on path independence and conservative fields (note 1). Simply connected domains; conservative fields and irrotational fields.
Lecture 12: Problem sheet 3 tutorial class.
Lecture 13: Surface integrals, definitions.
Lecture 14: Surface integrals, examples.
Lecture 15: Problem sheet 4 tutorial class.
Lecture 16: Stokes' theorem. Green's theorem, proof of Green's theorem.
Lecture 17: More on the proof of Green's theorem, the good part of the proof of Stokes' theorem.
Lecture 18: Problem sheet 5 tutorial class.
Lecture 19: More tutorial class. The rest of the proof of Stokes' theorem.
Lecture 20: Gauss theorem, introduction and proof.
Lecture 21: Problem sheet 6 tutorial class.
Lecture 22: More tutorial class. Gauss theorem, discussion of proof and applications. The vector potential
Lecture 23: More on the vector potential, proof of existence. The Hodge decomposition.
Lecture 24: Official class survey. Organizational discussion. Start of problem sheet 7 tutorial class.
Lecture 25: Finishing up the problem sheets.
Lecture 26: Free lecture
Lecture 27: Christmas Quiz
2 Fourier series and Fourier transforms
Lecture 28: Introduction to the Fourier series.
Lecture 29: Examples and calculations, the complex series
Lecture 30: Problem Sheet 9 tutorial class.
Lecture 31: Introduction to the Fourier integral.
Lecture 32: Examples of Fourier integrals.
Lecture 33: Problem Sheet 10 tutorial class.
Lecture 34: Finishing up sheet 10. Plancherel's formula. Comparing Fourier series and Fourier integrals
Lecture 35: The dirac delta function.
Lecture 36: Problem Sheet 11 tutorial class.
Lecture 37: Properties of the delta function.
Lecture 38: The delta function and Fourier series, delta functions in higher dimensions.
Lecture 39: Problem Sheet 12 tutorial class.
3 Ordinary differential equations
Lecture 40: Multidimensional delta functions, first order ODEs.
Lecture 41: Second order ODEs, the forced damped harmonic oscillator.
Lecture 42: Problem Sheet 13 tutorial class.
Lecture 43: The damped hamonic oscillator, solving it.
Lecture 44: Now solving it with a forcing term.
Lecture 45: Class discussion of teaching programme.