PX275 - 0 - mathematical methods

*image: sleepy neno *

term 1

A - differentiation

PX275 - A1 - partial differentiation
PX275 - A1a - total differentiation
PX275 - A2 - exact and inexact differentials
PX275 - A2a - inexact differentials
PX275 - A2b - testing for exact differentials
PX275 - A2c- maxwell's thermodynamic relation
PX275 - A3a - the chain rule
PX275 - A3b - example
PX275 - A3c - change of variables and coordinate system
PX275 - A3e - chain rule additional
PX275 - A4a - directional derivatives
PX275 - A4c - the laplacian
PX275 - A5 - lagrange multipliers
PX275 - A5b - definition using gradient

B - coordinate systems and integration

B1 - coordinate systems

PX275 - B1a - cylindrical polar coordinates
PX275 - B1b - cylindrical polar basis vectors
PX275 - B1c - cylindrical polar representation
PX275 - B1d - time derivatives
PX275 - B1e - spherical polar coordinates
PX275 - B1f - spherical basis vectors
PX275 - B1g - spherical polar representation

B2-4 - integration

PX275 - B2 - multiple integration
PX275 - B2a - 1D integrals in 2D and 3D
PX275 - B2b - simple examples of line integrals
PX275 - B3 - jacobian
PX275 - B3a - hyperbolic coordinates
PX275 - B4a - surface of revolution
PX275 - B4b - example - a cone
PX275 - B4c - example - a torus
PX275 - B4d - volumes of revolution
PX275 - B4e - pappus's first theorem

C - vector calculus

PX275 - C1a - gradient or gra
PX275 - C1b - divergence or div
PX275 - C1c - interpretation of div
PX275 - C1d - curl
PX275 - C1e - examples of curl
PX275 - C1f - example
PX275 - C1g - interpretation of curl
PX275 - C2a - grad squared
PX275 - C2b - examples of using the laplacian

D - vector integration

PX275 - D1 - vector displacements
PX275 - D2a - line integrals with fields
PX275 - D2b - conservative vector fields

green's theorem

PX275 - D3a - green's theorem in the plane
PX275 - D3b - proof of green's theorem
PX275 - D3c - application of green's theorem
PX275 - D3d - area of an ellipse
PX275 - D3e - the divergence theorem in 2D
PX275 - D3f - physical significance of the divergence theorem
PX275 - D3g - the physical significance of green's theorem in the plane

vector surface integrals

PX275 - D4a - flux of a vector field through a loop
PX275 - D4b - the vector surface area and element
PX275 - D4c - total vector surface area
PX275 - D4d - comments

E - stoke's theorem and the divergence theorem

PX275 - E1 - stoke's theorem
PX275 - E1b - examples
PX275 - E2 - divergence theorem
PX275 - E2b - dirac delta
PX275 - E2c - example
PX275 - E2d - scalar fields

F - tensors and summation conventions

PX275 - F1 - tensors
PX275 - F2 - summation conventions
PX275 - F3 - levi-civita symbol

image credits: most images in these notes are taken from Prof. Steve Dixon's typed notes for the module

term 2

G - partial differential equations

PX275 - G1 - functions of a single variable
PX275 - G2 - functions of a multiple variables
PX275 - G3a - diffusion equation
PX275 - G3b - heat equation
PX275 - G4 - wave equation
PX275 - G5 - method of separation of variables
PX275 - G6 - boundary and initial conditions
PX275 - G7 - dirac delta function
PX275 - G8 - complex exponential form of solution in 3D
PX275 - G9 - orthogonality relations
PX275 - G10 - plane waves
PX275 - G11a - schrödinger equation
PX275 -G11b - other examples

H - fourier series and transforms

PX275 - H1 - complex fourier series
PX275 - H2 - calculating coefficients
PX275 - H3 - fourier transforms
PX275 - H4 - example of fourier transform
PX275 - H5 - parseval's theorem
PX275 - H6 - fourier transforms in the time domain
PX275 - H7a - convolutions
PX275 - H7b - examples of convolutions
PX275 - H8 - multidimensional convolutions
PX275 - H9 - FT and differential equations
PX275 - H10 - FT and PDEs
PX275 - H11 - FT in data analysis

I - optics

PX275 - I0 - introduction to waves
PX275 - I1 - diffraction and interference
PX275 - I2 - huygen's principle and point sources
PX275 - I3 - point sources in 3D
PX275 - I4 - key geometry
PX275 - I5 - different path lengths
PX275 - I6a - square aperture
PX275 - I6b - rectangular aperture
PX275 - I6c - multiple slits
PX275 - I6d - circular aperture
PX275 - I7 - rayleigh criterion for resolution
PX275 - I8 - optical elements in instruments