Mathematical Methods Lectures 1-3 Dr Mark Naylor (pretending to be Prof Wyn Williams)

Organisation ~10 Lectures : Wks 3 & (Oct 5 & 26; Nov 2, 9, 16) Monday 2-3 pm Wks 3 & 6 - 9; (Oct 8 &29, Nov 5, 12 & 19) Thursday 3-4 pm ~6 Problem classes: Wks 3, 4, 6, 7, 8 & 9 (Oct 8, 15 & 29, Nov 5, 12 & 19) Thursday 4-5 pm Lectures based on material in notes Practical classes are to read through notes upto point reached in lectures and complete problem questions Demonstrators on hand to help and go through solutions in class Assessment: By problem set out of class

Recommended books Turcotte and Schubert, Geodynamics R Snieder, A Guided Tour of Mathematical Methods: For the Physical Sciences L Lyons, All you wanted to know about mathematics but were afraid to ask Vol 1&2

Ground rules As with any math problem: –Make it easy for us to give you marks –Making a sketch demonstrates your understanding –Showing working demonstrates your ability –Answers only will not receive full marks –Copying will be penalised Mobile phones off

Scalar field

Scalar fields – Magnitudes Temperature Pressure Gravity anomaly Resistivity Elevation Maximum wind speed (without directional info) Energy Potential Density Time…

Vector fields

Vector fields – Magnitude and direction Magnetic field (Scale Earth or mineral) Electric field Water velocity field Wind direction on a weather map Includes displacement, velocity, acceleration, force, momentum…

Vector fields coloured by a scalar field

Coordinate systems SphericalCartesian

Coordinates systems Cylindrical (3D)Polar (2D)

Why change coordinate system? More physical representation of the problem Can make the maths easier Reduce computation time

Appropriate coordinate systems Flying a plane

Appropriate coordinate systems Injector well with 100m completion interval

Earthquakes Appropriate coordinate systems

Volcanoes Fissure eruptions Appropriate coordinate systems

Planetary magnetic field

Magnetic field direction

Declination

Inclination

Measuring Earth’s magnetic field Total field

Application – directional drilling

Application: Magnetisation of rocks (Paleo-magnetism) For more info see:

Dot product example work done = a.b = |a||b| cos t = magnitude of the force x distance moved in the direction of the force magnitude of the force distance moved

Position on the Earth’s surface

Great circles

Great circles, small circles and orthodromes

Great circles – Finding the angle angle a b

Cross product

Great circles – Finding the pole a b P

Spherical triangles a b P Angle between 2 planes which intersect each other and the Earths surface

On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180. The surface of a sphere can be represented by a collection of two dimensional maps. Therefore it is a two dimensional manifold.

The angle between 2 planes is the angle between their poles Dot product a b P

Euler’s Theorem Any line on the surface of a sphere can be moved to any other position and orientation on the sphere by a single rotation about a suitably chosen axis passing through the centre of the sphere: The Euler Pole

Plate rotation and GPS

Pole of a plate motion