Mathematical Methods
Lectures 1-3
Dr Mark Naylor
(pretending to be Prof Wyn Williams)
Organisation
• ~10 Lectures :
Wks 3 & 6 - 9 (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
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Temperature
Pressure
Gravity anomaly
Resistivity
Elevation
Maximum wind speed (without directional info)
Energy
Potential
Density
Time…
Vector fields
Vector fields – Magnitude and direction
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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
Spherical
Cartesian
Coordinates systems
Polar (2D)
Cylindrical (3D)
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
Appropriate coordinate systems
• Earthquakes
Appropriate coordinate systems
• Volcanoes
• Fissure eruptions
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: http://gravmag.ou.edu/mag_rock/mag_rock.html
Dot product example
magnitude of the force
distance moved
work done = a.b
= |a||b| cos t
= magnitude of the force x distance moved in the direction of the force
Position on the Earth’s surface
Great circles
Great circles , small
circles and orthodromes
http://mathworld.wolfram.com/GreatCircle.html
Great circles – Finding the angle
a
b
angle
Cross product
Great circles – Finding the pole
a
b
P
Spherical triangles
b
a
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
b
a
P
Dot product
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