1. A little math, a little physics
Geophysics: the study of Earth, its condition and workings, from the perspective of and through the use of mathematical physics.
We shall require a little math and a little physics for our course.

2. Measurements and Units
In order to compare our physical analyses with others, we require a common language of measurement.
Since 1976, in Canada, we have been using the SI (Système Internationale) units for comparisons.
The basic units we use in describing measurements are: metres (m), kilograms (kg), seconds (s) and amperes (A). (Almost) all other units can be reduced to ratios of these.
What are these reference quantities?

4. The metre (m)
Originally, the metre (m) was defined as 1/ of the distance from the equator to the north pole along the Paris meridian.
For convenience, that metre was standardized as the distance (equivalent) between two inscribed lines on a platinum bar held in Paris.
Now the metre has been further standardized as the distance travelled by a beam of light in a vacuum during 1/ seconds... but what is a second?

5. The second (s)
The second was defined as 1/86400 of one rotation of the Earth relative to the Sun (i.e. From noon to the next noon).
The standard rotation period of the Earth was defined as the average “solar day” during the period But the Earth's rotation is slowing and so doesn't offer us a high-accuracy standard.
Now the second is defined as periods of oscillation determined by the transition between the two hyperfine levels of the 133Cs atom at 0K and at rest.

6. The kilogram (kg)
The kilogram was originally defined as the mass equivalent of 1 litre (1000 cm3) of pure H20 at 4C under normal pressure. A mass equivalent, the International Prototype Kilogram was formed with this mass determined as accurately as possible as an Ir-Pt cylinder which is stored under triple vacuums at Sèvres, France.
Several secondary standards are still stored in various laboratories around the world.

7. The kilogram (kg) II
The IPK is known to be evaporating and some of the secondary standards are known to be absorbing mass. This is obviously not a good standard for high-accuracy physical measurement.
The International Committee for Weights and Measures (CIPM) finally proposes to standardize the kilogram and will do so during its 2018 General Conference.

8. The kilogram (kg) III
The standardized metre (m) and second (s) have already been defined in terms of accessible natural phenomena. These are thought to be similarly accessible to every physicist wherever she might be in the Universe and so become universal standards.
Probably, CIPM will define the kilogram in terms of Planck's constant, h, which itself becomes a “defined constant”, equivalently and exactly:
h = × J · s .
1 kg = 1034/ s2 · m-2 h

9. The ampere (A)
1 A is currently defined operationally as that “constant current which will produce an attractive force of 2 × 10−7kg · m · s−2 (newtons) per metre of length between two straight, parallel conductors of infinite length and negligible circular cross section placed one metre apart in a vacuum”
The 2018 CIPM proposal defines the charge on one electron as e = × 10−19 A · s exactly and hence the derived definition of the ampere.

10. Other secondary units
Other units in common use by physicists and chemists can be constructed of these basic units or determined, simply, by count.
The definitions of the kelvin (K), the mole (mol) and the candela (cd) will be reviewed and more precisely defined in 2018.

11. Geographical coordinates on Earth
The common geographical coordinate system was defined to accord with a north pole penetrating the Earth’s surface shell at a point which represented the mean axis of rotation of the Earth during the interval between January 0, 1900 and December 32, 1905.
The physical shell of the Earth has moved since this period but we keep to the inscribed coordinate system... we don't want latitudes and longitudes of places on Earth to change over time (in spite of geophysical and geological processes).

12. Where is the rotation axis now? Where is the coordinate system now?
The axis of rotation of the Earth is displaced, now (September 2017) by about 14.5 m from the north geographical pole along the W64.5o meridian (that is, towards New Brunswick)... or better, the geographical coordinate system inscribed on the Earth has slipped 14.5 m from the rotation axis northward along this meridian.
Montreal’s position relative to the rotation axis is now about 14 m closer (north toward rotation axis) than it was when the geographical coordinate frame was fixed.

13. longitudes?
Also, relative to the rotation axis and the geographical 0-meridian established in the window, the effective geographical 0-meridian has slipped about about 0.42 seconds of arc to the west. Montreal has slipped about 10 m westward delaying sundial noon-time by about 0.03 s relative to the fixing of coordinate time. The Earth is a changeable and dynamic body! Geophysics!

14. Tectonics?
We recognize as well that as the shell (crust, lithosphere) of the Earth is in constant vertical and lateral adjustment, meaning that places on the Earth are locally moving through the geographical coordinate system as well.
Moreover, and while we have not considered plate motions, Montreal and the North American tectonic plate is moving roughly westward at about 2 m per century relative to Greenwich (London). Geology!

15. Now for some math! Tensors and their special cases
Beyond their units of measure, one might recognize that all physical quantities are described in terms of tensors. Most physical quantities that we commonly measure, though, are determined in the form of special cases of tensors, namely scalars and vectors.
Before dwelling on the character of tensors, we should understand another construction of mathematical and physical convenience: a coordinate system.

16. Coordinate systems
We usually employ the simplest coordinate system that can locate us and the objects and phenomena that we are studying most easily. On the surface of the near-spherical Earth, we usually locate ourselves by latitude and longitude referenced to that geographical coordinate system established in 1905.
Earth is not perfectly spherical – it has highlands and lowlands and is flattened by about 1 part in 300 along the rotation axis. Earth has a real geometry, a shape.

17. Geometry
Earth's apparent geometry does not perfectly conform to a spherical polar coordinate system. Geometry is “real”; the coordinate system we use is a “tool” for location.
Earth also exists within a underlying geometry that is affected by mass and motion... General Relativity or as it is now known “Gravitation Theory”. See: Pythagoras and more.
We won't delve into properly relativistic geophysics in this course but I will allude to Gravitation Theory from time to time as it might interest you.

18. Euclidean Geometry, Cartesian Coordinates and Pythagoras
How far is it from here to there? The displacement from here to there is described by a vector quantity composed of a distance and a direction. If we choose to originate a Cartesian right here and if the geometry in which we are measuring is Euclidian, we then cast out an orthogonal x−y−z coordinate system from “here” to determine the vector place of “there” as simply the particular x, y and z placements in our coordinate system.

19. location coordinates
Note that our coordinate system is chosen to be scaled in distance units – say metres. If we, “here” sit at the coordinate origin (x = Xh = 0, y = Yh = 0, z = Zh = 0), then “there” is located at x = Xt , y = Yt and z = Zt.
The subscripts h and t describe “here” and “there”. The vector place of “there” is (Xt , Yt , Zt) and of here (0, 0, 0) as I have located us here at the coordinate origin.

20. lt = (Xt2 + Yt2 + Zt2)1/2 . Pythagoras
If our space is Euclidian (sometimes called “flat”), a wonderful theorem due to Pythagoras gives us the answer. The distance, say lt , from here to there is simply obtained as
lt2= Xt2 + Yt2 + Zt2 or
lt = (Xt2 + Yt2 + Zt2)1/2 .

21. Pythagoras and metric tensor
We can recast this Pythagorean formula into a rather nice linear-algebraic form:
Our linear-algebraic form is more than just vector- matrix-vector multiplication; it is a tensor-algebraic multiplication. The “metric tensor”, here shown as the identity matrix, “operates” mathematically, just like an ordinary matrix. It is though a real tensor in that it describes and measures a real physical quantity: the local geometry of our space.

22. Tensor rank
The metric tensor is a 2-rank (not to be confused with matrix rank) tensor. That is, each element is determined by 2 coordinates. Tensors can be of any rank. The vectors, whether row or column, have elements that are determined by only 1 coordinate.
Vectors are 1-rank tensors.
In continuum mechanics, elasticity theory and seismology, we encounter an important 4-rank tensor, the Elastic Coefficients tensor that describes the simplest of elasticity conditions for linear, homogeneous materials. Rocks are not!

23. Higher rank tensor quantities
Displacement is a vector quantity in physics; distance is scalar quantity. A scalar is a 0-rank tensor. Displacement is distance plus some vector direction. Speed is scalar; velocity is vector.
Stress and strain are 2-rank tensors. Relating stress to strain requires, properly, a multiplication via a 4- rank tensor, the Elastic Coefficients Tensor. It is hard to conceive of the equivalent 3 × 3 × 3 × 3 matrix-like description.
We may use indicial notation for convenience or dyadic notation.

24. Example: Stress-strain relationship
Component or indicial notation:
where pij is the 9-coefficient stress tensor, elm is the strain tensor and Ciljm is the 81-coefficient Elastic Coefficients tensor.

25. Example: Stress-strain relationship II
Dyadic (symbolic) notation:
This notation is analogous to common vector-matrix notation of linear algebra.
The number of overlines indicates the tensor rank.

26. Pythagoras – dyadic notation
Note that, properly, all components of a vector described in Cartesian coordinates or of a tensor carry the same physical units. This is not true for other choices of coordinate systems.
We can use dyadic notation for the here-to-there distance above that was described in vector-matrix tensor form:
We traditionally use an over-arrow for vectors.

27. An interest in Pythagoras and Gravitation Theory?
If we have time and if you are interested later in the course, we might look into a simple entrance to Gravitation Theory (that is, General Relativity) through Pythagoras Theorem. I link a PDF noteset and a short PowerPoint presentation introducing Gravitational Waves. Pythagoras and more Gravitational Waves

28. Tensor-vector calculus; Div, Grad, Curl and all that...
For the purposes of this course, we shall live in a 3- space which is characterized by an independent time measure. That is, we shall not describe any of our geophysics in terms of the 4-space-time structure that Einstein introduced with his Special Relativity and which he later generalized as Gravitation Theory through the inclusion of the space-time distortions determined by mass. For most geophysics, we can ignore space-time dilations due to speed and mass. We shall keep time independent of space in the Newtonian way.

29. Necessity of PDEs
Many of the problems that we deal with involve variations in 3-space and time. This leads us to descriptions in terms of partial differential equations. I am less interested in your ability to manipulate mathematics than I am in your understanding of what the mathematical formalisms that we shall be using mean! Don’t sweat the elaborate math but do try to understand what the math means.

30. Gradient
The slope of the simple line, 𝑓(𝑥)=𝑚𝑥+𝑏, when generalized to the slope of a function in two variables, 𝑓 𝑥, 𝑦 =𝑚𝑥+𝑛𝑦+𝑏 is described locally through the gradient operator: The gradient has vector components and direction.

31. Divergence
For a vector field in 3-space, we determine the field’s local divergence as The field’s local divergence is a scalar measure and may be greater or less than zero.

32. A simple model…
But, Gauss showed that, That is, there are no sources or sinks of the magnetic field: there are no magnetic monopoles.

33. Curl
Vector fields may twist and turn… consider the vector field of particles in a Tornado or of particles swirling down a drain. Locally, we can measure their “rotational” components: The curl of vector field yields a vector measure of local vorticity.

34. Curl and Divergence
If one determines the divergence and curl of the vector field of elastic motions in a solid material, we separate the motions into two distinct wave types: The divergence extracts the scalar dilatational P- wave component, that part of the motion that is vorticity free. The curl extracts the vector rotational S-wave component that is dilatation free.

35. What of the divergence of the curl of a vector field, the curl of the divergence of a vector field or the curl of the gradient of some scalar field or field of vector components? Usually (though special conditions must hold for some of these):

36. Laplacian
The divergence of that vector quantity that we obtain as the gradient of some scalar or vector component. We might obtain the gradient of the divergence of some vector quantity. Again, though and with some restrictions: We might also obtain the curl of the curl of some vector field. Again, surprisingly:

37. Protypical PDEs of Geophysics
Laplace’s Equation: Poisson’s Equation: Diffusion Equation: Wave Equation: Diffusive-Wave Equation:

38. Potential Fields
In physics, we have learned to construct forms called potential functions which, upon some operation, describe measurables. For example, we can construct a gravitation potential due to a point mass, M, as We cannot directly measure this potential, U, but we can measure its gradient:

39. And again,
I am much less interested in your ability to manipulate these “tools” than I am in your understanding of their use and what you can do with them. Please read the noteset that accompanies this lecture.