Structural Geology
Lecture 2
The mathematics of structural geology
(vectors, tensors, Einstein summation)

Scalar - A physical quantity expressed by a single number;  a tensor of zero rank.
Gradient of a Scalar  -  The direction at which a quantity increases the fastest.

	(2-1)

Vector - A physical quantity expressed by a three numbers; a tensor of rank one. Coordinate system for a vector:  RIGHT HANDED - looking up the x3 axis from the origin in positive direction, the x1 axis will rotate into the x2 axis in 90° in a clockwise direction. The components of a vector, , are:

	(2-2)

 

The mathematical tool necessary to model Newton's laws of motion is the vector.  Newton invented calculus to manipulate vectors.  Needless to say, quantitative structural geology is based on the use of vector calculus.  Remember that a vector is a quantity having both magnitude and direction whereas a scalar has just magnitude.   Within this text a vector is represented by variables printed in bold letters (e.g. a).  A knowledge of direction implies that a coordinate system is known.  For structural geology a rectangular or Cartesian coordinate system  represented by the direction of three positive unit vectors i,j, and k will do (e.g. i = 1, etc) (Fig. 2-1).  Any vector can then be represented with scalar  components f1, f2, and f3  which represent distances along the axes denoted by the three unit vectors.  Every vector in space many be written as a linear combination of unit vectors .   The vector f is then

	f  =  f1i + f2 j + f3k	(2-3)
                                                
where f1, f2, and f3 are components of the vector f.  These three components are independent of the choice of the origin point of the coordinate system.  Vectors such as af, bf can be added  by summing the individual components so that the resultant vector F is:

	F  =  af + bf = (af1 + bf1 )i + (af2 + bf2)j + (af3 +  bf3)k

Magnitude of a vector is

	(2-4)

	(2-5)

Vectors are commutative:

	(2-6) 

Vectors obey the associative law:

	(2-7)

Vector times a scalar is a larger vector.

Force f is a vector commonly used in explaining the causes of geological structures.   Force is defined as the cause of an acceleration.  Newton's second law  states that if f is the net force acting on an object of mass m moving with a velocity v, then force (f) equals mass (m) times acceleration (a).  This fundamental equation of classical mechanics is an example of a vector multiplied by a scalar to derive another vector:

	f =   m(dv/dt)  =  ma.	

Two common operations on a vector include the dot and cross products, respectively  (Fig 2-2).   The dot product of two vectors is the projection of vector A on vector B:

	A*B  =  AB cos q
                                                         
where q is the angle between the two vectors.  An example of the dot product is the work done by a force.   Consider a rock on which a constant force f acts.  Let the rock be given a displacement d. Then the work W done  during the displacement of the rock is defined as the product of |d| and the component of |f| in the direction of d, that is, 

	W  =  |f||d|cos a  =  f * d.
                                            
where a is the angle between f and d.

Scalar product or Dot product of a vector - magnitude of one vector times the magnitude of projection of 2nd vector on the first vector.  This number of a scalar.

	(2-8)

	(2-9)

Repeated suffixes means summation on i.    i = 1to 3.

If the vectors are orthogonal then

	(2-10) 

The cross product of two vectors is the magnitude of A times B times the sine of the angle between then.  The direction (u) is perpendicular to the plane of A and B.

	A x B =   AB sin q u   
                                                  
where q is the angle between A and B.   According to plate tectonics theory, as continental plates move over the mantle a moment of force is generated.   The moment of force is also known as a torque t which is a vector quantity (Figure 2-3). 

 In mechanics the magnitude of the moment t of a force f about a point Q is defined as the product t = |f|.d where d is the perpendicular distance between Q and the line of action L of f.  If r is the vector from Q to any point A on L, then d = |r|sin g and

	t  =  |r||p|.sin g  =  |r  X  p|. 
                  
Vector product or Cross product  -  The magnitude of H is the area of the parallelogram whose sides are F and G.

	(2-11)

The cross product is not associative
	
	(2-12)

With each square matrix we associate a number called its determinant. If the matrix is A, then we demote its determinant by det A.

A   =      det A   = 	(2-13)

det A =   	(2-14)

where

, etc.
	(2-15)

Einstein Summation illustrated using heat flow:

Heat flow, q (cal cm-2 sec-1), in homogeneous, isotropic rock is indicated by a thermal gradient, .

	(2-16)

where k is the thermal conductivity.

For Granite: k = 8 x 10-3  cal cm-1 sec-1 °C-1.

For Steel: k = 180 x 10-3  cal cm-1 sec-1 °C-1.

Heat flow is parallel to a thermal gradient so it can be expresses in a vector form where for an isotropic homogeneous rock.

	(2-17a)

	(2-17b)

	(2-17c)

If the rock is anisotropic then the thermal conductivity is not the same in each of the three directions.  If , then

	(2-18a)

	(2-18b)

	(2-18c)

where each q is linearly related to a.  If we consider heat flow through a cube, then the first subscript relates to the direction of the heat flow and the second subscript relates to the plane on which the heat flow operates.  Remember that the ‘1” plane is defined by the 2- and 3- axes of the coordinae system defining the planes

If the temperature gradient is parallel to the x1 axis, then

	(2-19)

but heat flow still occurs in three directions according to

	(2-20a)

	(2-20b)

	(2-20c)

Because of anisotropy of the rock, there is heat flow in all three directions of the rock even though the gradient was applied parallel to the x1 axis.

kij is a tensor of second rank represented by the matrix

	(2-21)

For a temperature gradient which is not parallel to a coordinate axis in an anisotropic material:

               for i = 1, 3	(2-22)

If a dummy suffix is repeated with respect to any one term, then that term assumes summation over the term.  A free suffix must occur once on each side of the equation.   

The expression for hear flow in an anisotropic material may be written in its full form as 

	(2-23a)

	(2-23b)

	(2-23c)

Tensor transformation --  The physical quantity, such as thermal conductivity, kij, is the same regardless of the set of axis chosen.   This is the chief characteristic of a tensor.  Transformation from one coordinate system to another depends on the direction cosine relating the axes of each of the coordinate systems.     The transformation process will be discussed in detail during a later lecture.

Structural geology requires an understanding of vector differentiation, the first aspect of vector calculus.  Vector differentiation is best illustrated by considering the description of a curve in space.  If a particular vector P is the position vector p(t) joining the origin O of a coordinate system and any point (x,y,z,), then the vector function u(t) defines x,y, and z as functions of t

	p(t)  =   x(t)i + y(t)j  +  z(t)k

As t changes, the terminal point of u describes a space curve having parametric equations: 

	 x = x(t), y  =  y(t),  z  =  z (t).     
                                       
Then  

	 Dp/Dt = {[p(t + Dt)] - p(t)}/Dt    
                                   
 is a vector in the direction of Dp (see next figure).

Then the limit 

	dp/dt as Dt › 0                                                              

will be a vector  in the direction of the tangent to the space curve at (x,y,z) and is given by

	dp            dx           dy          dz
	__     =     __ i  +    __ j   +   __ k
	dt             dt           dt           dt                                        

The derivative of the vector of position p with respect to time t is velocity v.
  
Let u(t) be the position vector of a moving block of rock P in the earth where t is the time.  The u(t) represents the path C of P.  We know that the vector

	v  =  dp(t)/dt                                                                        

is tangent to C and, therefore points in the instantaneous direction of motion of P.  The derivative of the velocity vector is called the acceleration vector a; thus

	a(t)  =  dv(t)/dt  =  d2p(t)/d2t                           

Acceleration is the second derivative of position with time.  Newton's second law includes a derivative of the velocity vector  v with respect to time dv/dt.   Acceleration a is velocity changing with time.  Second derivatives are must useful in structural geology.
Another useful operation of calculus is the so-called chain rule of differentiation.  If w is a differentiable function of x and x is a differentiable function of t, then 

	dw/dt  =  dw/dx.dx/dt.  

This chain rule can be generalized for a function of two variables such as w = f(x,y) and x  =  x(t) and y  =  y(t)  then

	dw/dt  =  ∂w/∂x.dx/dt  +  ∂w/∂y.dy/dt.                         

 The operation ∂/∂x is the well known partial derative of a function.  If x and y are functions of position within a coordinate system and t is a function of time, then w varies with position and time.  If in contrast to the previous example  x  =  x(p,v) and y  =  y(p,v)  then

	∂w/∂p  =  ∂w/∂x.∂x/∂p  +  ∂w/∂y.∂y/∂p  
and
 
	∂w/∂v  =  ∂w/∂x.∂x/∂v  +  ∂w/∂y.∂y/∂v.

Partial differential equations are time consuming to write so a convention called Einstein summation is used.  To illustrate Einstein summation lets consider a property of a rock that relates two vectors.   A position within a rock x is related to a displacement u by displacement gradients E.  Position and displacment are both vectors with components x1, x2, x3 and u1, u2, u3, respectively.  x and  are related in the following way:

	u1  =  E11x1 +  E12x2  +  E13x3
 
	u2  =  E21x1  +  E22x2  +  E23x3

	u3  =  E31x1  +  E32x2  +  E33x3                                         

Rather than writing out all of these equations we can shorten then in the following manner:

	           3
	u1  =  S E1jxj
	         j = 1

	          3
	u2  =  S E2jxj
	         j = 1
	         
	            3
	u3  =  S E3jxj
	         j = 1                                                                               

or more compactly, as

	          3
	ui  =  S Eijxj
	        i = 1                                                                             

We now leave out the summation sign:

	ui  =  Eijxj                                                                     

and introduce the Einstein summation convention: When a letter suffix occurs twice in the same term, summation with respect to that suffix is to be automaticaly understood.  In the above example the suffixes can be any letters.  i is the free suffix and j is the dummy suffix.  By this we mean that j can take any variable

	ui  =  Eijxj  =   Eikxk.