Transcript Math/Notation "Review": Vectors, Derivatives

Geology 5640/6640 16 Jan 2015
Introduction to Seismology
Last time: “Review” of Basic Principles; Vectors
• Reflections are wave energy that “bounce off” a surface
dividing media with different velocity properties, remaining
in the original medium
• Refractions are energy transmitted through the boundary
into the second medium
• Conversions from P to S particle motion (or vice versa)
almost always occur at these boundaries
• All of these obey Snell’s Law…
• Diffractions are predicted by Huygen’s Principle
• Vectors: We will use several different notations for
Cartesian coordinates interchangeably in this class….
Read for Wed 21 Jan: S&W 29-52 (§2.1-2.3)
© A.R. Lowry 2015
Seismo-Math & Notation
(“Introduction” to Vectors)
Vectors:
z
y
x
kˆ
x3
x2
x1

ˆj
iˆ
We’ll use these notations for Cartesian coordinate 
systems
interchangeably…
Vectors have direction and magnitude;
 here we may
denote as boldface u or with an overbar u or arrow u .
A hat-sign denotes a unit direction vector uˆ.
A vector v from point Q at coordinate (0, 0, 0) to point P at
coordinate (a, b, c) may be denoted by any of
z
a 
 
v  v  v  axˆ  byˆ  czˆ  b

c 

P (a,b,c)
v
Q (0,0,0)
c
a
y
b
x


More generally, v between Q at (X1,Y1,Z1) and P at (X2,Y2,Z2)
can be expressed as any of:
X2  X1 


v  v  v  X2  X1 xˆ  Y2  Y1 yˆ  Z 2  Z1 zˆ  Y2  Y1 

Z 2  Z1 

Length of a vector is the square-root of the summed-squared
elements, i.e.,

v v 
X2  X1  Y2  Y1  Z2  Z1
2
2
2
Addition and Subtraction of vectors is performed
element-wise. If:
a 
 
v  axˆ  byˆ  czˆ  b

c 

Then:

and
d 
 
u  dxˆ  eyˆ  fzˆ  e 

f 

d  a 


u - v  d  a xˆ  
e  byˆ  f  c zˆ  e  b 

f  c 

(You could do this graphically too).
A scalar
multiplied by a vector gives:

cd 
 
cu  cdxˆ  ceyˆ  cfzˆ  ce 

cf 

Properties of Vector Addition and Multiplication:
Commutative:
Associative:
uv vu
u  v  w u  v  w
Distributive:

cu  v  cu cv
Some Special Vectors:
 0
Null:
 
0 
0

0

Unit: has length 1:

vˆ 1 Note a unit vector in any
direction can be defined as:
v
vˆ 
v
Example Unit Vectors:
1
 
iˆ  xˆ  0

0

0
ˆj  yˆ  1
 

0

0
 
kˆ  zˆ  0

1

Thus we can write any vector in the form:

a 

 
v  b aiˆ  bˆj  ckˆ  iˆ

c 


 a 
ˆj kˆ b
 

c 


Part of this course will include an “introduction” to
linearalgebra…
This sort of representation of vector multiplication is
standard and will crop up a lot in this class!
Note that other notations are common…
This is figure A.3-1 from the appendices
of Stein & Wysession. eˆ1, eˆ2 and eˆ3 are
often used to denote three (arbitrarilyoriented) orthogonal directions, which
may or may not equal xˆ1, xˆ 2 and xˆ 3.
 

 

A normal nˆ is a unit vector that is
perpendicular to a plane or surface.
Recall that means nˆ 1!

Multiplying Vectors:
A dot product represents the length of one vector multiplied
by length of its projection onto another (i.e., a measure of
a1 
b1  how parallel
 
 
a

a
b

Given  2  and b2 , they are).

a 3 

a  b  a1

b3 

b1 
 
a 3 b2  a1b1  a2 b2  a 3 b3

b3 

a2


a b  a b cos
Hence, the dot product is a scalar value.
We could also write this as:

3
ab
a b  a b
i i
i1
i i
The latter, in which the repeated index implies the summation,
is called Indicial Notation (or Einstein summation
notation) & shows
 up a lot in seismology!
The dot product is easily shown to have associative,
commutative and distributive properties…
It is also useful for calculating the angle between two vectors:
 a  b
  cos 
 a b

1
We can also use it to calculate the
projection of some vector b onto
 another vector a:
the direction of
ab
ˆa  b 
a




The Cross Product of two vectors is defined as:
a1  b1  a 2 b3  a 3 b2 
    

a  b  a2  b2  a 3 b1  a1b3 

a3 
 
b3 
 
a1b2  a 2 b1 

This produces a new vector that is orthogonal (perpendicular)
to the plane spanned by a and b (with length = area of the

parallelogram
described by a and b).
We can also express this as the determinant of the matrix:
iˆ

a  b  a1
b
 1
ˆj
a2
b2
kˆ  (Here, the straight lines surrounding the

a 3  matrix denote the determinant).
b3 

derived from
2x2 matrix
determinants as
shown at left:
The determinant of a 3x3 matrix can be
The cross product c = ab obeys the right-hand rule:
And the length of c is defined by ||c|| = ||ab|| = ||a||||b||sin.
Hence, the cross-product can be thought of as a measure
of how orthogonal two vectors are.
The length ||ab|| also equals the
area of a parallelogram defined
by a and b:
||c||
The cross-product is:
Anti-commutative:
Distributive:
Associative:
ab = –ba
a(b + c) = ab + ac
(ab)c = a(bc)
“Introduction” to Derivatives:
Since calculus is a pre-req for the course, I assume you’ve
seen derivatives and integrals before, so I will only
review a few basics. (If you haven’t, you’ll need to pick
it up on your own).
Derivatives are used for finding minima and maxima of
functions as well as describing rates of change.
Consider an example function:
f(x,y) = x2 + 2y2
Then we have the following:
partial derivatives:
indicial notation:
f
 2x
x
f
 4y
y
f
 x f
x
f
 y f
y
If a function has directional elements, e.g., uz, then indicial
notation
 for a partial derivative
 in the y direction would
be:
u z
 y u z
y
Another common notation is the gradient operator:
u1 u2 u3
 u 



x1 x 2 x 3