BASIC
MATHEMATICS
Memory Joggers
IMAGINARY
NUMBERS
Real numbers include both rational numbers, such as 42 and −23/129,
and irrational numbers, such as pi and the square root of two. A
real number can be given by an infinite decimal representation, such
as 2.4871773339..., where the digits continue in some way. Real
numbers may be thought of as points on an infinitely long number
line.
Imaginary numbers are in the form "bi" where "b" is a nonzero, real
number and "i" is defined by iČ = 1, called the "imaginary
unit". An Imaginary number bi can be added to a real number a, to
form a "complex number" of the form (a + bi), where a and b are
called respectively, the "real part" and the "imaginary part" of the
complex number. Imaginary numbers can therefore be thought of as
complex numbers where the real part is zero. The square of an
imaginary number is a negative real number.
The complex conjugate of a complex number has the
reverse sign of the original imaginary part. The
complex conjugate of (a + bi) is (a  bi).
Real and imaginary parts are additive: (a + bi) + (c
+di) = (a + c) + (b + d)i, and are multiplicative:
(a + bi) * (c + di) = ac + bci +adi + bdi^{2}
= (ac  bd) + (bc + ad)i.
Division is accomplished by multiplying the
numerator and denominator by the complex conjugate
of the denominator:
BASIC
ALGEBRA
Elementary
algebra is one of the main branches of pure
mathematics and introduces the concept of variables
representing numbers. Statements based on these
variables are manipulated using the rules of
operations that apply to numbers, such as addition.
This can be done for a variety of reasons, including
equation solving.
While the word algebra comes from the Arabic
languageand much of its methods from Arabic/Islamic
mathematics, its roots can be traced to earlier
traditions, most notably ancient Indian mathematics.
A polynomial is an expression that is
constructed from one or more variables and
constants, using only the operations of addition,
subtraction, multiplication, and exponentiation. For
example, x^{2} + 2x − 3 is a
polynomial in the single variable x.
Factoring of polynomials is the process of
expressing a given polynomial as a product of other
polynomials. The example polynomial above can be
factored as (x − 1)(x + 3).
A
quadratic equation is one in which a term is raised
to the power of 2 and other terms are raised to a
lower power or a are constants:
x^{2} + 2x − 3 =0 is a
quadradic equation.
A
root, or zero, of a polynomial in a single
variable is the value that of x that forces the
polynomial to produce a result equal to zero. In the
example above, x = 1 and x = 3 are roots or zeros
of the polynomial
x^{2} + 2x − 3 =
(x
− 1)(x + 3) = 0.
All real polynomials of odd degree have a real
number as a root. Many real polynomials of even
degree do not have a real root.
A
polynomial of degree n will have n roots. Roots are
not always obvious. The roots of a quadratic
equation are found with:
Roots of higher order equations are solved by
graphing or iterative "solver" software.
CRAMER'S
RULE
Cramer's Rule is a handy way to solve for any one of the
variables in a set of linear simultaneous equations without
having to solve the whole system of equations. Or it can be
used to solve for all the unknowns. For example:
a1 * X + a2 * Y + a3 * Z = PHID
b1 * X + b2 * Y + b3 * Z =
PHIN
1.0 * X + 1.0 * Y + 1.0 * Z = 1.00
WHERE:
a1, a2, a3 = density log porosity readings for rock
components X, Y, Z
b1, b2, b3 = neutron log porosity readings for rock
components X, Y, Z
X, Y, Z = rock volumes of the three components
(fractional units)
The lefthand side
of the equations with the variables is the coefficient matrix
and the righthand side is the answer matrix.
Coefficient Matrix D
Answer Matrix
 a1 a2 a3 
 PHID 
 b1 b2 b3 
 PHIN 
 1.0 1.0 1.0 
 1.0 
 a3 b3 1.0 
Dx
is the determinant formed by replacing the Xcolumn values with
the answercolumn values. Similarly, the Dy and Dz}
determinants are formed by replacing the Ycolumn and the
Zcolumn, as shown below.
X Determinant Dx Y Determinant Dy
Z Determinant Dz
 PHID a2 a3 
 a1 PHID a3 
 a1 a2 PHID 
 PHIN b2 b3 
 b1 PHIN b3 
 b1 b2 PHIN 
 1.0 1.0 1.0 
 1.0 1.0 1.0 
 1.0 1.0 1.0 
Cramer's Rule says that
1: X = Dx / D}
2: Y = Dy / D
3: Z = Dz / D.
The next step is to evaluate each determinant and calculate X,
Y, and Z.
RESOLVING DETERMINANTS
Solving for the value of
a determinant is a matter of properly applying the arithmetic
needed. Start with a sample, such as Dx. Extend the matrix by
rewriting all the columns except the last one, as below. Then
multiply the values in each "full" diagonal (coloured cells) and add
these products together (honour the signs). This gives the sum of
the "Down" diagonals, Dd.
X Determinant Dx
X Determinant EXTENDED
 PHID a2 a3 
 PHID
a2
a3  PHID a2
 PHIN b2 b3 
 PHIN
b2
b3 
PHIN
b2
 1.0 1.0 1.0 
 1.0 1.0
1.0

1.0
1.0
a3 * PHIN * 1.0
\
\== a2 * b3 * 1.0
\== PHID * b2 *1.0
ADD Products together = Dd
Then do the same with the opposite diagonals. This gives the sum of
the "Up" diagonals, Du.
X Determinant Dx
X Determinant EXTENDED
 PHID a2 a3 
 PHID a2
a3
 PHID
a2
 PHIN b2 b3 
 PHIN
b2
b3 
PHIN
b2
 1.0 1.0 1.0 

1.0
1.0
1.0
 1.0 1.0
Obtain the products of the "Up" diagonals and ADD the products = Du.
THEN Dx = Dd  Du. Follow the same procedure for Dy, Dz, and
D, then use Cramer's Rule to solve for X, Y, Z.
If you give the coefficients appropriate numerical values that
correspond to calcite, dolomite, and water for example,, with
particular values of PHID and PHIN, you will get the fraction of
each component in the reservoir. Porosity will equal the volume of
water.
To generalize the model, form the approptiate equations for each
determinant using variable names instead of actual numerical values.
The equations will look a little messy, but will work with any
rational inputs. Negative answers for X, Y, or Z are illegal and
suggest bad data or bad parameters. Small negative values can be
trimmed to zero, but large negative answers will need more help.
BASIC
GEOMETRY
Plane
geometry is the study of angles and triangles, perimeter, area and
volume. It differs from algebra in that it develops a logical
structure where mathematical relationships are proved and applied.
Euclid (c. 300 BCE) introduced certain axioms, or postulates,
expressing primary or selfevident properties of points, lines, and
planes. He proceeded to rigorously deduce other properties by
mathematical reasoning. Some of these are listed below.
A point shows position.
A line is infinite and
straight and is a set of continuous points that extend
indefinitely in either of its direction. A line segment is part of
the straight line between two points.
Parallel lines do not
cross.
A ray is the part of the
line which consists of a given point and the set of all points on
one side of that point.
A circle is a set of
points equidistant from another point, known as the center of the
circle.
An angle can be defined
as two rays or two line segments having a common end point. The
endpoint becomes known as the vertex.
An angle occurs when two rays meet or unite at the same
endpoint.
A plane is a flat surface containing three points
that are not all in a straight line.
The intersection of two
plones is a line.
Areas
OF GEOMETRIC FIGURES
Rectangle: A = a * b
Parallelogram: A = b * h
Trapezoid: A = h * (b_{1} + b_{2})
/ 2
Circle: A = pi * r^{ 2}
Ellipse:
A = pi * r_{1 *} r_{2}
Triangle:
A = 0.5 * b * h
Equilateral triangle A = 0.25 *
(3)
* a^{2}
Triangle given SAS: A = 0.5 * a * b * sin C
Triangle given a,b,c: A =
[s
* (s  a) * (s  b) * (s  c)]
where s = (a + b + c) / 2
Regular polygon: A = 0.5 * n * sin(360°/n) * S^{2}
where: n = # of sides and S = length from center to a corner
Volumes
OF GEOMETRICAL SHAPES
Cube: V = a^{3}
Rectangular prism: V = a * b * c
Irregular prism: V = b * h
Cylinder: V = b * h =
pi * r^{2}
* h
Pyramid: V = 0.5 * b * h
Cone: V = (1/3) * b * h = (1/3) * pi
* r^{2} * h
Sphere: V = (4/3)
* pi * r^{3}
Ellipsoid: V= (4/3)
* pi * r_{1 *} r_{2 *}
r_{3}
Surface
Areas OF GEOMETRIC SHAPES
Cube: S = 6 * a^{2}
Prism:
Lateral
Area S = perimeter(b) * L
Total Area S =
perimeter(b) * L + 2 * b
Sphere: S = 4 * pi * r^{2}
BASIC
TRIGONOMETRY
Trigonometry
is a branch of mathematics that studies triangles on a plane
surfaces and deals with relationships between the sides and the
angles of triangles and with the trigonometric functions, which
describe those relationship.
Conversions
PI = 3.141 592... (approximately 22/7 = 3.1428)
radians = degress x PI / 180
degress = radians x 180 / PI
e = 2.718 282....
Functions
The
trigonometry functions are defined by a right triangle with angle
theta, adjacent side b, oposite side a, and hypotenuse c.
sin(theta) = a / c
csc(theta)
= 1 / sin(theta) = c / a
cos(theta) = b / c
sec(theta) = 1 / cos(theta) = c / b
tan(theta) = sin(theta) / cos(theta) = a / b
cot(theta) = 1/ tan(theta) = b / a
USEFUL EQUALITIES
sin^{^2}(x) + cos^{^2}(x) = 1
tan^{^2}(x)
+ 1 = sec^{^2}(x)
cot^{^2}(x) + 1 = csc^{^2}(x)
sin(x
y) = sin x cos y
cos x sin y
cos(x
y) = cos x cos y
sin x sin y
sin(x) = sin(x)
csc(x) =
csc(x)
cos(x) = cos(x)
sec(x) = sec(x)
tan(x) = tan(x)
cot(x) = cot(x)
tan(x
y) = (tan x
tan y) / (1
tan x tan y)
sin(2x) = 2 sin x cos x
cos(2x) =
cos^{^2}(x)  sin^{^2}(x) = 2 cos^{^2}(x) 
1 = 1  2 sin^{^2}(x)
tan(2x) = 2 tan(x) / (1 
tan^{^2}(x))
sin^{^2}(x) = 1/2  1/2
cos(2x)
cos^{^2}(x) = 1/2 + 1/2 cos(2x)
sin x  sin y = 2 sin( (x  y)/2 ) cos( (x + y)/2 )
cos x
 cos y = 2 sin( (xy)/2 ) sin( (x + y)/2 )
Given Triangle
abc, with angles A,B,C; a is opposite to A, b oppositite B, c
opposite C:
Then:
a / sin(A) = b
/ sin(B) = c / sin(C)
c^{^2}
= a^{^2} + b^{^2}  2ab cos(C) Law of Sines
b^{^2} = a^{^2}
+ c^{^2}  2ac cos(B)
a^{^2} = b^{^2}
+ c^{^2}  2bc cos(A) Law of Cosines
(a  b)/(a +
b) = tan 1/2(AB) / tan 1/2(A+B) (Law of Tangents)
Hyperbolic
Definitions
sinh(x)
= ( e^{x}  e^{x} )/2
csch(x) =
1/sinh(x) = 2/( e^{x}  e^{x} )
cosh(x)
= ( e^{ x} + e^{ x} )/2
sech(x) =
1/cosh(x) = 2/( e^{x} + e^{x} )
tanh(x)
= sinh(x)/cosh(x) = ( e^{x}  e^{x} )/( e^{x}
+ e^{x} )
coth(x) = 1/tanh(x) = ( e^{x}
+ e^{x})/( e^{x}
 e^{x} )
cosh^{2}(x)
 sinh^{2}(x) = 1
tanh^{2}(x) + sech^{2}(x)
= 1
coth^{2}(x)  csch^{2}(x) = 1
Inverse Hyperbolic
Defintions
arcsinh(z) = ln( z +
(z^{2}
+ 1) )
arccosh(z) = ln( z
(z^{2}
 1) )
arctanh(z) = 1/2 ln( (1+z)/(1z) )
arccsch(z) = ln( (1+(1+z^{2})
)/z )
arcsech(z) = ln( (1(1z^{2})
)/z )
arccoth(z) = 1/2 ln( (z+1)/(z1) )
Relations to
Trigonometric Functions
sinh(z) = i sin(iz)
csch(z) = i csc(iz)
cosh(z) = cos(iz)
sech(z) = sec(iz)
tanh(z) =
i tan(iz)
coth(z) = i cot(iz)
Spherical
Trigonometry
Spherical trigonometry is
a branch of spherical geometry which deals with polygons (especially
triangles) on the sphere and the relationships between the sides and
the angles. This is of great importance for calculations in
astronomy and earthsurface, orbital and space navigation.
On the surface of a sphere, the closest analogue to straight lines
are great circles, i.e. circles whose center coincide with the
center of the sphere. As with a line segment in a plane, an arc of a
great circle (subtending less than 180°) on a sphere is the shortest
path lying on the sphere between its two endpoints. An area on the
sphere, bounded by arcs of great circles, is called a spherical
polygon.
The sides of these polygons are specified not by their lengths, but
by the angles at the sphere's center subtended by the endpoints of
the sides. Note that this arc angle, measured in radians, when
multiplied by the sphere's radius equals the arc length.
The
sum of the vertex angles of spherical triangles is always larger
than the sum of the angles of plane triangles, which is exactly
180°. The amount E by which the sum of the angles exceeds 180° is
called spherical excess:
E = A + B + C  pi
where A, B, and C denote the angles.
This surplus determines the surface area of any spherical triangle:
A = R^{2} * E
where R is the radius of the sphere.
Identities
Click Here
(mathworld.wolfram.com)