Download 1 Some basic and useful mathematics

Mar109: part 1; basic mathematics
Lecture 2012-12-05: Revised 2012-12-04.
Göran Broström
1 Some basic and useful mathematics
Introduction
In natural science mathematics plays an important role.
Coordinate systems
There are some fundamental assumptions we make in biogeoscience. Often we use the same
Cartesian coordinate system, i.e., coordinates (x, y, z) that are perpendicular to each other. X
is the thumb, y is the “pekfinger” and z is “långfingret”. Polar (r, , z) and spherical
coordinates (r,, ) are sometimes used.
Length: The length of an object is the distance between two points in space, it can be along
the x-axis but that is not necessary.
What are variables and parameters: In the “mathematical division” of natural science one
often talks about variables and parameters, but what are they. Well, in principle they seem to
have precise definitions but the words are perhaps not always used as intended. A variable is
what varies and parameters are fixed; however, in our somewhat sloppy field of science
parameters may change value depending on the situation (for instance drag coefficient). This
is more likely a parameterization (if the drag coefficient is constant it is a parameter, if it
depends on e.g., the strength of the flow it is a parameterization and its value depends on the
strength of the flow field. Variables vary as the name indicates, for time-dependent situations
it is often used for the quantities which is described by a time derivative.
Example1: f ( x)  ax 2  bx  c ; x is the variable, a, b, c are constants or parameters. The
tricky bit is if for instance c depends on x, is it a variable or a parameter?
Example2: dC dt  C ; C is the variable,  is a constants or parameters.
Basic functions
Functions & algebra
Algebra is simply the art of do calculations with letter instead of numbers. This very useful to
derive general expressions for how different things relate. Often one uses f or F to denone a
function: if it is a function of x we write f(x), if it is a function of y we write f(y), and if it is a
function of x and y we write f(x,y) etc. Many functions has analytical solutions, but far from
all has analytical solutions. However, there are more-or-less powerful methods to find
numerical values of a solution; often they are available in mathematical programs. One can
also find solutions using graphical methods; one simply plots the graph and zoom in on the
figure to estimate the solution.
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Example: if
f ( x)  ax  b
we expect that it is a times x, and that the equation
f ( x)  0
has solution
f ( x)  ax  b  0
x  b / a
Besides solving equations, the mathematical joggling in equation manipulations and
simplifications of expressions using useful estimates is a part of algebra. Consider the
following equation
a(U a  u ) 2  b(U w  u ) 2  0
that we want to solve for u- This is a second order equation in u and we get a relatively
complicated expression. However, knowing that typical values of Ua is 10 m/s and typical
values of Uw is 0.3 m/s we realize that u<<Ua and can be neglected such that
a(U a ) 2  b(U w  u ) 2  0
Algebraic equation of different orders.
Polynomial (sv polynom). From Wikipedia, the free encyclopedia: In mathematics, a
”polynomia” is an expression of finite length constructed from variables(also called
indeterminates) and constants, using only the operations of addition, subtraction, multiplication,
and non-negative integer exponents. However, the division by a constant is allowed, because
the multiplicative inverse of a non zero constant is also a constant. For example, x2 − x/4 + 7
is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the
variable x (4/x), and also because its third term contains an exponent that is not a non-negative
integer (3/2). The term "polynomial" can also be used as an adjective, for quantities that can
be expressed as a polynomial of some parameter, as in polynomial time, which is used in
computational complexity theory.
Example:
First order: f ( x)  ax  b
Second order: f ( x)  ax 2  bx  c
Third order: f ( x)  ax 3  bx 2  cx  d
Note that polynomials of n’th order always has n solutions (more about that later)!
Trigonometry: Sinus, Cosinus; arcsinus, arc cosines
From Wikipedia, the free encyclopedia
In mathematics, the trigonometric functions (also called circular functions) are functions of an
angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle.
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Trigonometric functions are important in the study of triangles and modeling periodic phenomena,
among many other applications.
The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the
standard unit circle with radius 1, where a triangle is formed by a ray originating at the origin and
making some angle with the x-axis, the sine of the angle gives the length of the y-component (rise) of
the triangle, the cosine gives the length of the x-component (run), and the tangent function gives the
slope (y-component divided by the x-component). More precise definitions are detailed below.
Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the
angle, and can equivalently be defined as the lengths of various line segments from a unit circle.
Figure: Base of trigonometry: if two right triangles have equal acute angles, they are similar, so their side lengths are
proportional. Proportionality constants are written within the image: sin θ, cos θ, tan θ, where θ is the common
measure of five acute angles.
Sinus and cosines are oscillating functions and remind of each other. The sinus and cosines
are often used to describe oscillating flows and motions that are typical for wave motions. An
important concept in wave theory is the phase of the wave, that is how large is the angle when
e.g. t=0, this is called the phase of the wave. It should be noted that Sin and Cos is defined
using the unit circle (Fig. 2)
Sinus and Cosinus oscillates with phase 2 (rad) or 360 degrees. The fact that one uses 2 is
that one often uses the unit circle to describe the angels, and a full circle is 2.
The sine and cosine functions graphed on the Cartesian plane.
Figure 3: Left; the Sin(x) and Cos(x) function, right same functions but Sin(x+/2) and
Cos(x). We see that when we use a phase shift in Sin function we get the Cos function.
There are functions called hyperbolicus functions: they are
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The reason mathematicians name these functions as a trigonometric function is that they share
many similar properties: a more detail explanation can be found when using imaginary
numbers.
The significance of radians (From Wikipedia)
Radians specify an angle by measuring the length around the path of the unit circle and
constitute a special argument to the sine and cosine functions. In particular, only sines and
cosines that map radians to ratios satisfy the differential equations that classically describe
them. If an argument to sine or cosine in radians is scaled by frequency,
then the derivatives will scale by amplitude.
Here, k is a constant that represents a mapping between units. If x is in degrees, then
This means that the second derivative of a sine in degrees does not satisfy the differential
equation
but rather
The cosine's second derivative behaves similarly. This means that these sines and cosines are
different functions, and that the fourth derivative of sine will be sine again only if the
argument is in radians.
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Exponentials, logarithms
In mathematics, the exponential function is the function ex, where e is the number (approximately
2.718281828) such that the function ex is its own derivative. The exponential function is used to model
a relationship in which a constant change in the independent variable gives the same proportional
change (i.e. percentage increase or decrease) in the dependent variable. The function is often written
as exp(x), especially when it is impractical to write the independent variable as a superscript. The
exponential function is widely used in physics, chemistry and mathematics.
The graph of y = ex is upward-sloping, and increases faster as x increases. The graph always lies
above the x-axis but can get arbitrarily close to it for negative x; thus, thex-axis is a horizontal
asymptote. The slope of the tangent to the graph at each point is equal to its y coordinate at that point.
In general, the variable x can be any real or complex number,
The derivative (rate of change) of the exponential function is the exponential function itself. More
generally, a function with a rate of change proportional to the function itself (rather than equal to it) is
expressible in terms of the exponential function. This function property leads to exponential growth
and exponential decay.
The natural exponential function
Derivatives and differential equations
The importance of the exponential function in mathematics and the sciences stems mainly
from properties of its derivative. In particular,
That is, ex is its own derivative and hence is a simple example of a Pfaffian function.
Functions of the form cex for constant c are the only functions with that property (by the
Picard–Lindelöf theorem). Thus, the function solves the differential equation y ′ = y.
exp is a fixed point of derivative as a functional. This represents the case of a variable's
growth or decay rate is proportional to its size
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Hyperbolicus functions
Derivation
From Wikipedia, the free encyclopedia
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as
its input changes. Loosely speaking, a derivative can be thought of as how much one quantity
is changing in response to changes in some other quantity; for example, the derivative of the
position of a moving object with respect to time is the object's instantaneous velocity.
The process of finding a derivative is called differentiation. The reverse process is called
antidifferentiation or integration
The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the
derivative of the function at the marked point.
Differentiation is a method to compute the rate at which a dependent output y changes with respect
to the change in the independent input x. This rate of change is called the derivative of y with respect
to x: here we use functional relationship is often denoted y = f(x), where f denotes the function
The simplest case is when y is a linear function of x, meaning that the graph of y divided by x is a
straight line. In this case, y = f(x) = m x + b, for real numbers m and b, and the slope m is given by
where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in."
This gives an exact value for the slope of a straight line. If the function f is not linear (i.e. its graph is
not a straight line), however, then the change in y divided by the change in x varies: differentiation is a
method to find an exact value for this rate of change at any given value of x.
Rate of change as a limiting value
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Figure 1. The tangent line at (x, f(x))
In Leibniz's notation, such an infinitesimal change in x is denoted by dx, and the derivative of y with
respect to x is written
suggesting the ratio of two infinitesimal quantities. (The above expression is read as "the derivative of
y with respect to x", "d y by d x", or "d y over d x". The oral form "d y d x" is often used
conversationally, although it may lead to confusion.)
Definition via difference quotients
Let f be a real valued function. The slope m of the secant line is the difference between the y
values of these points divided by the difference between the x values, that is,
This expression is Newton's difference quotient. The derivative is the value of the difference
quotient as the secant lines approach the tangent line. Formally, the derivative of the function
f at x is the limit
which can also be written as
of the difference quotient as h approaches zero, if this limit exists. If the limit exists, then f is
differentiable at a. Here f′ (a) is one of several common notations for the derivative.
Continuity and differentiability
This function does not have a derivative at the marked point, as the function is not continuous
there.
Higher derivatives
Let f be a differentiable function, and let f′(x) be its derivative. The derivative of f′(x) (if it has one) is
written f′′(x) and is called the second derivative of f. Similarly, the derivative of a second derivative, if
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it exists, is written f′′′(x) and is called the third derivative of f. These repeated derivatives are called
higher-order derivatives.
Notations for differentiation
The notation for derivatives introduced by Gottfried Leibniz is one of the earliest. It is still
commonly used when the equationy = f(x) is viewed as a functional relationship between
dependent and independent variables. Then the first derivative is denoted by
and was once thought of as an infinitesimal quotient. Higher derivatives are expressed using
the notation
for the nth derivative of y = f(x) (with respect to x). These are abbreviations for multiple
applications of the derivative operator. For example,
With Leibniz's notation, we can write the derivative of y at the point x = a in two different
ways:
Leibniz's notation allows one to specify the variable for differentiation (in the denominator).
This is especially relevant for partial differentiation. It also makes the chain rule easy to
remember:
Lagrange's notation
Sometimes referred to as prime notation, one of the most common modern notations for
differentiation is due to Joseph-Louis Lagrange and uses the prime mark, so that the
derivative of a function f(x) is denoted f′(x) or simply f′. Similarly, the second and third
derivatives are denoted
and
To denote the number of derivatives beyond this point, some authors use Roman numerals in
superscript, whereas others place the number in parentheses:
or
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The latter notation generalizes to yield the notation f (n) for the nth derivative of f — this
notation is most useful when we wish to talk about the derivative as being a function itself, as
in this case the Leibniz notation can become cumbersome.
1.4 Integrals
From Wikipedia, the free encyclopedia
Integrals appear in many practical situations. If a swimming pool is rectangular with a flat
bottom, then from its length, width, and depth we can easily determine the volume of water it
can contain (to fill it), the area of its surface (to cover it), and the length of its edge (to rope
it). But if it is oval with a rounded bottom, all of these quantities call for integrals. Practical
approximations may suffice for such trivial examples, but precision engineering (of any
discipline) requires exact and rigorous values for these elements.
Approximations to integral of √x from 0 to 1, with ■ 5 right samples (above) and ■ 12 left
samples (below)
Given a function f of areal variable x and an interval [a, b] of the real line, the definite integral
is defined informally to be the area of the region in the xy-plane bounded by the graph of f, the
x-axis, and the vertical lines x = a and x = b, such that area above the x-axis adds to the total,
and that below the x-axis subtracts from the total.
The term integral may also refer to the notion of the antiderivative, a function Fwhose
derivative is the given function f. In this case, it is called an indefinite integral and is written:
The integrals discussed in this article are termed definite integrals.
If f is a continuous real-valued function defined on a closed interval[a, b], then, once an
antiderivative F of f is known, the definite integral of f over that interval is given by
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Integrals and derivatives became the basic tools of calculus, with numerous applications in
science and engineering. The founders of the calculus thought of the integral as an infinite
sum of rectangles of infinitesimal width
A definite integral of a function can be represented as the signed area of the region bounded
by its graph.
Terminology and notation
The simplest case, the integral over x of a real-valued function f(x), is written as
The integral sign ∫ represents integration. The dx indicates that we are integrating over x; dx is
called the variable of integration. In correct mathematical typography, the dx is separated
from the integrand by a space (as shown). Inside the ∫...dx is the expression to be integrated,
called the integrand. In this case the integrand is the function f(x). Because there is no domain
specified, the integral is called an indefinite integral.
When integrating over a specified domain, we speak of a definite integral. Integrating over a
domain D is written as
or
if the domain is an interval [a, b] of x;
The domain D or the interval [a, b] is called the domain of integration.
If a function has an integral, it is said to be integrable. In general, the integrand may be a
function of more than one variable, and the domain of integration may be an area, volume, a
higher dimensional region, or even an abstract space that does not have a geometric structure
in any usual sense (such as a sample space in probability theory).
The variable of integration dx has different interpretations depending on the theory being
used. It can be seen as strictly a notation indicating that x is a dummy variable of integration;
if the integral is seen as a Riemann sum, dx is a reflection of the weights or widths d of the
intervals of x; in Lebesgue integration and its extensions, dx is a measure; in non-standard
analysis, it is an infinitesimal; or it can be seen as an independent mathematical quantity,
a differential form. More complicated cases may vary the notation slightly. In Leibniz's
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notation, dx is interpreted an infinitesimal change in x, but his interpretation lacks rigour in
the end. Nonetheless Leibniz's notation is the most common one today; and as few people are
in need of full rigour, even his interpretation is still used in many settings.
1.5 Imaginary numbers
Relationship to exponential function and complex numbers
It can be shown from the series definitions that the sine and cosine functions are the
imaginary and real parts, respectively, of the complex exponential function when its argument
is purely imaginary:
This identity is called Euler's formula. In this way, trigonometric functions become essential
in the geometric interpretation of complex analysis.
Ordinary sine and cosine can be written using complex numbers as:
This is quite similar to the definition of the sinh and cosh, and this is the underlying reason
that one define
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1.6 Finding approximate simple solutions
We deal with very difficult equations in ocean science. In fact, there are very few fields that
deal with more complicated equations and dynamical systems. So be aware, you have entered
a difficult but interesting field of science. Anyway, we cannot hope to find real solutions to
our problems within quite a few years to come.
Scaling
Lets again consider the Navier-Stokes equation
  2u  2u  2u 
u
u
u
u
1 p
u
v
 w  fv  
   2  2  2 
t
x
y
z
 x
y
z 
 x
This equation is to complicated for a analysis and we need to find out how we can understand
this eqution.
Lets say we have velocity U and scale of an object L (say 0.1 ms-1 and L=0.1 m): we can now
estimate how large the terms are
u U 2
~
x
L
2
 u U
 2 ~ 2
x
L
u
The first term is thus large for large velocities and relatively small objects, the second term
becomes very large for small objects and relatively large velocities.
u
u
1 p

x
 x
 u 2
2 x

p
x
If we approximate the differentials with a certain change (for instance a given distance x)
 u 2
2 x

p
x
‘or
p  

2
u 2
This equation tells us that the pressure decreases if the velocity increases. If you do not
believe this, never enter an aircraft again. It is this principle that makes plane flying: the
velocity is larger on the top side of the wind then on the low side of the wing. The relt is that
pressure is lower on the upper side of the wing than on the lower side, thus pressure is higer
below than above and this lifts the plane. This can be derived in more a more general manner
and then it is known as the Bernoulli equation or principle.
If one assumes that the difference in u  ( H ,W )u , that is it depends on the width and height
of an object, and the free stream velocity we arrive at
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p  

2
Au 2
Or the relation we arrived at by dimensional analysis.
Bernouli equation:
From Wikipedia, the free encyclopedia
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the
speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the
fluid's potential energy. Bernoulli's principle is named after the Swiss scientist Daniel
Bernoulli who published his principle in his book Hydrodynamica in 1738.
Bernoulli's principle can be derived from the principle of conservation of energy. This states
that, in a steady flow, the sum of all forms of mechanical energy in a fluid along a streamline
is the same at all points on that streamline. This requires that the sum of kinetic energy and
potential energy remain constant. Thus an increase in the speed of the fluid occurs
proportionately with an increase in both its dynamic pressure and kinetic energy, and a
decrease in its static pressure and potential energy. If the fluid is flowing out of a reservoir,
the sum of all forms of energy is the same on all streamlines because in a reservoir the energy
per unit volume (the sum of pressure and gravitational potential ρ g h) is the same
everywhere.
Bernoulli's principle can also be derived directly from Newton's 2nd law. If a small volume of
fluid is flowing horizontally from a region of high pressure to a region of low pressure, then
there is more pressure behind than in front. This gives a net force on the volume, accelerating
it along the streamline.
Fluid particles are subject only to pressure and their own weight. If a fluid is flowing
horizontally and along a section of a streamline, where the speed increases it can only be
because the fluid on that section has moved from a region of higher pressure to a region of
lower pressure; and if its speed decreases, it can only be because it has moved from a region
of lower pressure to a region of higher pressure. Consequently, within a fluid flowing
horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed
occurs where the pressure is highest.
Incompressible flow equation
A common form of Bernoulli's equation, valid at any arbitrary point along a streamline, is:
where:
is the fluid flow speed at a point on a streamline,
is the acceleration due to gravity,
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is the elevation of the point above a reference plane, with the positive z-direction
pointing upward – so in the direction opposite to the gravitational acceleration,
is the pressure at the chosen point, and
is the density of the fluid at all points in the fluid.
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