ANALYSIS
5 MARKS
1
.
History of Vector Analysis
A History of Vector Analysis
(1967) is a book on the history of
vector analysis
by Michael J. Crowe,
originally published by
the
University of Notre Dame Press
. As a scholarly treatment of a reformation in
technical communication
, the text is a
contribution to the
history of science
. In 2002, Crowe gave a talk
[1]
summarizing the book, including an entertaining introduction
in which he covered its publication history and related the award of a Jean Scott prize of $4000. Crowe h
ad entered the book in a
competition for "a study on the history of complex and hypercomplex numbers" twenty

five years after his book was first
published.
2
.
Vector operations
Algebraic operations
The basic algebraic (non

differential) operations in vector
calculus are referred to as
vector algebra
, being defined
for a vector space and then globally applied to a vector field, and consist of:
scalar multiplication
m
ultiplication of a scalar field and a vector field, yielding a vector field:
;
vector addition
addition of two vector fields, yielding a vector field:
;
dot product
multiplication of two vector fields, yielding a scalar field:
;
cross product
multiplication
of two vector fields, yielding a vector field:
;
There are also two
triple products
:
scalar triple product
the dot product of a vector and a cross product of two vectors:
;
vector triple product
the
cross product of a vector and a cross product of two vectors:
or
;
although these are less often used as basic operations, as they can be expressed in terms
of the dot and cross products.
[
.
]
Differential operations
Vector calculus studies various
differential operators
defined on scalar or vector fields,
which are typically expressed in terms of the
d
el
operator (
), also known as "nabla".
The four most important differential operations in vector calculus are:
3
.
Generalizations
Nonlinear generalizations
Linear PCA versus nonlinear Principal Manifolds
[22]
for
visualization
of
breast cancer
microarray
data: a) Configuration of nodes and 2D Principal
Surface in the 3D PCA linear manifold. The dataset is curved and cannot b
e mapped adequately on a 2D principal plane; b) The distribution in
the internal 2D non

linear principal surface coordinates (ELMap2D) together with an estimation of the density of points; c) The same as b), but
for the linear 2D PCA manifold (PCA2D). The
"basal" breast cancer subtype is visualized more adequately with ELMap2D and some features of
the distribution become better resolved in comparison to PCA2D. Principal manifolds are produced by the
elastic maps
algorithm. Data are
available for public competition.
[23]
Software is available for free non

commercial use.
[24]
Most of the modern methods for
nonlinear dimensionality reduction
find their theoretical and algorithmic roots in
PCA or K

means. Pearson's original idea was to take a straight line (or plane) which will be "the best fit" to a set of
data points.
Principal
curv
es
and
manifolds
[25]
give the natural geometric framework for PCA generalization and
extend the
geometric interpretation of PCA by explicitly constructing an embedded manifold for
data
approximation
, and by encoding using standard geometric
projection
onto the manifold, as it is illustrated by
Fig. See also the
elastic map
algorithm and
principal geodesic analysis
.
Multilinear generalizations
In
multilinear subspace learning
, PCA is generalized to
multilinear PCA
(MPCA) that extracts features directly from
tensor rep
resentations. MPCA is solved by performing PCA in each mode of the tensor iteratively. MPCA has been
applied to face recognition, gait recognition, etc. MPCA is further extended to uncorrelated MPCA, non

negative
MPCA and robust MPCA.
Higher order
N

way pr
incipal component analysis may be performed with models such as
Tucker decomposition
,
PARAFAC
,
mul
tiple factor analysis, co

inertia analysis, STATIS, and DISTATIS.
Robustness

Weighted PCA
While PCA finds the mathematically optimal method (as in minimizing the squared error), it is sensitive
to
outliers
in the data that produce large errors PCA tries to avoid. It therefore is common practice to remove outliers
before computing PCA. However, in some contexts, outliers can be difficult
to identify. For example in
data
mining
algorithms like
correlation clustering
, the assignment of points to clusters and outliers is not known
beforehand. A recently proposed generalization of PCA
[26]
based on a
Weighted PCA
increases robustnes
s by
assigning different weights to data objects based on their estimated relevancy.
20MARKS
1
.
Definition
A vector space over a
field
F
is a
set
V
together with two
binary operations
that satisfy the eight axioms listed
below. El
ements of
V
are called
vectors
. Elements of
F
are called
scalars
. In this article, vectors are distinguished
from scalars by boldface.
[nb 1]
In the two examples above, our set consis
ts of the planar arrows with fixed starting
point and of pairs of real numbers, respectively, while our field is the real numbers. The first operation,
vector
addition
, takes
any two vectors
v
and
w
and assigns to them a third vector which is commonly written as
v
+
w
,
and
called the sum of these two vectors. The second operation takes any scalar
a
and any vector
v
and gives
another
vector
a
v
. In view of the first example,
where the multiplication is done by rescaling the vector
v
by a
scalar
a
, the multiplication is called
scalar multiplication
of
v
by
a
.
To qualify as a vector spa
ce, the set
V
and the operations of addition and multiplication must adhere to a number of
requirements called
axioms
.
[1]
In the list below, let
u
,
v
and
w
be arbitrary vectors in
V
, and
a
and
b
scalars in
F
.
Axiom
Meaning
Associativity
of addition
u
+ (
v
+
w
) = (
u
+
v
) +
w
Commutativity
of addition
u
+
v
=
v
+
u
Identity element
of addition
There exists an element
0
∈
V
, 捡汬敤 瑨e
穥ro v散瑯r
, su捨 th慴
v
+
0
=
v
for all
v
∈
V
.
Inv敲s攠敬emen瑳
of 慤d楴楯n
䙯r
敶敲y
v
∈
V, there exists an element −
v
∈
V
, 捡汬敤 瑨e
add楴楶攠楮v敲se
潦
v
, such that
v
+ (−
v
) =
0
Distributivity
of scalar
multiplication with respect to vector
addition
a
(
u
+
v
) =
a
u
+
a
v
Distributivity of scalar
multiplication with respect to field
addition
(
a
+
b
)
v
=
a
v
+
b
v
Compatibility of scalar
multiplication with field
multiplication
a
(
b
v
) = (
ab
)
v
[nb 2]
Identity element of scalar
multiplication
1
v
=
v
, where 1 denotes the
multiplicative identity
in
F
.
These axioms generalize properties of the vectors introduced in the above examples. Indeed, the result of addition of
two ordered
pairs (as in the second example above) does not depend on the order of the summands:
(
x
v
,
y
v
) + (
x
w
,
y
w
) = (
x
w
,
y
w
) + (
x
v
,
y
v
),
Likewise, in the geometric example of vectors as arrows,
v
+
w
=
w
+
v
,
since the parallelogram defining the
sum of the vectors
is independent of the order of the vectors. All other axioms can be checked in a similar
manner in both examples. Thus, by disregarding the concrete nature of the particular type of vectors, the
definition incorporates these two and many more examples in
one notion of vector space.
Subtraction of two vectors and division by a (non

zero) scalar can be defined as
v
−
w
=
v
+ (−
w
),
v
/
a
= (1/
a
)
v
.
When the scalar field
F
is the
real
numbers
R
, the vector space is called a
real vector space
. When the
scalar field is the
complex numbers
, it is called a
complex vector space
. These two cases are the ones
used
most often in engineering. The most general definition of a vector space allows scalars to be
elements of any fixed
field
F
. The notion is then known as an
F

vector
spaces
or a
vector space over
F
. A field is, essentially, a set of numbers
possessing
addition
,
subtraction
,
multiplication
and
division
operations.
[nb 3]
For example,
rational
numbers
also form a field.
In contrast to the intuition stemming from vectors in the plane and higher

dimensional ca
ses, there is,
in general vector spaces, no notion of
nearness
,
angles
or
distances
. To deal with such matters,
particular types of vector spaces are introduced; see
below
.
Altern
ative formulations and elementary consequences
The requirement that vector addition and scalar multiplication be binary operations includes (by
definition of binary operations) a property called
closure
: that
u
+
v
and
a
v
are in
V
for all
a
in
F
,
and
u
,
v
in
V
. Some older sources mention these properties as separate axioms.
[2]
In the parlance
of
abstract algebra
, the first four axioms can be subsumed by requiring the set of
vectors to be an
abelian group
under addition. The remaining axioms give this group an
F

module
structure. In other words there is a
ring homomorphism
ƒ
from the field
F
into
the
endomorphism ring
of the group of vectors. Then scalar multiplication
a
v
is defin
ed as (
ƒ
(
a
))(
v
).
[3]
There are a number of direct consequences of the vector space axioms. Some of them derive
from
elementary group theory
, applied to the additive group of vectors: for example the zero
vector
0
of
V
and the additive inverse −
v
of any vector
v
are unique. Ot
her properties follow from the
distributive law, for example
a
v
equals
0
if and only if
a
equals 0 or
v
equals
0
.
Examples
.
:
Examples of vector spaces
Coordinate spaces
The simplest example of a vector space over a field
F
is the field itself, equipped with its standard addition and
multiplication. More generally, a vector space can be composed of
n

tuples
(sequences of length
n
) of elements of
F
,
such as
(
a
1
,
a
2
, ...,
a
n
), where each
a
i
is an element of
F
.
[12]
A
vector space composed of all the
n

tuples of a field
F
is known as a
coordinate space
, usually denoted
F
n
. The
case
n
= 1 is the above mentioned simplest example, in which
the field
F
is also regarded as a vector space over
itself. The case
F
=
R
and
n
= 2 was discussed in the introduction above.
The complex numbers and other field extensions
The set of
complex numbers
C
, i.e., numbers that can be written in the form
x
+
i
y
for
real
numbers
x
and
y
wh
ere
is the
imaginary unit
, form a vector space over the reals with the usual
addition and multiplication: (
x
+
i
y
) + (
a
+
i
b
) = (
x
+
a
) +
i
(
y
+
b
)
and
for real numbers
x
,
y
,
a
,
b
and
c
. The various axioms of a
vector space follow from the fact that the same rules hold for complex number arithmetic.
In fact, the example of complex numbers is essentially the same (i.e., it is
isomorphic
) to the vector space of
ordered pairs
of real numbers mentioned above: if we think of the complex number
x
+
i
y
as representing the
ordered pair (
x
,
y
) in the
complex plane
then we see that the rules for sum and
scalar product correspond
exactly to those in the earlier example.
More generally,
field extensions
provide another class of examples of vector spaces, particularly in algebr
a
and
algebraic number theory
: a field
F
containing a
smaller field
E
is an
E

vector space,
by the given
multiplication and addition operations of
F
.
[13]
For example, the complex numbers are a vector space over
R
,
and the field extension
is a vector space over
Q
.
Function spaces
Functions from any fixed set Ω to a field
F
also form vector spaces, by performing addition and scalar
multiplication pointwise. That is, the sum of two functions
ƒ
and
g
is the function (
f
+
g
) given by
(
ƒ
+
g
)(
w
) =
ƒ
(
w
) +
g
(
w
),
and simila
rly for multiplication. Such
function spaces
occur in many geometric situations, when Ω is
the
real line
or an
interval
, or other
subsets
of
R
. Many notions in topology and analysis, such
as
continuity
,
integrability
or
differentiability
are well

behaved with respect to linearity: sums and scalar
multiples of functions possessing such a property still have that property.
[14]
Therefore, the se
t of such
functions are vector spaces. They are studied in greater detail using the methods of
functional analysis
,
see
below
. Algebraic constraints also yield vector spaces: the
vector space
F[x]
is given by
polynomial
functions
:
ƒ
(
x
) =
r
0
+
r
1
x
+ ... +
r
n
−
1
x
n
−
1
+
r
n
x
n
, where the
coefficients
r
0
, ...,
r
n
are in
F
.
[15]
Linear equations
.
s:
Linear equation
,
Linear differential equation
, and
Systems of
linear equations
Systems of
homogeneous linear equations
are closely tied to vector spaces.
[16]
For example, the
solutions of
a
+
3
b
+
c
= 0
4
a
+
2
b
+
2
c
= 0
are given by triples with arbitrary
a
,
b
=
a
/2, and
c
= −5
a
/2. They form a vector space: sums and
scalar
multiples of such triples still satisfy the same ratios of the three variables; thus they are
solutions, too.
Matrices
can be used to condense multiple linear equat
ions as above into
one
vector equation
, namely
A
x
=
0
,
where
A
=
is the matrix containing the coefficients of the given
equations,
x
is the vector
(
a
,
b
,
c
),
A
x
denotes the
matrix product
and
0
= (0, 0) is the zero
vector. In a similar vein, the solutions of homogeneous
linear differential equations
form
vector spaces. For example
ƒ''(x) + 2ƒ'(x)
+
ƒ(x) = 0
yields
ƒ
(
x
) =
a e
−
x
+
bx e
−
x
,
where
a
and
b
are arbitrary constants, and
e
x
is the
natural
exponential function
.
2
Applicants
Chemistry
The field of
chemistry
uses analysis in at least three ways: to identify the components of a
particular
chemical compound
(qualitative analysis), to identify the proportions of components in
a
mixture
(quantitative analysis), and to break down
chemical processes
and examine
chemical
reactions
between
elements
of
matter
. For an example of its use, analysis of the concentration of
elements is important in managing a
nuclear reactor
, so
nuclear scientists
will analyze
neutron
activation
to develop discrete measurements within vast samples. A
matrix
ca
n have a
considerable effect on the way a chemical analysis is conducted and the quality of its results.
Analysis can be done manually or with a
device
. Chemical analysis is an important e
lement
of
national security
among the
major world powers
with
materials
measurement and signature
intelligence
(MASINT) capabilities.
Isotop
es
Chemists can use
isotope analysis
to assist analysts with issues
in
anthropology
,
archeology
,
food chemistry
,
forensic
s
,
geology
, and a host of other questions
of
physical science
. Analysts can discern the origins of natural
and man

made isotopes in the
study of
environmental radioactivity
.
Business
Financial statement analysis
–
the analysis of the accounts and the economic prospects of a firm
Fundamental analysis
–
a stock
valuation method that uses financial analysis
Technical analysis
–
the study of price action in securities markets in order to forecast future prices
Business analysis
–
involves identifying the needs and determining the solutions to business
problems
Price
analysis
–
involves the breakdown of a price to a unit figure
Market analysis
–
consists of suppliers and customers, and price is determined by the interaction
of
supply and demand
Computer science
Requirements analysis
–
encompasses those tasks that
go into determining the needs or conditions to
meet for a new or altered product, taking account of the possibly conflicting requirements of the
various stakeholders, such as beneficiaries or users.
Competitive analysis (online algorithm)
–
shows how online algorithms perform and demonstrates
the power of randomization in algorithms
Lexical analysis
–
the process of processing an input sequence of characters and producing as output
a sequence of symbols
Object

oriented analysis and design
–
à la
Booch
Program analysis (computer science)
–
the process of automatically analyzing the behavior of
computer programs
Semantic analysis (computer science)
–
a pass by a compiler that adds semantical information to the
parse tree and performs certain checks
Static code analysis
–
the analysis of computer software that is performed without actually executing
programs built from that
Structured systems analysis and design methodology
–
à la
Yourdon
Syntax analysis
–
a process in compilers that recognizes the structure of programming languages, also
known as parsing
Worst

case execution time
–
determines the longest time that a piece of software can take to run
Economics
Agroecosystem analysis
Input

output model
if applied to a region, is called Regional Impact Multiplier System
Principal components analysis
–
a technique that can be used to simplify a dataset
Engineering
See also:
Engineering analysis
and
Systems analysis
Analysts in the field of
engineering
look at
requirements
,
structures
,
mechanisms,
systems
and
dimensions
.
Electrical engineers
analyze
systems
in
electronics
.
Life
cycles
and
system failures
are broken down and studied by engineers. It is also looking at
different factors incorporated within the design.
Intelligence
The field
of
intelligence
employs analysts to break down and understand a wide array of
questions.
Intelligence agencies
may use
heuristics
,
induc
tive
and
deductive reasoning
,
social
network analysis
,
dynamic network analysis
,
link analysis
, and
brainstorming
to sort through
problems they face.
Military intelligence
may explore issues through the use of
game theory
,
Red
Teaming
, and
wargaming
.
Signals intelligence
applies
crypt
analysis
and
frequency analysis
to
break
codes
and
ciphers
.
Business intelligence
applies theories of
competitive intelligence
analysis
and
competitor analysis
to resolve questions in the
marketplace
.
Law
enforcement
intelligence applies a number of theories in
crime analysis
.
Linguistics
Linguistics began with the analysis of
Sanskrit
and
Tamil
; today it looks at individual lan
guages
and
language
in general. It breaks language down and analyzes its component
parts:
theory
,
sounds and their meaning
,
utterance usage
,
word origins
, the
history of words
, the
meaning of
words
and
word combinations
,
sentence construction
,
basic construction beyond the
sentence level
,
stylistics
, and
conversation
. It examines the above using
statistics and modeling
,
and
semantics
. It analyzes language in context
of
anthropology
,
biology
,
evolution
,
geography
,
history
,
neurology
,
psychology
, and
sociology
. It
also takes the
applied
approach, looking at
individual language development
and
clinic
al
issues.
Literature
Literary theory
is the analysis of
literature
. Some say that
literary criticism
is a subset of literary
theory. The focus can be as diverse as the analysis of
Homer
or
Freud
. This is mainly to do with
the breaking up of a topic to make it easier to understand.
Mathematics
Mathematical
analysis is the study of infinite processes. It is the branch of mathematics that
includes calculus. It can be applied in the study of
classical
concepts of
mathematics, such as
real
numbers
,
complex variables
,
trigonometric functions
, and
algorithms
, or of
non

classical
concepts like
constructivism
,
harmonics
,
infinity
, and
vectors
.
Music
Musical analysis
–
a process attempting to answer the question "How does this music work?"
Schenkerian analysis
Philosophy
Philosophical analysis
–
a general term for the techniques used by philosophers
Analysis
is the name of a prominent journal in philosophy.
Psychotherapy
Psychoanalysis
–
seeks to elucidate connections among
unconscious components of patients' mental
processes
Transactional analysis
Signal processing
Finite element analysis
–
a computer simulation technique used in engineering analysis
Independent component
analysis
Link quality analysis
–
the analysis of signal quality
Path
quality analysis
Statistics
In
statistics
, the term
analysis
may refer to any method used for
data analys
is
. Among the many
such methods, some are:
Analysis of variance
(ANOVA)
–
a collection of statistical models and their associated procedures
which compare means by
splitting the overall observed variance into different parts
Boolean analysis
–
a method to find deterministic dependencies between variables in a sample,
mostly used in
exploratory data analysis
Cluster analysis
–
techniques for grouping objects into a collection of groups (called clusters), based
on some measure of proximity or similarity
Factor analysis
–
a method to construct models describing a data set of observed variables in terms of
a smaller set of unobserved variables (called factors)
Meta

analysis
–
combines the results of several studies that address a set of related research
hypotheses
Multivariate analysis
–
analysis of data involving several variables, such as by factor analysis,
regression analysis, or principal component analysis
Principal component analysis
–
transformation of a sample of correlated variables into uncorrelated
variables (called principal components), mostly used in exploratory data analysis
Regression analysis
–
techniques for analyzing the relationships between several variables in the data
Scale analysis (stati
stics)
–
methods to analyse survey data by scoring responses on a numeric scale
Sensitivity analysis
–
the study of how the variation in the output of a model depen
ds on variations in
the inputs
Sequential analysis
–
evaluation of sampled data as it is collected, until the criterion of a stopping
rule is met
Spatial analysis
–
the study of entities using geometric or geographic properties
Time

series analys
is
–
methods that attempt to understand a sequence of data points spaced apart at
uniform time intervals
Other
Aura analysis
–
a technique in which supporters of the method claim
that the body's aura, or energy
field is analyzed
Bowling analysis
–
a notation summarizing a cricket bowler's performance
Lithic analysis
–
the analysis of stone tools using basic scientific techniques
Protocol a
nalysis
–
a means for extracting persons' thoughts while they are performing a task
3
.
Scalar potential
A
scalar
potential
is a fundamental concept in
vector analysis
and
physics
(the adjective
scalar
is frequently omitted if there is
no danger of confusion wit
h
vector potential
). The scalar potential is an example of a
scalar field
. Given a
vector field
F
, the
scalar potential
P
is defined such that:
,
[1]
where
∇
P 楳⁴he
g牡d楥it
of P and 瑨e 獥cond pa牴rof 瑨e equa瑩on 楳im楮u猠瑨e g牡d楥i琠fo爠a func瑩on of 瑨e
䍡牴敳楡i
coo牤ina瑥t
x,y,z.
[2]
In some cases, mathematicians may use a positive sign in front of the gradient to define the
potential.
[3]
Because of this definition of P in terms of the gradient, the direction of
F
at any point is the direction of the
steepest decrease of P at that point, its magnitude is the rate of that decrease per unit le
ngth.
In order for
F
to be described in terms of a scalar potential only, the following have to be true:
1.
, where the integration is over a
Jordan arc
passing from
location
a
to
location
b
and P(
b
) is P evaluated at location
b
.
2.
, where the integral is over any simple closed path, otherwise known as a
Jordan curve
.
3.
The first of these conditions represen
ts the
fundamental theorem of the gradient
and is true for any vector field that is a
gradient of a
differentiable
single valued
scalar field P. The second condition is a requirement of
F
so that it can be
expressed as the gradient of a scalar function
. The third condition re

expresses the second condition in terms of
the
curl
of
F
using the
fundamental theorem of the curl
. A vector field
F
that satisfies these conditions is said to
be
irrotational
(Conservative).
Scalar potentials play a prominent role in many areas of physics and engineering. The
gravity potential
is the scalar
p
otential associated with the gravity per unit mass, i.e., the
acceleration
due to the field, as a function of position. The
gravity potential is the gravitational
potential energy
per unit mass. In
electrostatics
the
electric potential
is the scalar
potential associated with the
electric field
, i.e., with the
electrostatic force
per unit
charge
. The electric potential is in this
case the electrostatic potential energy per unit charge. In
fluid dynamics
, irrotational
lamellar fields
have a scalar potential
only in the special case when
it is a
Laplacian field
. Certain aspects of the
nuclear force
can be described by a
Yukawa
potential
. The potential play a prominent role in the
Lagrangian
and
Hamiltonian
formulations of
classical mechanics
.
Further, the scalar potential is the fundam
ental quantity in
quantum mechanics
.
Not every vector field has a scalar potential. Those that do are called
conservative
, corresponding to the notion
of
conservative force
in physics. Examples of non

conservative forces include frictio
nal forces, magnetic forces, and in fluid
mechanics a
solenoidal field
velocity field. By the
Helmholtz decomposition
theorem however, all vector fields can be
describable in terms of a scalar potential and corresponding
vector potential
. In electrodynamics the electromagnetic scalar
and vector potentials are known together as the
electromagnetic four

potential
.
Integrability conditions
If
F
is a
conservative vector field
(also called
irrotational
,
curl

free
, or
potential
), and its components
have
continuous
partial derivatives
, the
potential
of
F
with respect to a reference point
is defined in terms of the
line
integral
:
where
C
is a parametrized path from
to
The fact that the line integral depends on the path
C
only through its terminal points
and
is, in essence,
the
path independence property
of a conservative vector field. The
fundamental theorem of calculus
for line
integrals implies that if
V
is defined in this way, then
so that
V
is a scalar potential of the
conse
rvative vector field
F
. Scalar potential is not determined by the vector field alone: indeed, the gradient of a
function is unaffected if a constant is added to it. If
V
is defined in terms of the line integral, the ambiguity
of
V
reflects the freedom in t
he choice of the reference point
Altitude as gravitational potential energy
An example is the (nearly) uniform
gravitational field
near the Earth's surface. It has
a potential energy
where
U
is the gravitational potential energy and
h
is the height above the surface. This means that
gravitational potential energy on a
contour map
is proportio
nal to altitude. On a contour map, the two

dimensional negative gradient of the altitude is a two

dimensional vector field, whose vectors are always
perpendicular to the contours and also perpendicular to the direction of gravity. But on the hilly region
r
epresented by the contour map, the three

dimensional negative gradient of
U
always points straight
downwards in the direction of gravity;
F
. However, a ball rolling down a hill cannot move directly
downwards due to the normal force of the hill's surface, w
hich cancels out the component of gravity
perpendicular to the hill's surface. The component of gravity that remains to move the ball is parallel to the
surface:
where
θ
is the angle of inclination, and the component of
F
S
perpendicular to gravity is
This force
F
P
, parallel to the ground, is greatest when
θ
is 45 degrees.
Let Δ
h
be the uniform interval of altitude between contours on the contour map, and let Δ
x
be the
distance between two contours. Then
so that
However, on a contour map, the gradie
nt is inversely proportional to Δ
x
, which is not
similar to force
F
P
: altitude on a contour map is not exactly a two

dimensional
potential field. The magnitudes of forces are different, but the directions of the forces
are the same on a contour map as well
as on the hilly region of the Earth's surface
represented by the contour map.
Pressure as buoyant potential
In
fluid mechanics
, a fluid in equilibrium, but in the presence o
f a uniform
gravitational field is permeated by a uniform buoyant force that cancels out the
gravitational force: that is how the fluid maintains its equilibrium. This
buoyant
force
is the
negative gradient of
pressure
:
Since buoyant force points upwards, in the direction opposite to gravity, then
pressure in the fluid increases downwards. Pressure in a static body of wat
er
increases proportionally to the depth below the surface of the water. The surfaces
of constant pressure are planes parallel to the ground. The surface of the water
can be characterized as a plane with zero pressure.
If the liquid has a vertical
vortex
(whose axis of rotation is perpendicular to the
ground), then the vortex causes a depression in the pressure field. The surfaces of
constant pressure are parallel to the ground far away from th
e vortex, but near
and inside the vortex the surfaces of constant pressure are pulled downwards,
closer to the ground. This also happens to the surface of zero pressure. Therefore,
inside the vortex, the top surface of the liquid is pulled downwards into a
depression, or even into a tube (a solenoid).
The buoyant force due to a fluid on a solid object immersed and surrounded by
that fluid can be obtained by integrating the negative pressure gradient along the
surface of the object:
A moving airplane wing
makes the air pressure above it decrease relative to
the air pressure below it. This creates enough buoyant force to counteract
gravity.
[
.
]
Calculating the scalar potential
Given a vector field
E
, its scalar potential
Φ
can be calculated to be
where
τ
is volume. Then, if
E
is
irrotational
(Conservative),
This formula is known to be correct if
E
is
continuous
and
vanishes asymptotically to zero towards infinity, decaying faster
than 1/
r
and if the
divergence
of
E
likewise vanishes towards
infinity, decaying faster than 1/
r
2
.
4.
Applications
Computational geometry
The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed
in
computer graphics
. For example, the winding of polygon (clockwise or anticlockwise) about a point within the
polygon (i.e. the centroid or midpoint) can be calculated by tr
iangulating the polygon (like spoking a wheel) and
summing the angles (between the spokes) using the cross product to keep track of the sign of each angle.
In
c
omputational geometry
of
the plane
, the cross product is used to determine the sign of the
acute angle
defined b
y
three points
,
and
. It corresponds to the direction of the cross product of
the two coplanar
vectors
defined by the pairs of points
and
, i.e., by the sign of t
he
expression
. In the "right

handed" coordinate system, if the result is 0,
the points are collinear; if it is positive, the three points constitute a negative angle of rotation around
from
to
, otherwise a positive angle. From another point of view
, the sign of
tells whether
lies to the left or to
the right of line
.
Mechanics
Moment
of a force
applied at point B around point A is given as:
Other
The cross
product occurs in the formula for the
vector operator
curl
. It is also used to describ
e the
Lorentz
force
experienced by a moving electrical charge in a magnetic field. The definitions of
torque
and
angular
momentum
also involve the cross product.
The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in epipolar and
multi

view geo
metry, in particular when deriving matching constraints.
Cross product as an exterior product
The cross product can be viewed in terms of the
exterior product
. This view al
lows for a natural geometric
interpretation of the cross product. In
exterior algebra
the exterior product (or wedge product) of two vectors is
a
bivector
. A bivector is an oriented plane element, in much the same way that a vector is an oriented line
element. Given two vectors
a
and
b
, one can view the bivector
a
∧
b
as the oriented parallelogram spanned
by
a
and
b
. The cross product is then obtained by taking the
Hodge dual
of the bivector
a
∧
b
, mapping
2

vectors
to vecto
rs:
This can be thought of as the oriented multi

dimensional element "perpendicular" to the bivector. Only in
three dimensions is the result an oriented line element
–
a vector
–
whereas, for example, in 4 dimensions
the Hodge dual of a bivector
is two

dimensional
–
another oriented plane element. So, only in three
dimensions is the cross product of
a
and
b
the vector dual to the bivector
a
∧
b
: it is perpendicular to the
bivector, with orientation dependent on the coordinate system's handedness,
and has the same magnitude
relative to the unit normal vector as
a
∧
b
has relative to the unit bivector; precisely the properties
described above.
Cross product and handedness
When measurable quantities involve cross products, the
handedness
of the
coordinate systems used cannot
be arbitrary. However, when physics laws are written as equations, it should be possible to make an
arbitrary choice of the coordinate system (including handedness). To avoid problems, one should be
careful to never write dow
n an equation where the two sides do not behave equally under all
transformations that need to be considered. For example, if one side of the equation is a cross product of
two vectors, one must take into account that when the handedness of the coordinate
system is
not
fixed a
priori, the result is not a (true) vector but a
pseudovector
. Therefore, for consistency, the other
side
must
also be a pseudovector
.
More generally, the result
of a cross product may be either a vector or a
pseudovector, depending on the type of its operands (vectors or pseudovectors). Namely, vectors and
pseudovectors are interrelated in the following ways under application of the cross product:
vector × vector
= pseudovector
pseudovector × pseudovector = pseudovector
vector × pseudovector = vector
pseudovector × vector = vector.
So by the above relationships, the unit basis vectors
i
,
j
and
k
of an orthonormal, right

handed (Cartesian)
coordinate frame
must
all
be pseudovectors (if a basis of mixed vector types is disallowed, as it normally
is) since
i
×
j
=
k
,
j
×
k
=
i
and
k
×
i
=
j
.
Because the cross product may also be a (true) vector, it may not change direction with a mirror image
transformation. This happens, according to the above relationships, if one of the operands is a (true) vector
and the other one is a pseudovector (
e.g.
,
the cross product of two vectors). For instance, a
vector triple
product
involving three (true) vectors is a (true) vector.
A handedness

free approach is possible
using
exterior algebra
.
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