# MULTIVARIATE ANALYSIS-ANSx - Our University

AI and Robotics

Nov 18, 2013 (4 years and 5 months ago)

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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,

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:

;

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]

-
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

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
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
, 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

u

+ (
v

+

w
) = (
u

+

v
) +

w

Commutativity

u

+

v

=

v

+

u

Identity element

There exists an element

0

V
, 捡汬敤 瑨e

, 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

v

V, there exists an element −
v

V
, 捡汬敤 瑨e

v
, such that

v

+ (−
v
) =

0

Distributivity

of scalar
multiplication with respect to vector

a
(
u

+

v
) =

a
u

+

a
v

Distributivity of scalar
multiplication with respect to field

(
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

,

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

v

of any vector

v

are unique. Ot
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
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

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
-
study of

.

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

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

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

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
,

, 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
.

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

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

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

and is true for any vector field that is 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
-

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

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
.