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 had 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.
Summary of book
The book has eight chapters: the first on the origins of vector analysis including Ancient Greek and 16th and
17th century influences; the second on the 19th century William Rowan Hamilton and quaternions; the third on
other 19th and 18th century vectorial systems; the fourth on the general interest in the 19th century on vectorial
systems including analysis of journal publications as well as sections on major figures and their views
(e.g., Peter Guthrie Tait as an advocate of Quaternions and James Clerk Maxwell as a critic of Quaternions);
the fifth on Josiah Willard Gibbs and Oliver Heaviside and their development of a modern system of vector
In chapter six, "Struggle for existence", Michael J. Crowe delves into the zeitgeist that pruned down quaternion
theory into vector analysis on three-dimensional space. He makes clear the ambition of this effort by
considering five major texts as well as a couple dozen articles authored by participants in "The Great Vector
Debate". These are the books:
Elementary Treatise on Quaternions (1890) Peter Guthrie Tait
Elements of Vector Analysis (1881,1884) Josiah Willard Gibbs
Electromagnetic Theory (1893,1899,1912) Oliver Heaviside
Utility of Quaternions in Physics (1893) Alexander MacAulay
Vector Analysis and Quaternions (1906) Alexander Macfarlane
Twenty of the ancillary articles appeared in Nature; others were in Philosophical
Magazine, London or Edinburgh Proceedings of the Royal Society, Physical Review,
and Proceedings of the American Association for the Advancement of Science. The
authors included the book authors, Cargill Gilston Knott, and a half-dozen other hands.
The "struggle for existence" is a phrase from Charles Darwin’s Origin of Species and
Crowe quotes Darwin: "…young and rising naturalists,…will be able to view both sides
of the question with impartiality." After 1901 with the Gibbs/Wilson/Yale
publication Vector Analysis, the question was decided in favour of the vectorialists with
separate dot and cross products. The pragmatic temper of the times set aside the fourdimensional source of vector algebra.
Crowe's chapter seven is a survey of "Twelve major publications in Vector Analysis from
1894 to 1910". Of these twelve, seven are in German, two in Italian, one in Russian, and
two in English. Whereas the previous chapter examined a debate in English, the final
chapter notes the influence of Heinrich Hertz' results with radio and the rush of German
research using vectors. Joseph George Coffin of MIT and Clark University published
his Vector Analysis in 1909; it too leaned heavily into applications. Thus Crowe provides
a context for Gibbs and Wilson’s famous textbook of 1901.
The eighth chapter is the author's summary and conclusions.[2] The book relies on
references in chapter endnotes instead of a bibliography section. Crowe also states that
theBibliography of the Quaternion Society, and its supplements to 1912, already listed
all the primary literature for the study.
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
multiplication 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:
although these are less often used as basic operations, as they can be
expressed in terms of the dot and cross products.
Vector calculus studies various differential operators defined on scalar or vector
fields, which are typically expressed in terms of the del operator ( ), also
known as "nabla". The four most important differential operations in vector
calculus are:
Nonlinear generalizations
Linear PCA versus nonlinear Principal Manifolds [22] forvisualization 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 be 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 curves 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 andprincipal geodesic analysis.
Multilinear generalizations
In multilinear subspace learning, PCA is generalized to multilinear PCA (MPCA) that extracts features
directly from tensor representations. 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 principal component analysis may be performed with models such as Tucker
decomposition, PARAFAC, multiple 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 robustness by assigning different weights to data objects based on their estimated
A vector space over a field F is a set V together with two binary operations that satisfy the eight axioms
listed below. Elements of V are called vectors. Elements of F are calledscalars. In this article, vectors are
distinguished from scalars by boldface.[nb 1] In the two examples above, our set consists 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 av. 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 space, 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 wbe arbitrary vectors in V,
and a and b scalars in F.
Associativity of addition
u + (v + w) = (u + v) + w
Commutativity of addition
Identity element of addition
There exists an element 0 ∈ V, called the zero vect
Inverse elements of addition
For every v ∈ V, there exists an element −v ∈ V, ca
of scalar multiplication with respect to vector addition a(u + v) = au + av
Distributivity of scalar multiplication with respect to field addition
(a + b)v = av + bv
Compatibility of scalar multiplication with field multiplication
a(bv) = (ab)v [nb 2]
Identity element of scalar multiplication
1v = v, where 1 denotes the multiplicative identity i
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
(xv, yv) + (xw, yw) = (xw, yw) + (xv, yv),
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 cases,
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.
Alternative 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 av are in Vfor
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 av is defined as
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. Other
properties follow from the distributive law, for example av equals 0 if and only if a equals 0
or v equals0.
Main article: 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
(a1, a2, ..., an), where each ai 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 Fn. 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
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 where
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)
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
algebra and algebraic number theory: a field F containing a smaller field E is anE-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 similarly 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
set 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) = r0 + r1x + ... + rn−1xn−1 + rnxn, where the coefficients r0, ..., rn are in F.[15]
Linear equations
Main articles: 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 + 3b + c = 0
4a + 2b + 2c = 0
are given by triples with arbitrary a, b = a/2, and c = −5a/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
equations as above into one vector equation, namely
Ax = 0,
where A =
is the matrix containing the coefficients of the given
equations, x is the vector (a, b, c), Ax 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 ex is
the natural exponential function.
See also: Analytical chemistry and List of chemical analysis methods
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 can 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 element
of national security among the major world powers with materials measurement and signature
intelligence (MASINT) capabilities.
See also: Isotope analysis and Isotope geochemistry
Chemists can use isotope analysis to assist analysts with issues
in anthropology, archeology, food chemistry, forensics, 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.
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
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
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
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 andsystem failures are broken down and studied by engineers. It is also looking at
different factors incorporated within the design.
See also: Intelligence analysis
The field of intelligence employs analysts to break down and understand a wide array of
questions. Intelligence agencies may use heuristics, inductive 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 ofgame theory, Red
Teaming, and wargaming. Signals intelligence applies cryptanalysis and frequency analysis to
break codes and ciphers. Business intelligence applies theories ofcompetitive intelligence
analysis and competitor analysis to resolve questions in the marketplace. Law
enforcement intelligence applies a number of theories in crime analysis.
See also: Linguistics
Linguistics began with the analysis of Sanskrit and Tamil; today it looks at individual languages
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
ofanthropology, biology, evolution, geography, history, neurology, psychology, and sociology. It
also takes the applied approach, looking at individual language development andclinical issues.
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.
Main article: Mathematical analysis
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 nonclassical concepts like constructivism, harmonics, infinity, and vectors.
Musical analysis – a process attempting to answer the question "How does this music work?"
Schenkerian analysis
Philosophical analysis – a general term for the techniques used by philosophers
Analysis is the name of a prominent journal in philosophy.
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
In statistics, the term analysis may refer to any method used for data analysis. 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
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
Scale analysis (statistics) – 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 depends 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 analysis – methods that attempt to understand a sequence of data points spaced apart
at uniform time intervals
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 analysis – a means for extracting persons' thoughts while they are performing a task
Scalar potential
From Wikipedia, the free encyclopedia
This article is about a general description of a function used in mathematics and physics to describe
conservative fields. For the scalar potential of electromagnetism, seeelectric potential. For all other uses,
see 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 with vector potential). The scalar potential is an example of a scalar
field. Given a vector field F, the scalar potential Pis defined such that:
where ∇P is the gradient of P and the second part of the equation is minus the gradient for a function of
the Cartesian coordinatesx,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 length.
In order for F to be described in terms of a scalar potential only, the following have to be true:
, where the integration is over a Jordan arc passing from
location a to location b and P(b) is P evaluated at location b .
, where the integral is over any simple closed path, otherwise known as
a Jordan curve.
The first of these conditions represents 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 ofF 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 potential associated with the gravity per unit mass, i.e., theacceleration 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. Influid 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 aYukawa potential.
The potential play a prominent role in the Lagrangian and Hamiltonian formulations of classical mechanics.
Further, the scalar potential is the fundamental 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 frictional 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 theelectromagnetic 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
The fact that the line integral depends on the path C only through its terminal points
in essence, the path independence property of a conservative vector field. Thefundamental
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 conservative 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 the 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
where U is the gravitational potential energy and h is the height above the surface. This
means that gravitational potential energy on acontour map is proportional 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 represented by the contour map, the threedimensional 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, which 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
where θ is the angle of inclination, and the component of FS perpendicular to gravity is
This force FP, parallel to the ground, is greatest when θ is 45 degrees.
Let Δh be the uniform interval of altitude between contours on the contour map, and
let Δx be the distance between two contours. Then
so that
However, on a contour map, the gradient is inversely proportional to Δx,
which is not similar to force FP: 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 of 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 water 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 the 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.
the scalar potential
Given a vector field E, its scalar potential Φ can be calculated to
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/r2.
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 triangulating 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 computational geometry of the plane, the cross product is used to determine the sign of the acute
angle defined by three points
. It corresponds to the direction
of the cross product of the two coplanar vectors defined by the pairs of points
, i.e., by
the sign of the 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
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
Moment of a force
applied at point B around point A is given as:
The cross product occurs in the formula for the vector operator curl. It is also used to describe
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 geometry, 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 allows 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-vectorsto vectors:
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 down 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.[citation
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.

A History of Vector Analysis