''Vector Analysis'' is a textbook by

Edwin Bidwell Wilson Edwin Bidwell Wilson (April 25, 1879 – December 28, 1964) was an American mathematician, statistician, physicist and general polymath. He was the sole protégé of Yale University physicist Josiah Willard Gibbs and was mentor to MIT economist ...

, first published in 1901 and based on the lectures that

Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American mechanical engineer and scientist who made fundamental theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynami ...

had delivered on the subject at

Yale University Yale University is a Private university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...

. The book did much to standardize the

notation In linguistics and semiotics, a notation system is a system of graphics or symbols, Character_(symbol), characters and abbreviated Expression (language), expressions, used (for example) in Artistic disciplines, artistic and scientific disciplines ...

and vocabulary of three-dimensional

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

and

vector calculus Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...

, as used by physicists and mathematicians. It was reprinted by Yale in 1913, 1916, 1922, 1925, 1929, 1931, and 1943. The work is now in the public domain. It was reprinted by

Dover Publications Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, book ...

in 1960.

The book carries the subtitle "A text-book for the use of students of mathematics and physics. Founded upon the lectures of J. Willard Gibbs, Ph.D., LL.D." The first chapter covers vectors in three spatial dimensions, the concept of a (real)

scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...

, and the product of a scalar with a vector. The second chapter introduces the

dot A dot is usually a small, round spot. Dot, DoT or DOT may also refer to: Orthography * Full stop or "period", a sentence terminator * Dot (diacritic), a mark above or below a character (e.g. ȧ, ạ, İ, Ċ, ċ, etc.), usually to indicate sou ...

and

cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...

s for pairs of vectors. These are extended to a

scalar triple product In geometry and algebra, the triple product is a product of three 3- dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vect ...

and a quadruple product. Pages 77–81 cover the essentials of

spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, ge ...

, a topic of considerable interest at the time because of its use in

celestial navigation Celestial navigation, also known as astronavigation, is the practice of position fixing using stars and other celestial bodies that enables a navigator to accurately determine their actual current physical position in space or on the surface ...

. The third chapter introduces the vector calculus notation based on the

del operator Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes t ...

. The

Helmholtz decomposition In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational ( curl-free) vector field and a sole ...

of a

vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...

is given on page 237. The final eight pages develop

bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of ...

s as these were integral to the course on the

electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...

theory of

light Light, visible light, or visible radiation is electromagnetic radiation that can be visual perception, perceived by the human eye. Visible light spans the visible spectrum and is usually defined as having wavelengths in the range of 400– ...

that Professor Gibbs taught at Yale. First Wilson associates a bivector with an ellipse. The product of the bivector with a

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

on the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

is then called an ''elliptical rotation''. Wilson continues with a description of ''elliptic harmonic motion'' and the case of stationary waves.

Genesis

Hermann Grassmann Hermann Günther Grassmann (, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was littl ...

had introduced basic ideas of a

linear space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''. The operations of vector addition and sc ...

in 1844 and 1862, and W. K. Clifford published ''

Elements of Dynamic ''Elements of Dynamic'' is a book published by William Kingdon Clifford in 1878. In 1887 it was supplemented by a fourth part and an appendix. The subtitle is "An introduction to motion and rest in solid and fluid bodies". It was reviewed positiv ...

'' in 1878, so as Gibbs was teaching physics in the 1880s he took these developments into consideration for his students. A pamphlet that he printed for them acknowledges both Grassmann and Clifford. The influence of Grassmann is seen in the

s, and the influence of Clifford in the decomposition of the

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...

product into

scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused wit ...

and

. In 1888 Gibbs sent a copy of his pamphlet to

Oliver Heaviside Oliver Heaviside ( ; 18 May 1850 – 3 February 1925) was an English mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vector calculus, an ...

who was formulating his own vectorial system in the ''Transactions of the Royal Society'', praised Gibbs' "little book", saying it "deserves to be well known". However, he also noted that it was "much too condensed for a first introduction to the subject". On the occasion of the bicentennial of Yale University, a series of publications were to be issued to showcase Yale's role in the advancement of knowledge. Gibbs was authoring ''

Elementary Principles in Statistical Mechanics ''Elementary Principles in Statistical Mechanics'', published in March 1902, is a work of scientific literature by Josiah Willard Gibbs which is considered to be the foundation of modern statistical mechanics. Its full title was ''Elementary Prin ...

'' for that series. Mindful of the demand for innovative university textbooks, the editor of the series, Professor Morris, wished to include also a volume dedicated to Gibbs's lectures on vectors, but Gibbs's time and attention were entirely absorbed by the ''Statistical Mechanics''. E. B. Wilson was then a new graduate student in mathematics. He had learned about quaternions from

James Mills Peirce James Mills Peirce (May 1, 1834 – March 21, 1906) was an Americans, American mathematician and educator. He taught at Harvard University for almost 50 years. Early life and family Peirce was born May 1, 1834, in Cambridge, Massachusetts. He wa ...

at Harvard, but Dean A. W. Phillips persuaded him to take Gibbs's course on vectors, which treated similar problems from a rather different perspective. After Wilson had completed the course, Morris approached him about the project of producing a

textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions, but also of learners ( ...

. Wilson wrote the book by expanding his own class notes, providing

exercise Exercise or workout is physical activity that enhances or maintains fitness and overall health. It is performed for various reasons, including weight loss or maintenance, to aid growth and improve strength, develop muscles and the cardio ...

s, and consulting with others (including his father).

(1931
"Reminiscences of Gibbs by a student and colleague"
''Bulletin of the American Mathematical Society.'' Volume 37, Number 6, 401–416. File:Wilson-1.jpg, 1907 copy of Vector Analysis File:Wilson-3.jpg, Preface to Vector Analysis (1907) File:Wilson-4.jpg, Table of contents to Vector Analysis (1907) File:Wilson-5.jpg, First page of Vector Analysis (1907)

References

* Alexander Ziwet (1902
Review of ''Vector Analysis''
''Bulletin of the American Mathematical Society'' 8:207–15. * Anon. (review) ''Bulletin des sciences mathématiques'' 26:21–30. *

Victor Schlegel Victor Schlegel (4 March 1843 – 22 November 1905) was a German mathematician. He is remembered for promoting the geometric algebra of Hermann Grassmann and for a method of visualizing polytopes called Schlegel diagrams. In the nineteenth ce ...

(review) ''Jahrbuch über die Fortschritte der Mathematik'' 33:96–7. *

Cargill Gilston Knott Cargill Gilston Knott FRS, FRSE LLD (30 June 1856 – 26 October 1922) was a Scottish physicist and mathematician who was a pioneer in seismological research. He spent his early career in Japan. He later became a Fellow of the Royal Society, ...

(review) Philosophical Magazine'' 6th Ser, 4:614–22. * Michael J. Crowe (1967) ''

A 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 i ...

'', Notre Dame University Press.

External links

* E. B. Wilson (1901
Vector Analysis, based on the Lectures of J. W. Gibbs
at

Internet Archive The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...

. * {{cite book , others=Founded upon the lectures of J. William Gibbs , author=Edwin Bidwell Wilson , title=

Vector Analysis Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...

, publisher=Yale University Press , place=New Haven , year=1913 , via=

Wikimedia Commons Wikimedia Commons, or simply Commons, is a wiki-based Digital library, media repository of Open content, free-to-use images, sounds, videos and other media. It is a project of the Wikimedia Foundation. Files from Wikimedia Commons can be used ...

1901 non-fiction books Mathematics textbooks Vector calculus

Contents

Genesis

References

External links