''Vector Analysis'' is a textbook by
Edwin Bidwell Wilson, first published in 1901 and based on the lectures that
Josiah Willard Gibbs had delivered on the subject at
Yale University. The book did much to standardize the
notation and vocabulary of three-dimensional
linear algebra and
vector calculus, 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 in 1960.
Contents
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 ...
, and the product of a scalar with a vector. The second chapter introduces the
dot 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 is ...
s for pairs of vectors. These are extended to a
scalar triple product and a quadruple product. Pages 77–81 cover the essentials of
spherical trigonometry, 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 of ...
. The third chapter introduces the vector calculus notation based on the
del operator. The
Helmholtz decomposition of a
vector field is given on page 237.
The final eight pages develop
bivectors as these were integral to the course on the
electromagnetic theory of
light that Professor Gibbs taught at Yale. First Wilson associates a bivector with an ellipse. The product of the bivector with a
complex number on the
unit circle is then called an ''elliptical rotation''. Wilson continues with a description of ''elliptic harmonic motion'' and the case of
stationary wave
In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
s.
Genesis
Hermann Grassmann had introduced basic ideas of a
linear space in 1844 and 1862, and
W. K. Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in ...
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
bivectors, 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. Hamilton defined a quatern ...
product into
scalar product 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 is ...
.
In 1888 Gibbs sent a copy of his pamphlet to
Oliver Heaviside 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'' 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 American mathematician and educator. He taught at Harvard University for almost 50 years.
Early life and family
He was the eldest son of Sarah Hunt (Mills) Peirce and Benjamin Peirce (18 ...
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. Wilson wrote the book by expanding his own class notes, providing
exercise
Exercise is a body activity that enhances or maintains physical fitness and overall health and wellness.
It is performed for various reasons, to aid growth and improve strength, develop muscles and the cardiovascular system, hone athletic ...
s, and consulting with others (including his father).
[ Edwin Bidwell Wilson (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 (review) ''Jahrbuch über die Fortschritte der Mathematik'' 33:96–7.
*
Cargill Gilston Knott (review) Philosophical Magazine'' 6th Ser, 4:614–22.
*
Michael J. Crowe Michael J. Crowe (b. 1936) is Rev. John J. Cavanaugh Professor Emeritus in the Program of Liberal Studies and
Graduate Program in History and Philosophy of Science at the University of Notre Dame. He is best known for writing the influential book '' ...
(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. Gibbsat
Internet Archive.
* {{cite book , others=Founded upon the lectures of J. William Gibbs , author=Edwin Bidwell Wilson , title=
Vector Analysis , publisher=Yale University Press , place=New Haven , year=1913 , via=
Wikimedia Commons
1901 non-fiction books
Mathematics textbooks
Vector calculus