chapter 4, Vector Analysis

Excerpt

Historical Comments: The word “vector” comes from a Latin word meaning “to carry,” but actually entered mathematics through astronomy (where it had a different meaning). The word vector and the idea that a force is a vector quantity was known to Aristotle. Also, Galileo Galilei (1564–1642) explicitly stated the parallelogram law for the vector addition of forces. Nonetheless, vector algebra and vector analysis are both considered to be a product of the nineteenth century.

It was known in 1830 that complex numbers could be used to represent vectors in a plane. The search for a “three-dimensional complex number” eventually led to the invention of the “quaternion” in 1843 by William Rowan Hamilton (1805–1865). A quaternion is something of a cross between a complex number and a vector. For example, if the real part of a quaternion is zero, the part that is left can be identified with an ordinary vector. Hamilton felt that quaternions were well suited for solving problems in physics—in fact, strong efforts were made by several mathematicians over half a century to introduce quaternions into physics. Eventually, the quaternion concept was considered unacceptable for applications.

The next significant development in vector analysis took place when the idea of “curl” appeared in 1873 in the revolutionary work of James C. Maxwell (1831–79) on electromagnetic waves. Shortly afterward, W. L. Clifford (1845–79) coined the term “divergence.” The notation that is commonly used today in vector analysis is due primarily to the work of J. Willard Gibbs (1839–1903) in 1893, but also to some extent the work of Oliver Heaviside (1850–1925) on electromagnetic theory. The two basic integral theorems—that of Stokes and the divergence theorem—were actually developed before Gibbs. In 1831, for example, the Russian M. Ostogradsky (1801–61) converted a volume integral into a surface integral (equivalent to the divergence theorem). The divergence theorem can also be found in the work of Carl F. Gauss (1777–1855) and in that of the English mathematician George Green (1793–1841). Oddly enough, the theorem attributed to George G. Stokes (1819–1903) was used as a question in a prize examination at Cambridge in 1854.

The words “scalar,” “vector,” and “tensor” were all used by Hamilton. However, tensor calculus was perfected in 1889 and 1926–27, respectively, by the Italian mathematicians G. Ricci (1853–1925) and T. Levi-Civita (1873–1941). Albert Einstein (1879–1955) used the term “tensor” only in connection with the transformation laws, but he was first in applying the generalized calculus of tensors to problems in gravitation.