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Elements of Vector Analysis: Arranged for the Use of Students in Physics
Excerpt from Elements of Vector Arranged for the Use of Students in Physics
(The three letters, i, j, k, will make an exception, to be mentioned more particularly hereafter.
2. Def. - Vectors are said to be equal when they are the same both in direction and in magnitude. The reader will observe that this vector equation is the equivalent of three scalar equations.
A vector is said to be equal to zero, when its magnitude is zero. Such vectors may be set equal to one another, irrespectively of any considerations relating to direction.
3. Perhaps the most simple example of a vector is afforded by a directed straight line, as the line drawn from A to B. We may use the notation AB to denote this line as a vector, i. e., to denote its length and direction without regard to its position in other respects. The points A and B may be distinguished as the origin and the terminus of the vector. Since any magnitude may be represented by a length, any vector may be represented by a directed line; and it will often be convenient to use language relating to vectors, which refers to them as thus represented.
Reversal of Direction, Scalar Multiplication and Division.
4. The negative sign (-) reverses the direction of a vector. (Sometimes the sign + may be used to call attention to the fact that the vector has not the negative sign.)
Def. - A vector is said to be multiplied or divided by a scalar when its magnitude is multiplied or divided by the numerical value of the scalar and its direction is either unchanged or reversed according as the scalar is positive or negative. These operations are represented by the same methods as multiplication and division in algebra, and are to be regarded as substantially identical with them. The terms scalar multiplication and scalar division are used to denote multiplication and division by scalars, whether the quantity multiplied or divided is a scalar or a vector.
5. Def. - A un…
(The three letters, i, j, k, will make an exception, to be mentioned more particularly hereafter.
2. Def. - Vectors are said to be equal when they are the same both in direction and in magnitude. The reader will observe that this vector equation is the equivalent of three scalar equations.
A vector is said to be equal to zero, when its magnitude is zero. Such vectors may be set equal to one another, irrespectively of any considerations relating to direction.
3. Perhaps the most simple example of a vector is afforded by a directed straight line, as the line drawn from A to B. We may use the notation AB to denote this line as a vector, i. e., to denote its length and direction without regard to its position in other respects. The points A and B may be distinguished as the origin and the terminus of the vector. Since any magnitude may be represented by a length, any vector may be represented by a directed line; and it will often be convenient to use language relating to vectors, which refers to them as thus represented.
Reversal of Direction, Scalar Multiplication and Division.
4. The negative sign (-) reverses the direction of a vector. (Sometimes the sign + may be used to call attention to the fact that the vector has not the negative sign.)
Def. - A vector is said to be multiplied or divided by a scalar when its magnitude is multiplied or divided by the numerical value of the scalar and its direction is either unchanged or reversed according as the scalar is positive or negative. These operations are represented by the same methods as multiplication and division in algebra, and are to be regarded as substantially identical with them. The terms scalar multiplication and scalar division are used to denote multiplication and division by scalars, whether the quantity multiplied or divided is a scalar or a vector.
5. Def. - A un…
88 pages, Paperback
First published July 27, 2015
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About the author
Josiah Willard Gibbs
75 books11 followersJosiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American scientist who made important theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous deductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics (a term that he coined), explaining the laws of thermodynamics as consequences of the statistical properties of large ensembles of particles. Gibbs also worked on the application of Maxwell's equations to problems in physical optics. As a mathematician, he invented modern vector calculus (independently of the British scientist Oliver Heaviside, who carried out similar work during the same period).
In 1863, Yale awarded Gibbs the first American doctorate in engineering. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was professor of mathematical physics from 1871 until his death. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised by Albert Einstein as "the greatest mind in American history". In 1901 Gibbs received what was then considered the highest honor awarded by the international scientific community, the Copley Medal of the Royal Society of London, "for his contributions to mathematical physics".
Commentators and biographers have remarked on the contrast between Gibbs's quiet, solitary life in turn of the century New England and the great international impact of his ideas. Though his work was almost entirely theoretical, the practical value of Gibbs's contributions became evident with the development of industrial chemistry during the first half of the 20th century. According to Robert Andrews Millikan, in pure science Gibbs "did for statistical mechanics and for thermodynamics what Laplace did for celestial mechanics and Maxwell did for electrodynamics, namely, made his field a well-nigh finished theoretical structure."
In 1863, Yale awarded Gibbs the first American doctorate in engineering. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was professor of mathematical physics from 1871 until his death. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised by Albert Einstein as "the greatest mind in American history". In 1901 Gibbs received what was then considered the highest honor awarded by the international scientific community, the Copley Medal of the Royal Society of London, "for his contributions to mathematical physics".
Commentators and biographers have remarked on the contrast between Gibbs's quiet, solitary life in turn of the century New England and the great international impact of his ideas. Though his work was almost entirely theoretical, the practical value of Gibbs's contributions became evident with the development of industrial chemistry during the first half of the 20th century. According to Robert Andrews Millikan, in pure science Gibbs "did for statistical mechanics and for thermodynamics what Laplace did for celestial mechanics and Maxwell did for electrodynamics, namely, made his field a well-nigh finished theoretical structure."
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