Vectors in three-dimensional space
Encyclopedia
Vectors in three-dimensional space (1978) is a book concerned with physical quantities defined in "ordinary" 3-space. It was written by J.S.R.Chisholm
, an English mathematical physicist, and published by Cambridge University Press
. According to the author, such physical quantities are studied in Newtonian mechanics, fluid mechanics
, theories of elasticity
and plasticity
, non-relativistic quantum mechanics
, and many parts of solid state physics. The author further states that "the vector concept developed in two different ways: in a wide variety of physical applications, vector notation and techniques became, by the middle of this century, almost universal; on the other hand, pure mathematicians reduced vector algebra to an axiomatic system, and introduced wide generalisations
of the concept of a three-dimensional 'vector space'." Chisholm explains that since these two developments proceeded largely independently, there is a need to show how one can be applied to the other.
and displacements
including these sections: Introduction, Scalar multiplication of vectors, Addition and subtraction of vectors, Displacements in Euclidean space, Geometrical applications. The second on Scalar products
and components including these sections: Scalar products, Linear dependence and dimension, Components of a vector, Geometrical applications, Coordinate system
s. The third on Other products of vectors. The last three chapters round out Chisholm's integration of these two largely independent developments.
John Stephen Roy Chisholm
J. S. R. Chisholm is an English mathematical physicist. He is currently Professor Emeritus of Applied Mathematics at the University of Kent. His Erdős number is 4...
, an English mathematical physicist, and published by Cambridge University Press
Cambridge University Press
Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the world's oldest publishing house, and the second largest university press in the world...
. According to the author, such physical quantities are studied in Newtonian mechanics, fluid mechanics
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
, theories of elasticity
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
and plasticity
Plasticity (physics)
In physics and materials science, plasticity describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the...
, non-relativistic quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, and many parts of solid state physics. The author further states that "the vector concept developed in two different ways: in a wide variety of physical applications, vector notation and techniques became, by the middle of this century, almost universal; on the other hand, pure mathematicians reduced vector algebra to an axiomatic system, and introduced wide generalisations
Generalization
A generalization of a concept is an extension of the concept to less-specific criteria. It is a foundational element of logic and human reasoning. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements. As such, it...
of the concept of a three-dimensional 'vector space'." Chisholm explains that since these two developments proceeded largely independently, there is a need to show how one can be applied to the other.
Summary
Vectors in three-dimensional space has six chapters, each divided into five or more subsections. The first on linear spacesVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
and displacements
Displacement (vector)
A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...
including these sections: Introduction, Scalar multiplication of vectors, Addition and subtraction of vectors, Displacements in Euclidean space, Geometrical applications. The second on Scalar products
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
and components including these sections: Scalar products, Linear dependence and dimension, Components of a vector, Geometrical applications, Coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
s. The third on Other products of vectors. The last three chapters round out Chisholm's integration of these two largely independent developments.