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Vector Methods Applied to Differential Geometry, Mechanics, and Potential Theory
D. E. Rutherford
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eBook - ePub
Vector Methods Applied to Differential Geometry, Mechanics, and Potential Theory
D. E. Rutherford
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Designed to familiarize undergraduates with the methods of vector algebra and vector calculus, this text offers both a clear view of the abstract theory as well as a concise survey of the theory's applications to various branches of pure and applied mathematics. A chapter on differential geometry introduces readers to the study of this subject by the methods of vector algebra. The next section explores the many aspects of the theory of mechanics adaptable to the use of vectors, and a full discussion of the vector operator "nabla" proceeds to a treatment of potential theory and Laplace's equation. This includes applications to the theories of gravitation, hydrodynamics, and electricity. A brief chapter on four-dimensional vectors concludes the text.
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CHAPTER I
VECTOR ALGEBRA
§ 1. WE are all familiar with the fact that if B is two miles from A and if C is two miles from B, then C is not necessarily four miles from A. Only in very special circumstances are distances compounded according to the ordinary arithmetical law of addition. Actually there are many other entities which behave in this way as distances rather than as ordinary numbers; the study of such entities leads to the calculus of vectors.
A scalar is a quantity which has magnitude but which is not related to any definite direction in space. A scalar is completely specified by a number. A vector is an entity which obeys the same law of addition as a distance does. It has a magnitude and is also related to a definite direction in space. It follows that any vector may be represented by a straight line with an arrow head whose direction is that of the vector and whose length represents the magnitude of the vector according to a convenient scale. We shall denote vectors by Clarendon type, e.g., r. German textbooks usually use Gothic type to denote vectors. A vector of zero magnitude can have no direction associated with it. Such a vector is called the zero vector and is denoted by 0. The magnitude of a vector is called its length. It is frequently convenient to denote the length of a vector r by r. A similar convention is used when vectors are denoted by other letters.
Two vectors a...