VectorsandVectorFields 223 A VectorsandVectorFieldshome.iitk.ac.in/~peeyush/mth523/L1.pdf · 224...
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Vectors and Vector Fields 223 A Vectors and Vector Fields A.1 Introduction This Appendix outlines those aspects of vector algebra, vector calculus, and vector field theory that are helpful in the study of fluid dynamics. A.2 Scalars and Vectors Many physical entities (e.g., mass, energy) are entirely defined by a numerical magnitude (expressed in appropriate units). Such entities, which have no directional element, are known as scalars. Moreover, since scalars can be represented by real numbers it follows that they obey the laws of ordinary algebra. However, there exits a second class of physical entities (e.g., velocity, acceleration, force) that are only completely defined when both a numerical magnitude and a direction in space are specified. Such entities are known as vectors. By definition, a vector obeys the same algebra as a displacement in space, and may thus be represented geometrically by a straight-line, → PQ (say), where the arrow indicates the direction of the displacement (i.e., from point P to point Q). See Figure A.1. The magnitude of the vector is represented by the length of the straight-line. It is conventional to denote vectors by bold-faced symbols (e.g., a, F) and scalars by non-bold-faced symbols (e.g., r, S ). The magnitude of a general vector, a, is denoted |a|, or just a, and is, by definition, always greater than or equal to zero. It is convenient to define a vector with zero magnitude—this is denoted 0, and has no direction. Finally, two vectors, a and b, are said to be equal when their magnitudes and directions are both identical. A.3 Vector Algebra Suppose that the displacements → PQ and → QR represent the vectors a and b, respectively. See Figure A.2. It can be seen that the result of combining these two displacements is to give the net displacement → PR. Hence, if → PR represents the vector c then we can write c = a + b. (A.1) This defines vector addition. By completing the parallelogram PQRS , we can also see that → PR = → PQ + → QR = → PS + → SR . (A.2) However, → PS has the same length and direction as → QR, and, thus, represents the same vector, b. Likewise, → PQ and → SR both represent the vector a. Thus, the above equation is equivalent to c = a + b = b + a. (A.3) P Q Figure A.1: A vector.
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