MIT OpenCourseWare http://ocw.mit.edu Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, 1989. ISBN: 9780132490207. Please use the following citation format: Haus, Hermann A., and James R. Melcher, Electromagnetic Fields and Energy. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed [Date]). License: Creative Commons Attribution-NonCommercial-Share Alike. Also available from Prentice-Hall: Englewood Cliffs, NJ, 1989. ISBN: 9780132490207. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms¯ � 1 APPENDIX 1.1 VECTOR OPERATIONS A vector is a quantity which possesses magnitude and direction. In order to describe a vector mathematically, a coordinate system having orthogonal axes is usually cho-sen. In this text, use is made of the Cartesian, circular cylindrical, and spherical coordinate systems. In these three-dimensional systems, any vector is completely described by three scalar quantities. For example, in Cartesian coordinates, a vec-tor is described with reference to mutually orthogonal coordinate axes. Then the magnitude and orientation of the vector are described by specifying the three pro-jections of the vector onto the three coordinate axes. In representing a vector1 A mathematically, its direction along the three or-thogonal coordinate axes must be given. The direction of each axis is represented by a unit vector i, that is, a vector of unit magnitude directed along the axis. In Cartesian coordinates, the three unit vectors are denoted ix, iy, iz. In cylindrical coordinates, they are ir, iφ, iz, and in spherical coordinates, ir, iθ, iφ. A, then, has three vector components, each component corresponding to the projection of A onto the three axes. Expressed in Cartesian co ordinates, a vector is defined in terms of its components by A = Axix + Ayiy + Aziz (1) These components are shown in Fig. A.1.1. 1 Vectors are usually indicated either with boldface characters, such as A, or by drawing a line (or an arrow) above a character to indicate its vector nature, as in A or A. 12 Appendix Chapter 1 Fig. A.1.1 Vector A represented by its components in Cartesian coordinates and unit vectors i. Fig. A.1.2 (a) Graphical representation of vector addition in terms of spe-cific coordinates. (b) Representation of vector addition independent of specific coordinates. Vector Addition. The sum of two vectors A = Axix + Ayiy + Aziz and B = Bxix + Byiy + Bziz is effected by adding the coefficients of each of the components, as shown in two dimensions in Fig. A.1.2a. A + B = (Ax + Bx)ix + (Ay + By)iy + (Az + Bz)iz (2) From (2), then, it should be clear that vector addition is both commutative, A+B = B + A, and associative, (A + B) + C = A + (B + C). Graphically, vector summation can be performed without regard to the coor-dinate system, as shown in Fig. A.1.2b, by noticing that the sum A + B is a vector directed along the diagonal of a parallelogram formed by A and B. It should be noted that the representation of a vector in terms of its com-ponents is dependent on the coordinate system in which it is carried out. That is, changes of coordinate system will require an appropriate vector transformation. Fur-ther, the variables used must also be transformed. The transformation of variables and vectors from one coordinate system to another is illustrated by considering a transformation from Cartesian to spherical coordinates. Example 1.1.1. Transformation of Variables and Vectors We are given variables in terms of x, y, and z and vectors such as A = Axix + Ayiy + Aziz. We wish to obtain variables in terms of r, θ, and φ and vectors ex-pressed as A = Arir + Aθiθ + Aφiφ. In Fig. A.1.3a, we see that the point P has two3 Sec. 1.1 Appendix Fig. A.1.3 Specification of a point P in Cartesian and spherical co-ordinates. (b) Transformation from Cartesian coordinate x to spherical coordinates. (c) Transformation of unit vector in x direction into spher-ical coordinate coordinates. representations, one involving the variables x, y and z and the other, r, θ and φ. In particular, from Fig. A.1.3b, x is related to the spherical coordinates by x = r sin θ cos φ (3) In a similar way, the variables y and z evaluated in spherical coordinates can b e shown to be y = r sin θ sin φ (4) z = r cos θ (5) The vector A is transformed by resolving each of the unit vectors ix, iy, iz in terms of the unit vectors in spherical coordinates. For example, ix can first be4 Appendix Chapter 1 Fig. A.1.4 Illustration for definition of dot product. resolved into components in the orthogonal coordinates (x�, y�, z) shown in Fig. A.1.3c. By definition, y� is along the intersection of the φ = constant and the x − y planes. Also in the x− y plane is x�, which is perpendicular to the y� − z plane. Thus, sin φ, cos φ, and 0 are the components of ix along the x�, y�, and z axes resp ectively. These components are in turn resolved into components along the spherical coordi-nate directions by recognizing that the component sin φ along the x� axis is in the −iφ direction while the component of cos φ along the y� axis resolves into comp onents cos φ cos θ in the direction of iθ, and cos φ sin θ in the ir direction. Thus, ix = sin θ cos φir + cos θ cos φiθ − sin φiφ (6) Similarly, iy = sin θ sin φir + cos θ sin φiθ + cos φiφ (7) iz = cos θir − sin θiθ (8) It must be emphasized that the concept of a vector is independent of the coordinate system. (In the same sense, in Chaps. 2 and 4, vector operations are defined independently of the coordinate system in which they are expressed.) A vector can be visualized as
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