Vector Analysis 1_dl

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Unformatted text preview: THIS MATERIAL WILL HELP YOU TO PREPARE FOR THE BASIC CONCEPTS IN MATHEMATICS FOR ENGINEERING ELECTROMAGNETICS SOME BASIC MATHEMATICAL CONCEPTS FOR ENGINEERING ELECTROMAGNETICS 1 KEY TOPICS: x DEFINITIONS OF SCALAR AND VECTOR x CARTESIAN COORDINATE SYSTEM (x, y, z) x UNIT VECTOR x VECTOR ADDITION x VECTOR SUBTRACTION x VECTOR MULTIPLICATION BY A SCALAR x DOT PRODUCT &amp;amp; CROSS PRODUCT x RIGHT HAND RULE x CYLINDRICAL COORDINATE SYSTEM (r, , z) SOME BASIC MATHEMATICAL CONCEPTS FOR ENGINEERING ELECTROMAGNETICS 2 SCALARS AND VECTORS A scalar is a quantity which has only magnitude . Examples of Scalars: time (second); temperature; mass (kg); electric potential (volt). A vector has both magnitude and direction Examples of Vectors: velocity (m/s); force (Newton); electric field (Newton/ Coulomb). To distinguish between a scalar and a vector , it is a customary to denote a vector by a letter with an arrow on top of it , such as A jjjjjjjjjjjj k . A scalar is represented simply by a letter, for example: A . SOME BASIC MATHEMATICAL CONCEPTS FOR ENGINEERING ELECTROMAGNETICS 3 CARTESIAN COORDINATE SYSTEM (= RECTANGULAR COORDINATE SYSTEM) In Cartesian coordinate system, there are three coordinate axes which are mutually at right angles to each other, namely x, y and z axes. A vector A jjjjjjjjjjjj k which extends from the origin to the point A A x , A y , A z b c in Cartesian coordinates can be written as: A jjjjjjjjjjjj k = A x a x jjjjjjjjj k + A y a y jjjjjjjjjj k + A z a z jjjjjjjjjj k (or A jjjjjjjjjjjj k = A x i ^ + A y j ^ + A z k ^ ) in which a x jjjjjjjjj k , a y jjjjjjjjjj k and a z jjjjjjjjjj k (or i ^ , j ^ and k ^ ) are the unit vectors (base vectors) along x, y and z directions respectively. A x , A y and A z are the scalar components of A jjjjjjjj k along the x, y and z directions, respectively. In Engineering Electromagnetics, notation a x jjjjjjjjj k , a y jjjjjjjjjj k , a z jjjjjjjjjj k is used instead of i ^ , j ^ and k ^ . SOME BASIC MATHEMATICAL CONCEPTS FOR ENGINEERING ELECTROMAGNETICS 4 MAGNITUDE OF A VECTOR For a vector A jjjjjjjjjjjj k = A x a x jjjjjjjjj k + A y a y jjjjjjjjjj k + A z a z jjjjjjjjjj k in the Cartesian coordinates, its magnitude A jjjjjjj k M M M is: A jjjjjjj k M M M = A x 2 + A y 2 + A z 2 q wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww UNIT VECTOR A unit vector a A jjjjjjjjjjj k along A jjjjjjjjjjjj k is defined as a vector whose magnitude is unity (i.e.: 1) and the direction is along A jjjjjjjjjjjj k . a A jjjjjjjjjjjj k = A jjjjjjj k A jjjjjjj k L L L M M M fffffffff Thus, we also can write A jjjjjjjjjjjj k as: A jjjjjjj k = A a A jjjjjjjjjjjj k For a vector A jjjjjjjjjjjj k = A x a x jjjjjjjjj k + A y a y jjjjjjjjjj k + A z a z jjjjjjjjjj k in the Cartesian coordinates, its unit vector a A jjjjjjjjjjjj k is given by: a A jjjjjjjjjjjjjjjjjj k = A x a x jjjjjjjjj k + A y a y jjjjjjjjjj k + A z a z jjjjjjjjjj k A x 2 + A y 2 + A z 2 q wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww...
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