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Review of Vector Analysis EEL 3472 Review of Vector Analysis Review of Vector Analysis Vector analysis is a mathematical tool with which electromagnetic (EM) concepts are most conveniently expressed and best comprehended. A quantity is called a scalar if it has only magnitude (e.g., mass, temperature, electric potential, population). A quantity is called a vector if it has both magnitude and direction (e.g., velocity, force, electric field intensity). The magnitude of a vector A is a scalar written as A or A A 2 EEL 3472 Review of Vector Analysis A unit vector eA along is defined as a vector whose magnitude is unity (that is,1) and its direction is along eA A A A A ( eA 1) Thus A AeA which completely specifies A in terms of A and its direction eA 3 EEL 3472 Review of Vector Analysis A vector A in Cartesian (or rectangular) coordinates may be represented as (A x , A y , A z ) A x ex A y ey A z ez or where AX, Ay, and AZ are called the components of A in the x, y, and z directions, respectively; e x , e y , and e z are unit vectors in the x, y and z directions, respectively. 4 EEL 3472 Review of Vector Analysis Suppose a certain vector V is given by V 2ex 3ey 4ez The magnitude or absolute value of the vector V is V 22 32 42 5.385 (from the Pythagorean theorem) 5 EEL 3472 Review of Vector Analysis The Radius Vector A point P in Cartesian coordinates may be represented by specifying (x, y, z). The radius vector (or position vector) of point P is defined as the directed distance from the origin O to P; that is, r x ex y ey z ez The unit vector in the direction of r is er 6 x ex y ey z ez x 2 y 2 z2 r r EEL 3472 Review of Vector Analysis Vector Algebra Two vectors A and B can be added together to give another vector C ; that is , C AB Vectors are added by adding their individual components. Thus, if A x ex A y ey A z ez and B Bx ex By ey Bz ez C (A x Bx )ex (A y By )ey (A z Bz )ez 7 EEL 3472 Review of Vector Analysis Parallelogram rule Head to tail rule Vector subtraction is similarly carried out as D A B A (B ) D (A x Bx )ex (A y B y )ey (A z Bz )ez 8 EEL 3472 Review of Vector Analysis The three basic laws of algebra obeyed by any given vector A, B, and C, are summarized as follows: Law Commutative Addition AB BA Associative A (B C) (A B) C Distributive k(A B) kA k B Multiplication kA Ak k(lA) (kl)A where k and l are scalars 9 EEL 3472 Review of Vector Analysis When two vectors A and B are multiplied, the result is either a scalar or a vector depending on how they are multiplied. There are two types of vector multiplication: 1. Scalar (or dot) product: AB 2.Vector (or cross) product: AB The dot product of the two vectors A and B is defined geometrically as the product of the magnitude of B and the projection of A onto B (or vice versa): A B AB cos A B where A B is the smaller angle between A and B 10 EEL 3472 Review of Vector Analysis If and B (BX , BY , BZ ) then A B A XBX A YB Y AZBZ which is obtained by multiplying A and B component by component AB BA A (B C ) A B A C 2 A A A A2 eX ey ey ez eZ ex 0 A (AX , A Y , AZ , ) eX ex ey ey eZ ez 1 11 EEL 3472 Review of Vector Analysis The cross product of two vectors A and B is defined as A B AB sin A Ben where en is a unit vector normal to the plane containing A and B . The direction of en is determined using the righthand rule or the right-handed screw rule. Direction of A B and en using (a) right-hand rule, (b) right-handed screw rule 12 EEL 3472 Review of Vector Analysis If A (AX , A Y , AZ , ) and B (BX , BY , BZ ) ex A B Ax ey Ay ez Az Bx By Bz then (A yBz A zBy )ex (A zBx A xBz )ey (A xBy A yBx )ez 13 EEL 3472 Review of Vector Analysis Note that the cross product has the following basic properties: (i) It is not commutative: AB B A It is anticommutative: A B B A (ii) It is not associative: A (B C) (A B) C (iii) It is distributive: A (B C ) A B A C (iv) 14 AA 0 (sin 0) EEL 3472 Review of Vector Analysis Also note that ex ey ez ey ez ex ez ex ey which are obtained in cyclic permutation and illustrated below. Cross product using cyclic permutation: (a) moving clockwise leads to positive results; (b) moving counterclockwise leads to negative results 15 EEL 3472 Review of Vector Analysis Scalar and Vector Fields A field can be defined as a function that specifies a particular quantity everywhere in a region (e.g., temperature distribution in a building), or as a spatial distribution of a quantity, which may or may not be a function of time. Scalar quantity scalar function of position scalar field Vector quantity vector function of position vector field 16 EEL 3472 Review of Vector Analysis 17 EEL 3472 Review of Vector Analysis Line Integrals A line integral of a vector field can be calculated whenever a path has been specified through the field. The line integral of the field V along the path P is defined as P2 V dl V P 18 cos dl P1 EEL 3472 Review of Vector Analysis 19 EEL 3472 Review of Vector Analysis Example. The vector V is given by V Vo ex where Vo is a constant. Find the line integral I V dl P where the path P is the closed path below. It is convenient to break the path P up into the four parts P1, P2, P3 , and P4. 20 EEL 3472 Review of Vector Analysis V Voex For segment P1, dl dx ex x xo V dl P1 x 0 Thus xo (Vo ex ) (dx ex ) Vo (ex ex )dx Vo (xo 0) Voxo 0 For segment P2, dl dy e y y yo and V dl (V e ) (dy e ) 0 o x P2 21 y (since ex e y 0) y 0 EEL 3472 Review of Vector Analysis For segment P3, dl dxex (the differential length dl points to the left) x xo V dl (V e ) (dx e ) - V x o P3 x x o o x 0 V dl 0 P4 I V x o P1 22 P2 P3 o 0 Voxo 0 0 (conservative field) P4 EEL 3472 Review of Vector Analysis Example. Let the vector field V be given by V Vo ex. Find the line integral of V over the semicircular path shown below Consider the contribution of the path segment located at the angle dl dl cos ex dl sin ey Since - 90 cos cos( - 90) sin sin sin( - 90) cos dl dl sin ex dl cos ey ad { (sin ex cos ey ) dl 23 EEL 3472 Review of Vector Analysis 180 I (V e ) (sin e o x x cos e y )ad 0 180 aVo [sin (ex ex ) cos (ex e y )]d 0 1 0 180 aVo sin d aVo (cos 180 0) cos 0 1 1 2aVo 24 EEL 3472 Review of Vector Analysis Surface Integrals Surface integration amounts to adding up normal components of a vector field over a given surface S. The flux of a vector field A through surface S We break the surface S into small surface elements and assign to each element a vector ds ds en ds is equal to the area of the surface element en is the unit vector normal (perpendicular) to the surface element 25 EEL 3472 Review of Vector Analysis (If S is a closed surface, ds is by convention directed outward) Then we take the dot product of the vector field V at the position of the surface element with vector ds. The result is a differential scalar. The sum of these scalars over all the surface elements is the surface integral. V ds V S ds cos S V cos is the component of V in the direction of ds (normal to the surface). Therefore, the surface integral can be viewed as the flow (or flux) of the vector field through the surface S (the net outward flux in the case of a closed surface). 26 EEL 3472 Review of Vector Analysis Example. Let V be the radius vector V xex yey zez The surface S is defined by zc d x d d y d The normal to the surface is directed in the +z direction Find V ds S 27 EEL 3472 Review of Vector Analysis Surface S V is not perpendicular to S, except at one point on the Z axis 28 EEL 3472 Review of Vector Analysis V ds V ds cos S V x 2 y 2 c2 S ds dxdy V cos c x 2 y 2 c2 c os ds x d y d x d c 2 2 2 V ds x y c dydx c [d (d)]dx S 2 2 2 x y c x d y d x d 2dc[d - (-d)] 4d2c 29 EEL 3472 Review of Vector Analysis Introduction to Differential Operators An operator acts on a vector field at a point to produce some function of the vector field. It is like a function of a function. If O is an operator acting on a function f(x) of the single variable X , the result is written O[f(x)]; and means that first f acts on X and then O acts on f. Example. f(x) = x2 and the operator O is (d/dx+2) O[f(x)]=d/dx(x2 ) + 2(x2 ) = 2x +2(x2 ) = 2x(1+x) 30 EEL 3472 Review of Vector Analysis An operator acting on a vector field O[V(x, y, z)] can produce either a scalar or a vector. Example. O(A) A A (the length operator), V 3yex zey Evaluate O(V) at the point x=1, y=2, z=-2 O(V) V V 9y2 z2 40 6.32 scalar Thus, O is a scalar operator acting on a vector field. Example. O(A) A A A 2A , x=1, y=2, z=-2 V 3yex zey , O(V) (3y ex z ey ) 9y2 z2 6y ex 2z ey (6 ex 2ey ) 40 12ex 4ey 49.95 ex 16.65ey vector Thus, O is a vector operator acting on a vector field. 31 EEL 3472 Review of Vector Analysis Vector fields are often specified in terms of their rectangular components: V(x, y, z) Vx(x, y, z)ex Vy (x, y, x)ey Vz(x, y, z)ez where Vx , Vy , and Vz are three scalar features functions of position. Operators can then be specified in terms of Vx , Vy , and Vz . The divergence operator is defined as V 32 Vx Vy Vz x y z EEL 3472 Review of Vector Analysis Example V x2ex yey (2 x)ez point x=1, y=-1, z=2. Vx x2 Vx 2x x . Evaluate V at the Vy y Vz 2 x Vy 1 Vz 0 y z V 2x 1 3 Clearly the divergence operator is a scalar operator. 33 EEL 3472 Review of Vector Analysis 1. V 2. V - divergence, acts on a vector to produce a scalar - gradient, acts on a scalar to produce a vector 3. V - curl, acts on a vector to produce a vector 4. 2V -Laplacian, acts on a scalar to produce a scalar Each of these will be defined in detail in the subsequent sections. 34 EEL 3472 Review of Vector Analysis Coordinate Systems In order to define the position of a point in space, an appropriate coordinate system is needed. A considerable amount of work and time may be saved by choosing a coordinate system that best fits a given problem. A hard problem in one coordinate system may turn out to be easy in another system. We will consider the Cartesian, the circular cylindrical, and the spherical coordinate systems. All three are orthogonal (the coordinates are mutually perpendicular). 35 EEL 3472 Review of Vector Analysis Cartesian coordinates (x,y,z) The ranges of the coordinate variables are x y z A vector A in Cartesian coordinates can be written as (A x , A y , A z ) or A x ex A y ey A z ez The intersection of three orthogonal infinite places (x=const, y= const, and z = const) defines point P. Constant x, y and z surfaces 36 EEL 3472 Review of Vector Analysis dl dxex dyey dz ez d dxdydz Differential elements in the right handed Cartesian coordinate system 37 EEL 3472 Review of Vector Analysis dS dydz ax dxdz ay dxdy az 38 EEL 3472 Review of Vector Analysis Cylindrical Coordinates (, , z) . 0 0 2 z A vector - the radial distance from the z – axis - the azimuthal angle, measured from the xaxis in the xy – plane - the same as in the Cartesian system. in cylindrical coordinates can be written as (A , A Az ) or A e A e Az ez 2 2 2 A (A A Az )1 / 2 Cylindrical coordinates amount to a combination of rectangular coordinates and polar coordinates. 39 EEL 3472 Review of Vector Analysis Relationship between (x,y,z) and (, , z) Positions in the x-y plane are determined by the values of and y x2 y2 tan1 zz x 40 EEL 3472 Review of Vector Analysis Point P and unit vectors in the cylindrical coordinate system e e ez e ez e ez e e e e e e ez ez 1 e e e ez e e 0 41 EEL 3472 Review of Vector Analysis semi-infinite plane with its edge along the z - axis Constant , and z surfaces 42 EEL 3472 Review of Vector Analysis Metric coefficient dl d ap da dz az dv dddz Differential elements in cylindrical coordinates 43 EEL 3472 Review of Vector Analysis dS ddza ddza d d a z Cylindrical surface ( =const) Planar surface ( = const) Planar surface ( z =const) 44 EEL 3472 Review of Vector Analysis Spherical coordinates (r, , ) . 0r 0 Colatitude ( polar angle) 0 2 - the distance from the origin to the point P - the angle between the z-axis and the radius vector of P - the same as the azimuthal angle in cylindrical coordinates 45 EEL 3472 Review of Vector Analysis er e e e e er e er e er er e e e e 1 Point P and unit vectors in spherical coordinates er e e e e er 0 A vector A in spherical coordinates may be written as (Ar , A A ) or Ar er A e A e 2 2 2 A (Ar A A )1 / 2 46 EEL 3472 Review of Vector Analysis r 2 2 2 x y z 1 tan x2 y2 z tan1 tan-1 y cos1 x x x2 y2 z cos1 z r x r sin cos y r sin sin z r cos Relationships between space variables (x, y, z), (r, , ), and (, , z) 47 EEL 3472 Review of Vector Analysis Constant 48 r, , and surfaces EEL 3472 Review of Vector Analysis dl dr ar rda r sin d a dv r2 sindrdd Differential elements in the spherical coordinate system 49 EEL 3472 Review of Vector Analysis dS r 2 sin d d ar r sin dr d a rdr d a 50 EEL 3472 Review of Vector Analysis 51 EEL 3472 Review of Vector Analysis 52 EEL 3472 Review of Vector Analysis 53 EEL 3472 Review of Vector Analysis 54 EEL 3472 Review of Vector Analysis POINTS TO REMEMBER 1. 2. 3. 55 EEL 3472 Review of Vector Analysis 4. 5. 6. 7. 56 EEL 3472