Chapter 1. Vector Analysis Hayt; 8/31/2009; 1-1 1.1 Scalars and Vectors Scalar : Vector: A quantity represented by a single real number Distance, time, temperature, voltage, etc Magnitude and direction Force, velocity, flux, etc At a given position and time a scalar field (function) A magnitude (Temperature distribution in a room) a vector field (function) A magnitude and a direction (Movement of smoke particles) Function conversion between different coordinate systems Physical laws should be independent of the coordinate systems. Coordinate system is chosen by convenience Three main topics (1) Vector algebra : addition, subtraction, multiplication (2) Orthogonal coordinate system : Cartesian, Cylindrical, Spherical (3) Vector calculus : differentiation, integration(gradient, divergence, curl) 1.2 Vector Algebra A vector has a magnitude and a direction A = Aa ˆA magnitude, A unit vector, A A Graphical representation Two vectors are equal if they have the same magnitude and direction, even though they may be displaced in space.
Vector addition, C = A+ B Two vectors, A and B, form a plane Parallelogram rule : C is the diagonal of the parallelogram Head-To-Tail rule : The head of A touches the tail of B. C is drawn from the tail of A to the head of B. Hayt; 8/31/2009; 1-2 Parallelogram rule Head-To-Tail rule Note C = A + B = B + A Commutative law A + ( B + F) = ( A + B) + F Associative law The location of a vector is at the tail of the arrow. Vector subtraction A B = A + B, where B = ( aˆ ) B ( ) B Vector multiplication by a scalar Multiplication of a vector by a positive scalar ka = ( ka) a ˆ : Change in the magnitude but not the direction A Associative and distributive laws apply r + s A + B = r A + B + s A + B = ra + rb + sa + sb ( )( ) ( ) ( ) Vector division by a scalar Multiplication by the reciprocal of that scalar Two vectors are equal, A = B, if A B = 0 Addition and subtraction of vector fields should be done locally. C( r ) = A( r ) + B( r ) : Addition at the same position r. Example; magnetic fields of the earth and a horseshoe magnet.
1.3 The Rectangular (Cartesian) Coordinate System Hayt; 8/31/2009; 1-3 Right-hand rule for x, y, z directions, Fig (a). A point is represented by x, y, z coordinates, Fig (b). Distance from the origin along x-axis A point can be also represented by a common intersection of three planes, x=x1, y=y1 and z=z1. Three differential lengths, dx, dy and dz, from P to P Three coordinates of P are x+dx, y+dy and z+dz in Fig(c). dx 2 + dy 2 + dz 2 The distance between P and P = ( ) ( ) ( ) dx, dy and dz form a rectangular parallelepiped with a differential volume and six differential areas. dv=dxdydz dxdy, dxdz, dydz
1.4 Vector Components and Unit Vectors Hayt; 8/31/2009; 1-4 A point in space is represented by a position vector r = x + y + z. Three component vectors along each corresponding axis. Component vectors are given by unit vectors as x = x aˆ, y = y aˆ and z = z aˆ x-unit vector x-component The position vector r p is from the origin to the point P(1,2,3). rˆ = aˆ + 2aˆ + 3aˆ p x y z Example A vector from P to Q R = r r = 2aˆ 2aˆ + aˆ aˆ + 2aˆ + 3aˆ = aˆ 4aˆ 2aˆ ( ) ( ) PQ Q P x y z x y z x y z x y z Any vector B = B ˆ ˆ ˆ xax + Byay + Bzaz The magnitude of B 2 2 2 : B = Bx + By + Bz The unit vector of B B B : aˆb = = B B + B + B 2 2 2 x y z
Hayt; 8/31/2009; 1-5 1.5 The Vector Field A vector field v( r) is a vector function of a position vector r. If we move from the origin to a point in space following the position vector r, the vector field gives a vector at that point. Using component vectors v r = v r aˆ + v r aˆ + v r aˆ ( ) ( ) ( ) ( ) x x y y z z 1.6 The Dot Product The dot product or scalar product of two vectors A and B are defined as A B = A B cosθ AB θ is the smaller angle between A and B : AB Note A B = B A The dot product obeys the distributive law A B= AB + AB + AB x x y y z z Using a ia = 0for i j, a i a = 1fori = j Note 2 A A = A = A i j i j 2
Hayt; 8/31/2009; 1-6 Projection of a vector B in â direction It is defined as B aˆ = B aˆ cosθ = B cosθ Ba Ba Example Law of Cosine Use vectors to prove C = B A 2 C = C C ( B A) ( B A) B B + A A A B B A 2 2 A + B 2ABcosα
1.7 The Cross Product The cross product or vector product of two vectors A and B are defined as Hayt; 8/31/2009; 1-7 A B = aˆ A B sinθ n AB a ˆn is a unit vector normal to the surface formed by A and B Note A B represents the area of the parallelepiped. B A = ( A B) aˆ aˆ = aˆ, aˆ aˆ = aˆ, a a = a x y z Evaluation of A B A B = y z x z x y ( ˆ ˆ ) ( ˆ ˆ ) ( ˆ ˆ ) ( ˆ ˆ ) ( ˆ ˆ ) ( ˆ ˆ ) ( ˆ ˆ ) + ( ˆ ˆ ) + ( ˆ ˆ ) A B a a + A B a a + A B a a + x x x x x y x y x z x z A B a a + A B a a + A B a a + y x y x y y y y y z x z A B a a A B a a A B a a z x z x z y z y z z z z (1) Using (1) A B = aˆ A B A B + aˆ A B A B + aˆ A B A B or ( ) ( ) ( ) x y z z y y z x x z z x y y x aˆ ˆ ˆ x ay az A B = A A A x y z B B B x y z
Hayt; 8/31/2009; 1-8 1.8 Cylindrical Coordinate System A point is given by the intersection of three mutually perpendicular surfaces. A circular cylinder, ρ = constant A half plane, φ = constant A plane, z = constant Unit vectors are defined along the direction of increasing coordinate values, Fig (b). Its direction is perpendicular to the surface of constant coordinate value. â ρ at P( 1, 1, z1) â φ at P( 1, 1, z1) a at P(,, z ) ˆz ρ φ is directed radially outward and normal to the cylindrical surface ρ = ρ1. ρ φ is normal to the plane φ = φ1. ρ1 φ 1 1 is normal to the plane z z1 =. Note â ρ and â φ are functions of position, NOT CONSTANTS. Three unit vectors are mutually perpendicular with cyclic order aˆ aˆ = aˆz ρ φ Differential volume Increase ρ ρ + dρ, φ φ + dφ, z z + dz The six surfaces enclose a differential volume, Fig(c). It can be approximated as a rectangular parallelepiped. The lengths of the sides are dρ, ρdφ anddz. dφ itself is an angle not a length The volume is given by ρdρdφ dz Areas of side surfaces are given by ρdρdφ, ρdφ dz and dρ dz.
Hayt; 8/31/2009; 1-9 Coordinate transformation of a point 2 2 x = ρcosφ ρ = x + y y = ρsinφ z = z 1 φ = tan z = z y/ x ( ) Coordinate transformation of a vector A = A aˆ + A aˆ + A aˆ Aaˆ + Aaˆ + Aaˆ x x y y z z ρ ρ φ φ A ˆ A ˆ A a a ρ a φ z z ˆz A aˆ = A ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ xax + Ayay + Azaz a = Axax a + Ayay a A aˆ = Aaˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x x + Aa y y + Aa z z a = Aa x x a + Aa y y a A aˆ = A aˆ + A aˆ + A aˆ aˆ = A aˆ aˆ ( ) ( ) ( ) ρ ρ ρ ρ φ φ φ φ z x x y y z z z z z z
1.9 Spherical Coordinate System Hayt; 8/31/2009; 1-10 A point is given by the intersection of three mutually perpendicular surfaces. A sphere, r = constant A cone, θ = constant A half plane, φ = constant Unit vectors are normal to these surfaces and directed toward increasing r, θ and φ. They are mutually perpendicular and have a cyclic order aˆ aˆ = aˆ r θ φ Differential volume Increase r r + dr, θ θ + dθ, φ φ + dφ The distance between two spheres is dr two cones is rdθ two half panes is r sinθdφ The differential volume is r 2 sinθ drdθdφ. The differential areas are r drdθ, r sinθ dr dφ and r 2 sinθ dθdφ.
Hayt; 8/31/2009; 1-11 Coordinate transformation of a point x = r sinθ cosφ y = r sinθ sinφ z = r cosθ r = x + y + z θ = cos φ = 2 2 2 1 2 2 2 ( y x) 1 tan / z x + y + z Coordinate transformation of a vector It can be done by using the projection of a vector