Chapter 9 Vectr Differential Calculus, Grad, Div, Curl 9.1 Vectrs in 2-Space and 3-Space 9.2 Inner Prduct (Dt Prduct) 9.3 Vectr Prduct (Crss Prduct, Outer Prduct) 9.4 Vectr and Scalar Functins and Fields Kreyszig ; 9-1
Chapter 9 Vectr Differential Calculus, Grad, Div, Curl Kreyszig ; 9-2 9.1 Vectrs in 2-Space and 3-Space Tw kinds f quantities used in physics, engineering and s n. A scalar : A quantity representing magnitude. A vectr : A quantity representing magnitude and directin. Terminal pint A vectr is represented by an arrw. The tail is called initial pint. The head (r the tip) is called terminal pint Initial pint The distance between the initial and the terminal pints is called the distance, magnitude, r nrm. A velcity is a vectr, v, and its nrm is v. A vectr f length 1 is called a unit vectr. Definitin Equality f Vectrs Tw vectrs a and b are equal, a= b, if they have the same length and the same directin. A translatin des nt change a vectr. Cmpnents f a Vectr A vectr a is given with initial pint P: ( x, y, z ) and terminal pint :(,, ) 1 1 1 The vectr a has three cmpnents alng x, y, and z crdinates. a= a, a, a [ ] where a1 = x2 x1, a2 = y2 y1, a3 = z2 z1 The length a is given by 2 2 2 a = a + a + a Q x y z. 2 2 2
Example 1 Cmpnents and Length f a Vectr : 4, 0, 2 A vectr a with initial pint P ( ) and terminal pint :( 6, 1, 2) Q. Kreyszig ; 9-3 Its cmpnents are a = 6 4 2, a = 1 0 1, a = 2 2 0 Hence a = [ 2, 1, 0] a = + + Its length is ( ) 2 2 2 2 1 0 5 Psitin vectr A pint A:(x, y, z) is given in Cartesian crdinate system. The psitin vectr f the pint A is a vectr drawn frm the rigin t the pint A. r= xyz,, [ ] Therem 1 Vectrs as Ordered Real Triple Numbers A vectr in a Cartesian crdinate system can be represented by three real numbers (,, ) which crrespnds t three cmpnents. Hence a= b means that a1 = b1, a2 = b2 and a3 = b3. a a a, Vectr Additin, Scalar Multiplicatin Definitin Additin f tw vectrs a= [ a1, a2, a3] and b = [ b1, b2, b3] a+ b = [ a + b, a + b, a + b ] 1 1 2 2 3 3 Parallelgram rule Head-T-Tail rule
Basic Prperties f Vectr Additin a+ b = b+ a ( a + b) + c = a + ( b + c) a + 0= 0+ a = a a+ a = ( ) 0 : cmmutative : assciative Kreyszig ; 9-4 A vectr a has the same length a as a but with the ppsite directin. Definitin In scalar multiplicatin, a vectr a is multiplied by a scalar c. ca = ca, ca, ca [ ] ca has the same directin with a with increased length fr c >0, ppsite directin fr c <0 Basic Prperties f Scalar Multiplicatin c( a + b) = ca + cb ( c + k) a = ca + ka c( ka) = ( ck) a = cka 1a = a, 0 a = 0 1 a= a, ( ) b+ a = b a ( ) Unit Vectrs iˆ, ˆj, k ˆ A vectr can be represented as a= a, a, a = ai ˆ + a ˆ j+ ak [ ] ˆ using three unit vectrs ˆi, ˆj, k ˆ alng x, y, z axes ˆ 1,0,0 ˆ 0,1,0 k ˆ = 0,0,1 i = [ ], j = [ ], [ ]
Kreyszig ; 9-5 Example 2 Vectr Additin. Multiplicatin by Scalars Let 1 a = [ 4, 0,1] and b = 2, 5, 3 Then a = [ 4, 0, 1], 7a = [ 28, 0, 7] 4 a+ b = 6, 5, 3 2 2( a b) = 2 2, 5, = 2a 2b 3 Example 3 ˆi, ˆj, k ˆ Ntatin fr Vectrs The tw vectrs in Example 2 are a= 4ˆ i + kˆ ˆ ˆ 1 b = 2i 5j+ k 3 ˆ 9.2 Inner Prduct (Dt Prduct) Definitin Inner Prduct (Dt Prduct) f Vectrs is defined as a b= a bcsγ= ab + ab + ab 1 1 2 2 3 3 0 γ π, the angle between a and b Orthgnality a is rthgnal t b if a b = 0. γ shuld be π /2 when a 0 and b 0 Therem 1 Orthgnality a b = 0 if and nly if a and b are perpendicular t each ther Length and Angle a = a a Then a b a b csγ= = a b a a b b
Basic Prperties f Inner Prduct ( qa 1 + qb 2 ) c= qa 1 c+ qb 2 c a b = b a a a 0 a a = 0 if and nly if a = 0 ( a+ b) c = a c + b c a b a b a+ b a + b 2 2 2 2 a+ b + a b = 2 a + b ( ) : linearity : symmetry : psitive definiteness : psitive definiteness : distributive : Cauchy-Schwarz inequality : triangle inequality : parallelgram equality Kreyszig ; 9-6 Example 1 Inner Prduct. Angle between Vectrs a = 1, 2, 0 b = 3, 2, 1 are given Tw vectrs [ ] and [ ] a b = 13 + 2(2) + 01 1 2 2 2 a = a a 1 + 2 + 0 5 2 ( ) 2 2 b = b b 3 + 2 + 1 14 a b 1 1 γ= arccs = cs = 1.69061 = 96.865 a b 5 14 Example 2 Wrk Dne by a Frce A cnstant frce p is exerted n a bdy. But the bdy is displaced alng a vectr d. Then the wrk dne by the frce in the displacement f the bdy is W = p csα d p d ( ) Inner prduct is used nicely here. Example 3 Cmpnent f a Frce in a Given Directin What frce in the rpe will hld the car n a 25 ramp. The weight f the car is 5000 lb. Since the weight pints dwnward, it can be represented by a vectr as a = 0, 5000, 0 [ ] a can be given by a sum f tw vectrs a= c + p Frce exerted t the rpe by the car Frce exerted t the ramp by the car p = a cs 65 2113 lb Frm the figure, ( ) A vectr in the directin f the rpe b = 1, tan25, 0 b The frce n the rpe p = a 2113 lb b
Prjectin (r cmpnent) f a in the directin f b a b p= a csγ= b Kreyszig ; 9-7 p is the length f the rthgnal prjectin f a nt b. Orthnrmal Basis The rthgnal unit vectrs in Cartesian crdinates system frm an rthnrmal basis fr 3-space. iˆ, ˆj, k ˆ ( ) An arbitrary vectr is given by a linear cmbinatin f the rthnrmal basis. The cefficients f a vectr can be determined by the rthnrmality. ˆ ˆ v= li 1 + l2j+ lk 3ˆ ˆ ˆ l ˆ 1 = v i, l2 = v j, l3 = v k = l ˆ i i ˆ + l ˆ j i ˆ + l k i ˆ ˆ Example 5 Orthgnal Straight Lines in the Plane Find the straight line L1 passing thrugh the pint P: (1, 3) in the xy-plane and perpendicular t the straight line L2: x-2y+2=0 The equatin f the straight line L2 : b1x+ by 2 = k b r = k, in vectr frm. b = b1, b2, 0, r = x, y, 0 Cnsider anther straight line b r = 0 This line passes thrugh the rigin and parallel t L2 r is the psitin vectr frm the rigin t a pint n L2. Since b r = 0, b is nrmal t r and t this line and t L2. The equatin f a line parallel t b is a r = c Since b = [ 1, 2, 0 ] frm L2, a= [ 2, 1, 0] L : 2x+ y = c 1 L1 passes thrugh P: (1, 3) 2+3=c L : 2x+ y = 5 1 [ ] [ ] with a b = 0.
Example 6 Nrmal Vectr t a Plane Find a unit vectr nrmal t the plane 4x+ 2y+ 4z= 7 Kreyszig ; 9-8 Express the plane in vectr frm : a r = c Using the unit vectr f a a, nˆ = a ˆn r = p : p= c/ a, a cnstant. Prjectin f r nt ˆn A psitin vectr r is frm the rigin t a pint n the plane. The same prjectin p fr any r ˆn shuld be a surface nrmal. Since a = [ 4, 2, 4] is given, the surface nrmal is btained as a 1 nˆ = a a 6 9.3 Vectr Prduct (Crss Prduct, Outer Prduct) Definitin The vectr prduct f v = a b a and b is defined as : anther vectr Its magnitude v= a b= a b sinγ : γ, angle between a and b Its directin is perpendicular t bth a and b cnfrming t right-handed triple( r screw). Nte that a b represents the area f the parallelgram frmed by a and b.
Kreyszig ; 9-9 In cmpnents a b= ab ab, ab ab, ab ab [ ] 2 3 3 2 3 1 1 3 1 2 2 1 v = a b can be calculated as fllws iˆ ˆj kˆ a a a a ˆ ˆ ˆ a a a b = a1 a2 a3 i j + k b b b b b b b b b 2 3 1 3 1 2 2 3 1 3 1 2 Example 1 Vectr Prduct b = a = [ 1, 1, 0] The vectr prduct, [ 3, 0, 0] Example 2 Vectr Prducts f the Standard Basis Vectrs Therem 1 General Prperties f Vectr Prducts ( ka) b = k( a b) = a kb a ( b+ c) = ( a b) + ( a c) ( a + b) c = ( a c) + ( b c) b a= ( a b) a b c a b c ( ) ( ) : fr every scalar k : distributive : distributive : anticmmutative : nt assciative
Example 3 Mment f Frce Kreyszig ; 9-10 A frce p is exerted n a pint A. Pint Q and pint A is cnnected by a vectr r. The mment m abut a pint Q is defined as m= pd, where d is the perpendicular distance frm Q t L. m= pr sinγ In vectr frm m= r p : Mment vectr Example 5 Velcity f a Rtating Bdy A vectr w can describe a rtatin f a rigid bdy. Its directin the rtatin axis (right-hand rule) Its magnitude = angular speed ω (radian/sec) The linear speed at a pint P v=ωd w r sinγ w r In vectr frm v = w r
Scalar Triple Prduct Kreyszig ; 9-11 The scalar triple prduct is defined as abc = a b c ( ) ( ) It can be calculated as a a a a b c = b b b ( ) c c c Therem 2 Prperties and Applicatins f Scalar Triple Prducts (a) The dt and crss can be interchanged a ( b c) = ( a b) c (b) Its abslute value is the vlume f the parallelepiped frmed by a, b and c. (c) Any three vectrs are linearly independent if and nly if their scalar tipple prduct is nnzer. Prf: (a) It can be prved by direct calculatins a b c = a b c β a β b c (b) ( ) cs ( cs ) area f the base height f the parallelepiped (c) If three vectrs are in the same plane r n the same straight line, a b c. either the dt r crss prduct becmes zer in ( ) a b c ( ) 0 Three vectrs NOT in the same plane r n the same straight line. They are linearly independent. Example 6 Tetrahedrn A tetrahedrn is frmed by three edge vectrs, b = 0, 4, 1 c = 5, 6, 0 a = [ 2, 0, 3] Find its vlume., [ ], [ ] First, find the vlume f the parallelepiped using scalar triple prduct. The vlume f tetrahedrn is 1/6 f that f parallelepiped. Therefre, the answer is 12.
9.4 Vectr and Scalar Functins and Fields. Kreyszig ; 9-12 A vectr functin gives a vectr value fr a pint p in space In Cartesian Crd. v = v( p) = v1( p), v2( p), v3( p) v( x, y, z) = v1( x, y, z), v2( x, y, z), v3( x, y, z) A scalar functin gives scalar values : f = f( p) A vectr functin defines a vectr field. A scalar functin defines a scalar field. In Engineering Meaning f field = Meaning f functin. The field implies spatial distributin f a quantity. Example 1 Scalar functin The distance frm a fixed pint p t any pint p is a scalar functin, f( p ). f( p ) defines a scalar field in space It means that the scalar values are distributed in space. ( ) = ( ) = ( ) + ( ) + ( ) 2 2 2 f p f xyx,, x x y y z z In a different crdinate system p and f( p ) p have different frms, but ( ) is a scalar functin. f p has the same value. Directin csines f the line frm Nt a scalar functin. p t p depend n the chice f crdinate system.
Example 3 Vectr Field (Gravitatin field) Kreyszig ; 9-13 Newtn's law f gravitatin c F = 2 r 2 2 2 : r = ( x x ) + ( y y ) + ( z z ) The directin f F is frm p t p. F( xyz,, ) defines a vectr field in space. In vectr frm Define the psitin vectr, r x x ˆ i + y y ˆ j+ z z k ( ) ( ) ( ) ˆ : Its directin is frm p t p. Then c F = r 3 r Vectr Calculus Cnvergence An infinite sequence f vectrs a(1), a(2), a(3),... lim a( n) a = 0. n lima = a : a, limit vectr ( n ) n cnverges t a if Similarly, a vectr functin v( t) lim v( t) l = 0. t t limv( t) t t = l has the limit l at t if Cntinuity v( t) is cntinuus at limv t = v t t t ( ) ( ) t = t if v( t) is cntinuus at t = t if and nly if its three cmpnents are cntinuus at t v t = v t ˆi + v t ˆj+ v t kˆ ( ) ( ) ( ) ( )
Definitin Derivatives f a Vectr Functin v( t) is differentiable at t if the limit exists v( t+ t) v( t) v' ( t) = lim t 0 t v t Called derivative f ( ) Kreyszig ; 9-14 v( t) v' ( t) v '( t), v '( t), v '( t) is differentiable at t if and nly if its three cmpnents are differentiable at t. = Differentiatin rules ( cv )' = cv ' ( u+ v)' = u + v ( u v)' = u v + u v ( u v)' = u v + u v uvw ' = u' vw + uv' w + uvw' ( ) ( ) ( ) ( ) : c, cnstant Example 4 Derivative f a Vectr Functin f Cnstant Length Let v( t) v t v t 2 = v t v t = c 2 be a vectr functin with a cnstant length, ( ) ( ) ( ) ( ) The ttal derivative ( v v )' = 2 v v ' = 0 v ' = 0 r v v ' = c. Partial Derivatives f a Vectr Functin Let the cmpnents with tw r mre variables be differentiable (ex.wind directin w.r.t. time and altitude) v = v t, t,... t ˆi + v t, t,... t ˆj+ v t, t,... t kˆ ( ) ( ) ( ) 1 1 2 n 2 1 2 n 3 1 2 n The partial derivative f v with respect t t m v v1 ˆ v2 ˆ v3 = i + j+ kˆ t t t t m m m m The secnd partial derivative 2 2 2 2 v v1 ˆ v2 ˆ v3 = i + j+ kˆ t t t t t t t t l m l m l m l m Example 5 Partial derivatives