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Chap. 8. Vector Differential Calculus. Grad. Div. Curl Chap. 8. Vector Differential Calculus. Grad. Div. Curl 8.1. Vector Algebra in 2-Space and 3-Space - Scalar: a quantity only with its magnitude; temperature, speed, mass, volume, … - Vector: a quantity with its magnitude and its direction; velocity, acceleration, force, … (arrow & directed line segment) Initial and termination point Norm of a: length of a vector a. IaI =1: unit vector Equality of a Vectors: a=b: same length and direction. Components of a Vector: P(x1,y1,z1) Q(x2,y2,z2) in Cartesian coordinates. = = []− − − = a PQ x 2 x1, y2 y1,z2 z1 [a1,a 2 ,a3 ] = 2 + 2 + 2 Length in Terms of Components: a a1 a 2 a3 Position Vector: from origin (0,0,0) point A (x,y,z): r=[x,y,z] Vector Addition, Scalar Multiplication b (1) Addition: a + b = []a + b ,a + b ,a + b 1 1 2 2 3 3 a a+b a+ b= b+ a (u+ v) + w= u+ (v+ w) a+ 0= 0+ a= a a+ (-a) = 0 = [] (2) Multiplication: c a c a 1 , c a 2 ,ca3 c(a + b) = ca + cb (c + k) a = ca + ka c(ka) = cka 1a = a 0a = 0 (-1)a = -a = []= + + Unit Vectors: i, j, k a a1,a 2 ,a 3 a1i a 2 j a3 k i = [1,0,0], j=[0,1,0], k=[0,0,1] 8.2. Inner Product (Dot Product) Definition: a ⋅ b = a b cosγ if a ≠ 0, b ≠ 0 a ⋅ b = 0 if a = 0 or b = 0; cosγ = 0 3 ⋅ = + + = a b a1b1 a 2b2 a3b3 aibi i=1 a ⋅ b = 0 (a is orthogonal to b; a, b=orthogonal vectors) Theorem 1: The inner product of two nonzero vectors is zero iff these vectors are perpendicular. Length and Angle in Terms of Inner Product: length of a : a = a ⋅a a ⋅ b a ⋅ b angle btw two vectors: cosγ = = a b a ⋅a b ⋅ b Ex. 1) a=[1,2,0], b=[3,-2,1], angle btw a and b ? General Properties of Inner Products: []+ ⋅ = ⋅ + ⋅ ⋅ = ⋅ ⋅ ≥ q1a q2 b c q1a c q2 b c a b b a a a 0 []a + b ⋅c = a ⋅c + b ⋅c a ⋅ b ≤ a b Schwarz inequality a + b ≤ a + b Triangle inequality a + b 2 + a − b 2 = 2()a 2 + b 2 Parallelogram equality ⋅ = + + Derivation of a b a1b1 a 2b2 a3b3 = + + = + + a a1i a 2 j a3 k, b b1i b2 j b3 k ⋅ = δ = i j ij 1for i j 0 for i ≠ j ⋅ = ⋅ + ⋅ + ⋅ + + ⋅ = + + a b a1b1i i a1b2 i j a1b3 i k a 2b3 k k a1b1 a 2b2 a3b3 Application of Inner Products: Ex. 2) Work Ex. 3) Component of a force in a given direction Ex. 5) Orthogonal straight lines in the plane Ex. 6) Normal vector to a plane 8.3. Vector Product (Cross Product) v b Definition: v = a × b, length : v = a b sinγ In components: v = [v1,v2,v3] a = [a2b3-a3b2, a3b1-a1b3, a1b2-a2b1] Right-handed triple of vectors a, b, v i, j, k form a right-handed triple in the positive directions i j k How to memorize above formula: × = a b a1 a 2 a3 b1 b2 b3 3 × = ε ε = + = i j ijk k ijk 1 if ijk 123,231,312 k=1 ε = − = ijk 1 if ijk 321,132,213 ε = ijk 0 if any two indices are alike General Properties of Vector Products: (qa)× b = q(a × b) = a × (qb) a × (b + c) = (a × b) + (a × c) (a + b)× c = (a × c) + (b× c) × ≠ × × ≠ × × × ≠ × × ()× × ≠ × ()× a c c a (i j j i) a (b c) (a b) c ( i i j i i j ) Typical Applications of Vector Products Ex. 4) Moment of a force (I) Ex. 5) Moment of a force (II) Ex. 6) Velocity of a rotating body Scalar Triple Product: a = [a ,a ,a ], b = [b ,b ,b ], c = [c ,c ,c ] 1 2 3 1 2 3 1 2 3 bxc a a1 a 2 a3 = ⋅()× = β (a b c) a b c b1 b2 b3 c h c c c 1 2 3 b Volume of the parallelepiped with a, b, c (ka b c) = k(a b c) a ⋅(b × c) = (a × b) ⋅c = c ⋅(a × b) see matrix expression Theorem 1: Three vectors form a linearly independent set iff their scalar triple product is not zero. 8.4. Vector and Scalar Functions and Fields. Derivatives - Two kinds of functions = = (1) Vector functions: v v ( p ) [ v 1 ( p ), v 2 ( p ), v 3 ( p )] depending on the point p in space. a vector field = In Cartesian coordinates, v(x, y,z) [v1(x, y,z), v2 (x, y,z), v3 (x, y,z)] (2) Scalar functions: f = f ( p ) depending on p. a scalar field Ex. 1) Scalar function (Euclidean distance in space) Ex. 2, 3) Vector function (Velocity field, force field) Vector Calculus - Basic concepts of vector calculus − = = (1) Convergence: lim a n a 0 or lim a n a (a: limit vector) n→∞ n→∞ lim v(t) − u = 0 or lim v(t) = u → → t t0 t t0 (vector function v of a real variable t has limit u) lim v(t) = v(t ) (2) Continuity: → 0 vector function v(t) is continuous at t=t0. t t0 = = + + v(t) [v1(t),v2 (t),v3 (t)] v1(t)i v2 (t)j v3 (t)k three components are continuous at t0. Derivative of a vector function Definition: + Δ − V‘(t) = v(t t) v(t) v'(t) lim Δ Δt→0 Δt v(t+ t) v(t) = v'(t) [v1'(t),v2 '(t),v3 '(t)] Derivative v’(t) is obtained by differentiating each component separately. ()cv' = cv' ()v + u '= v'+u' ()u ⋅ v '= u'⋅v + u ⋅ v' ()u × v '= u'×v + u × v' ()u v w '= ()u' v w + ()u v' w + ()u v w' Partial Derivatives of a Vector Function v(t) = [v ,v ,v ] = v i + v j+ v k 1 2 3 1 2 3 differentiable functions of n variables, t1, …, tn. Partial derivative of v: ∂v ∂v ∂v ∂v ∂2 v ∂2v ∂2v ∂2v = 1 i + 2 j + 3 k = 1 i + 2 j + 3 k ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ t l t l t l t l t l t m t l t m t l t m t l t m.
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