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.
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