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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, …

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Chap. 8. Vector Differential Calculus. Grad. Div. Curl

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  1. 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) • 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. • Length in Terms of Components: • Position Vector: from origin (0,0,0)  point A (x,y,z): r=[x,y,z] Initial and termination point

  2. Vector Addition, Scalar Multiplication (1) Addition: a + b = b + a (u + v) + w = u + (v + w) a + 0 = 0 + a = aa + (-a) = 0 (2) Multiplication: 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 i = [1,0,0], j=[0,1,0], k=[0,0,1] 8.2. Inner Product (Dot Product) Definition: b a a+b (a is orthogonal to b; a, b=orthogonal vectors)

  3. Theorem 1: The inner product of two nonzero vectors is zero iff these vectors are perpendicular. Length and Angle in Terms of Inner Product: Ex. 1) a=[1,2,0], b=[3,-2,1], angle btw a and b ? General Properties of Inner Products:

  4. Derivation of 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

  5. v b a 8.3. Vector Product (Cross Product) Definition: In components: v = [v1,v2,v3] = [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 How to memorize above formula: General Properties of Vector Products:

  6. bxc a  c b 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: Volume of the parallelepiped with a, b, c Theorem 1: Three vectors form a linearly independent set iff their scalar triple product is not zero. h see matrix expression

  7. 8.4. Vector and Scalar Functions and Fields. Derivatives - Two kinds of functions (1) Vector functions: depending on the point p in space.  a vector field In Cartesian coordinates, (2) Scalar functions: 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: (a: limit vector) (vector function v of a real variable t has limit u) (2) Continuity: vector function v(t) is continuous at t=t0. three components are continuous at t0.

  8. Derivative of a vector function Definition: Partial Derivatives of a Vector Function differentiable functions of n variables, t1, …, tn. Partial derivative of v: V‘(t) v(t+t) v(t) Derivative v’(t) is obtained by differentiating each component separately.

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