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Unformatted text preview: Chapter 1. Vector Analysis Chapter 1. Vector Analysis Chapter 1: Discusses the language (or the math) that will be used the entire semester. Day 1 1. Scalars and vectors 2. Scalar and vector fields 3. Vector Algebra 4. Component vectors and unit vectors Chapter 1. Vector Analysis Chapter 1. Vector Analysis Scalars and Vectors Scalar a quantity whose value may be represented by a single (positive or negative) real number. Example: height, temperature, speed Vector a quantity with both magnitude and direction in space Example: force, velocity Convention: a quantity is a vector if it is written in bold font or has an arrow above, e.g. Vectors are usually represented by directed segments: → A A Chapter 1. Vector Analysis Chapter 1. Vector Analysis A field (scalar or vector) may be defined mathematically as some function of that vector which connects an arbitrary origin to a general point in space. “Function of position” Types of fields: 1. Scalar field Examples of scalar fields: – density at any point P within a volume, ρ ( P ) – elevation of a point with coordinates (x,y) from sea level, h(x,y) – temperature at any point A inside a container, T( A ) Chapter 1. Vector Analysis Chapter 1. Vector Analysis 2. Vector Field Examples of vector fields: wind strength and direction in a region: Magnetic field of the earth: Chapter 1. Vector Analysis Chapter 1. Vector Analysis Vector Algebra • Vector addition follows the parallelogram law and is commutative and associative A A B B A + B = B + A Chapter 1. Vector Analysis Chapter 1. Vector Analysis A + (B + C) = (A + B) + C B A C B A C B A C A + B (A+B)+C A+(B+C) B + C Associativity of Vector Addition Chapter 1. Vector Analysis Chapter 1. Vector Analysis • Negating a vector reverses its direction: “negative” B =  B A  B = A + (B) A BB A + B A B • Multiplication of vectors and scalars obey the associative and distributive laws (r + s)( A + B ) = r( A + B ) + s( A + B ) = r A + r B + s A + s B Chapter 1. Vector Analysis Chapter 1. Vector Analysis • A vector reverses its direction when multiplied by a negative scalar • Division of a vector by a scalar is multiplication by the reciprocal of the scalar A ÷ a = A x 1/a • Two vectors are equal if their difference is zero A = B if A  B = 0 Chapter 1. Vector Analysis Chapter 1. Vector Analysis The Cartesian Coordinate System • Also known as the rectangular coordinate system • The righthanded cartesian coordinate system : z = 0 plane xy plane x = 0 plane yz plane yaxis xaxis zaxis origin y = 0, xz plane Chapter 1. Vector Analysis Chapter 1. Vector Analysis A point in space may be described by stating its x, y and z coordinates . These coordinates are the distances from the origin to the intersection of a perpendicular dropped from the point and the x, y and z axes....
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 Winter '10
 JoelJosephMarciano

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