vector revision

# Differentiation of vector functions applications to

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Differentiation of vector functions, applications to mechanics 4. Scalar and vector fields. Line, surface and volume integrals, curvilinear co-ordinates 5. Vector operators — grad, div and curl 6. Vector Identities, curvilinear co-ordinate systems 7. Gauss’ and Stokes’ Theorems and extensions 8. Engineering Applications

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1. Vector Algebra In which we explore ... Free, sliding and position vectors Coord frames and Vector components Equality, magnitude, Addition, Subtraction Scalar products, Vector Projection, Inner products Vector Products
Vectors 1.2 In Linear Algebra vectors were lists of n numbers. Often in the physical world, the numbers specify – magnitude (1 number) & direction (1 number in 2D, 2 in 3D) There are three slightly different types of vectors: – Free vectors: Only mag & dirn are important. We can translate at will. – Sliding vectors: Line of action is important (eg. forces for moments) Vector can slide with 1 degree of freedom. – Bound or position vectors: “Tails” all originate at origin O . r r 2 3 O r 1 Free vectors Sliding vectors Position vectors

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Coordinate frames 1.3 An advantage of vector algebra: —frees analysis from arbitrarily imposed coordinate frames. Eg, two free vectors are equal if mags and dirns are equal. Can be done with a drawing that is independent of any coordinate system . Try to spot things in the notes that are independent of coordinate system. However, coordinate systems are useful, so introduce the idea of vector components .
Vector components in a coordinate frame 1.4 x 2 x 1 a 1 a 2 3 a k a i j In a Cartesian coordinate frame a = [ a 1 , a 2 , a 3 ] = [ x 2 x 1 , y 2 y 1 , z 2 z 1 ] Define ˆ ı , ˆ , ˆ k as unit vectors in the x, y, z dirns ˆ ı = [1 , 0 , 0] ˆ = [0 , 1 , 0] ˆ k = [0 , 0 , 1] then a = a 1 ˆ ı + a 2 ˆ + a 3 ˆ k . Remember,generalvectorsnotstuckin3dimensions!

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Notation 1.5 We will use bold font to represent vectors a , ω , In written work, underline the vector a , ω We shall use the hat ˆ a to denote a unit vector. a denotes the transpose of a vector iff means “if and only if” mag and dirn are my shorthands for magnitude and direction
Vector equality 1.6 Two free vectors are said to be equal iff their lengths and directions are the same. Using coordinates, two n -dimensional vectors are equal a = b iff a 1 = b 1 , a 2 = b 2 , . . . a n = b n This does for position vectors. But for sliding vectors we must add thelineofactionmustbethesame .

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Vector magnitude and unit vectors 1.7 Provided we use an orthogonal coordinate system, the magnitude of a 3-vector is a = | a | = radicalBig a 2 1 + a 2 2 + a 2 3 and of an n -vector a = | a | = radicalBigg summationdisplay i a 2 i To find the unit vector in the direction of a , simply divide the vector by its magnitude ˆ a = a | a | .
Vector Addition and Subtraction 1.8 Vectors are added/subtracted by adding/subtracting corresponding components (like matrices) a + b = [ a 1 + b 1 , a 2 + b 2 , a 3 + b 3 ] Addition follows the parallelogram construction.

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