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
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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 .
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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
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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 | .
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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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