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4-Eng-2P04-A - ENGINEERING MECHANICS ENG 2P04 Dieter Stolle...

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1 ENGINEERING MECHANICS ENG 2P04 Dieter Stolle, Professor Department of Civil Engineering McMaster University Mathematical Preliminaries Definitions - 1 Ordered set (coordinate) representation where the unit vector directions implied Scalar: Temperature, pressure, density frame invariant Vector: First order tensor characterized by direction and magnitude frame invariant Position Vector: r xi yj zk To solve problems, we must attach vector to a frame of reference, usually orthogonal , , x y z r , , i j k
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2 Index Notation: Given that we have a vector we may write Mathematical Preliminaries Definitions - 2 r 1 1 2 2 3 3 i i r x e x e x e r x e where summation is implied over repeated indices Dot Product: Given Cartesian vectors written as the dot product is defined as 1 2 3 1 2 3 , , and , , F F F F u u u u W = W = cos F u F u Mathematical Preliminaries Definitions - 3 Dot Product: Using index notation, we may write i i j j i j i j i j ij i i W Fe u e W Fu e e W Fu W Fu Kronecker delta Cross Product: Right Hand Rule sin M r F r F n
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3 Mathematical Preliminaries Definitions - 4 Cross Product: sin M r F r F n 1 2 3 1 2 3 1 2 3 = det e e e M r F r r r F F F x y F r F y F x For two dimensional: y x M F x F y k Often the direction of rotation is known, a priori and vectors are implied, but not shown; i.e. , y x M F x F y Newton’ Laws (1) 1.
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