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Lec.06 - 6. SCALARS, VECTORS, AND TENSORS(FOR...

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Unformatted text preview: 9/12/11 6. SCALARS, VECTORS, AND TENSORS (FOR ORTHOGONAL COORDINATE SYSTEMS) I Main Topics A What are scalars, vectors, and tensors? B Order of scalars, vectors, and tensors C Linear transformaOon of scalars and vectors (and tensors) D Matrix mulOplicaOon 9/12/11 GG303 1 6. SCALARS, VECTORS, AND TENSORS 9/12/11 GG303 2 1 9/12/11 6. SCALARS, VECTORS, AND TENSORS Final (deformed) state IniOal (undeformed) state Displacement vector 9/12/11 GG303 3 6. SCALARS, VECTORS, AND TENSORS II What are scalars, vectors, and tensors? A QuanOOes with associated direcOons B Tensors 1 Broaden our perspecOves; geologists unacquainted with them are handicapped 2 For mulO ­dimensional thinking and communicaOon 3  They can be extremely useful 4  h^p://www.grc.nasa.gov/WWW/k ­12/Numbers/ Math/documents/Tensors_TM2002211716.pdf 9/12/11 GG303 4 2 9/12/11 6. SCALARS, VECTORS, AND TENSORS III Order of scalars, vectors, and tensors A Scalars (magnitudes) 1 Numbers with no associated direcOon (zero ­order tensors) 2 No subscripts in notaOon 3 Examples: Time, mass, length volume 4 Matrix representaOon: 1x1 matrix [x] 9/12/11 GG303 5 6. SCALARS, VECTORS, AND TENSORS III Order of scalars, vectors, and tensors (cont.) B Vectors (magnitude and a direcOon) 1 QuanOOes with one associated direcOon (ﬁrst ­order tensors) 2 One subscript in notaOon (e.g., ux) 3  Examples: Displacement, velocity, acceleraOon 9/12/11 GG303 6 3 9/12/11 6. SCALARS, VECTORS, AND TENSORS III Order of scalars, vectors, and tensors (cont.) B Vectors (magnitude and a direcOon) (cont.) 4 Matrix representaOon: 1xn row matrix, or nx1 column matrix, with n components a Two ­dimensional vector (n=2 components): [x y] or [x1 x2] 1 row, 2 columns ⎡x⎤ ⎡ x1 ⎤ ⎢ ⎣ ⎢ ⎣ ⎢ ⎥ or ⎢ x ⎥ y ⎥ ⎦ 2 ⎥ ⎦ 2 rows, 1 column x = component in x ­direcOon, y = component in y ­direcOon x1 = component in x ­direcOon, x2 = component in y ­direcOon b Three ­dimensional vector (n=3 components): [x y z] or [x1 x2 x3] 1 row, 3 columns 5 Don't confuse the dimensionality of a tensor with its order 9/12/11 GG303 7 6. SCALARS, VECTORS, AND TENSORS III Order of scalars, vectors, and tensors (cont.) C Tensors (magnitude and two direcOons) (for the 2nd ­order tensors we will consider) 1 QuanOOes with two associated direcOon (second ­order tensors) 2 Two subscripts in notaOon (e.g., σxx) 3  Examples: Stress, strain, permeability 9/12/11 GG303 8 4 9/12/11 6. SCALARS, VECTORS, AND TENSORS III "Order" of scalars, vectors, and tensors (cont.) C Tensors (magnitude and two direcOons) (cont.) 4 Matrix representaOon: nxn matrix, with n2 components a Two ­dimensional tensor (4 components): ⎡σ σ xy ⎤ ⎡ σ 11 σ 12 ⎢ x x ⎥ or ⎢ ⎢ σ yx σ yy ⎥ ⎢ σ 21 σ 22 ⎣ ⎣ ⎦ ⎤ ⎥ ⎥ ⎦ 2 rows, 2 columns b Three ­dimensional tensor (3 components): ⎡σ σ σ ⎤ ⎡σ σ σ 13 ⎤ ⎢ xx xy xz ⎥ ⎢ 11 12 ⎥ 3 rows, 3 columns ⎢ σ yx σ yy σ yz ⎥ or ⎢ σ 21 σ 22 σ 23 ⎥ ⎢ ⎥⎢ ⎥ ⎢ σ zx σ zy σ zz ⎥ ⎣ σ 31 σ 32 σ 33 ⎦ ⎣ ⎦ 5 An n ­dimensional 2nd ­order tensor consists of n rows of n ­dimensional vectors 9/12/11 GG303 9 6. SCALARS, VECTORS, AND TENSORS IV Linear transformaOons A "TransformaOons" refers to how components change when the coordinate system changes. B "Linear" means the transformaOon depends on the length of the components, not, for example, on the square of the component lengths. C TransformaOons are used to when we change reference frames in order to present physical quanOOes from a diﬀerent (clearer) perspecOve. D TransformaOons of tensors not covered today 9/12/11 GG303 10 5 9/12/11 6. SCALARS, VECTORS, AND TENSORS IV Linear transformaOons (cont.) E Linear transformaOons of scalars 1 Scalar quanOOes don't change in response to a transformaOon of coordinates; they are invariant 2  Examples (independent of reference frame orientaOon) a Mass b Volume c Density 9/12/11 GG303 11 6. SCALARS, VECTORS, AND TENSORS IV Linear transformaOons (cont.) F Linear transformaOons of vectors (cont.) 1  Vector components change with a transformaOon of coordinates a V = vx + vy = vxi + vyj b V = vx’ + vy’ = vx’i’ + vy’j’ c Vector component: vc d Scalar component: vc Bold: vector components Unbolded: Scalar components 9/12/11 GG303 i, j, i’, and j’ are unit basis vectors along the x,y,x’, and y’ axes, respecOvely 12 6 9/12/11 6. SCALARS, VECTORS, AND TENSORS IV Linear transformaOons (cont.) F Linear transformaOons of vectors (cont.) 2  Every component in the unprimed reference frame contributes linearly to each component in the primed reference frame. vx’ = ax’x vx + ax’yvy vy’ = ay’x vx + ay’yvy 9/12/11 GG303 13 6. SCALARS, VECTORS, AND TENSORS IV Linear transformaOons (cont.) F Linear transformaOons of vectors (cont.) 3  The direcOon cosines are weighOng factors that specify how much each component in one reference frame contributes to a component in the other reference frame. axx’ = cos(θxx’) = cos(θx’x) = ax’x axy’ = cos(θxy’) = cos(θy’x) = ay’x ayx’ = cos(θyx’) = cos(θx’y) = ax’y ayy’ = cos(θyy’) = cos(θy’y) = ayy vx’ = ax’x vx + ax’yvy vy’ = ay’x vx + ay’yvy 9/12/11 GG303 14 7 9/12/11 6. SCALARS, VECTORS, AND TENSORS IV Linear transformaOons (cont.) F Linear transformaOons of vectors (cont.) 4 TransformaOon rule for vectors a vi’ = ai’j vj b Expanded form vx’ = ax’x vx + ax’yvy vy’ = ay’x vx + ay’yvy 9/12/11 GG303 15 6. SCALARS, VECTORS, AND TENSORS IV Linear transformaOon of scalars, vectors, and tensors (cont.) C Vectors (cont.) 4 TransformaOon rule for vectors a vi’ = ai’j vj b Expanded form (Note upper case) •  •  •  List what you know List what you want to know Add the projecOon terms ⎤ ⎡ ax ′x ⎥=⎢ ⎥ ⎢ a y ′x ⎦⎣ GG303 V’V ⎡ vx ′ ⎢ ⎢ vy ′ ⎣ VV’ vx’ = ax’x vx + ax’yvy vy’ = ay’x vx + ay’yvy 9/12/11 5  Matrix form [V’] = [A][V] ⎡ vx ⎤ ⎡ axx ′ ⎢ ⎥=⎢ ⎢ vy ⎥ ⎢ ayx ′ ⎣ ⎦⎣ ax ′y ⎤ ⎡ vx ⎤ ⎥⎢ ⎥ a y ′y ⎥ ⎢ v y ⎥ ⎦ ⎦⎣ axy ′ ⎤ ⎡ vx ' ⎥⎢ ayy ′ ⎥ ⎢ vy ' ⎦⎣ ⎤ ⎥ ⎥ ⎦ 16 8 9/12/11 6. SCALARS, VECTORS, AND TENSORS V Matrix MulOplicaOon  ­ Examples A General Rule: An nxm matrix Omes an mxp matrix gives a nxp matrix B Examples 1 A 1x2 matrix Omes a 2x1 matrix gives a 1x1 matrix 9/12/11 2 rows 1 column A 2x1 matrix Omes a 1x2 matrix gives a 2x2 matrix ⎡ ( 3)(1) ( 3)(2 ) ⎡3⎤ ⎢ ⎥⎡ 1 2 ⎤ = ⎢ ⎣ ⎦ ⎢ ( 4 )(1) ( 4 )(2 ) ⎣4⎦ ⎣ 2 rows 1 column ⎡ ⎤ ⎡ 1 2 ⎤ ⎢ 3 ⎥ = [ (1)( 3) + (2 )( 4 )] = [11] ⎣ ⎦4 ⎣ ⎦ 1 row 2 columns 2  1 row 2 columns ⎤ ⎡3 6⎤ ⎥=⎢ ⎥ ⎥ ⎣4 8⎦ ⎦ 2 rows 2 columns 3 A 2x2 matrix Omes a 2x2 matrix gives a 2x2 matrix ⎡ 1 2 ⎤ ⎡ 1 0 ⎤ ⎡ (1)(1) + (2 )(0 ) (1)(0 ) + (2 )(1) ⎤ ⎥ ⎢ ⎥⎢ ⎥=⎢ ⎢ ⎣ 3 4 ⎦ ⎣ 0 1 ⎦ ⎣ ( 3)(1) + ( 4 )(0 ) ( 3)(0 ) + ( 4 )(1) ⎥ ⎦ 1 row = ⎡ 1 2 ⎤ ⎢ ⎥ 1 column ⎣3 4⎦ GG303 17 9 ...
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