Matrix and Index NotationDavid RoylanceDepartment of Materials Science and EngineeringMassachusetts Institute of TechnologyCambridge, MA 02139September 18, 2000A vector can be described by listing its components along the xyz cartesian axes; for in-stance the displacement vector u can be denoted as ux,uy,uz, using letter subscripts to indicatethe individual components. The subscripts can employ numerical indices as well, with 1, 2,and 3 indicating the x, y,andzdirections; the displacement vector can therefore be writtenequivalently as u1,u2,u3.A common and useful shorthand is simply to write the displacement vector as ui,wheretheisubscript is an index that is assumed to range over 1,2,3 ( or simply 1 and 2 if the problem isa two-dimensional one). This is called the range convention for index notation. Using the rangeconvention, the vector equation ui= a implies three separate scalar equations:u1= au2= au3= aWe will often find it convenient to denote a vector by listing its components in a vertical listenclosed in braces, and this form will help us keep track of matrix-vector multiplications a bitmore easily. We therefore have the following equivalent forms of vector notation:u = ui=u1u2u3=uxuyuzSecond-rank quantities such as stress, strain, moment of inertia, and curvature can be de-noted as 3×3 matrix arrays; for instance the stress can be written using numerical indices as[σ]=σ11σ12σ13σ21σ22σ23σ31σ32σ33Here the first subscript index denotes the row and the second the column. The indices also havea physical meaning, for instance σ23indicates the stress on the 2 face (the plane whose normalis in the 2, or y, direction) and acting in the 3, or z, direction. To help distinguish them, we’lluse brackets for second-rank tensors and braces for vectors.Using the range convention for index notation, the stress can also be written as σij,whereboth the i and the j range from 1 to 3; this gives the nine components listed explicitly above.1(Since the stress matrix is symmetric, i.e. σij= σji, only six of these nine components areindependent.)A subscript that is repeated in a given term is understood to imply summation over the rangeof the repeated subscript; this is the summation convention for index notation. For instance, toindicate the sum of the diagonal elements of the stress matrix we can write:σkk=3Xk=1σkk= σ11+ σ22+ σ33The multiplication rule for matrices can be stated formally by taking A =(aij)tobean(M×N)matrixandB=(bij)tobean(R×P) matrix. The matrix product AB is definedonly when R = N,andisthe(M×P)matrixC=(cij)givenbycij=NXk=1aikbkj= ai1b1j+ ai2b2j+ ···+aiNbNkUsing the summation convention, this can be written simplycij= aikbkjwhere the summation is understood to be over the repeated index k.Inthecaseofa3×3matrix multiplying a 3 × 1 column vector we havea11a12a13a21a22a23a31a32a33b1b2b3=a11b1+ a12b2+ a13b3a21b1+ a22b2+ a23b3a31b1+ a32b2+ a33b3= aijbjThe comma convention uses a subscript comma to imply differentiation with respect to thevariable following, so f,2= ∂f/∂y and ui,j= ∂ui/∂xj. For instance, the expression σij,j=0uses all of the three previously defined index conventions: range on i, sum on j, and differentiate:∂σxx∂x+∂σxy∂y+∂σxz∂z=0∂σyx∂x+∂σyy∂y+∂σyz∂z=0∂σzx∂x+∂σzy∂y+∂σzz∂z=0The Kroenecker delta is a useful entity is defined asδij=(0,i6=j1,i=jThis is the index form of the unit matrix I:δij= I =100010001So, for instance2σkkδij=σkk000σkk000σkkwhere σkk= σ11+ σ22+
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