H.M. Schey,. Div, Grad, Curl and all that: An informal Text on Vector Calculus, W.W. Nortonand Co., (1973). (Good physical introduction to the subject) Mase, George. Theory and problems of Continuum Mechanics: Schaum’soutline Series. (Heavy on tensors but lots of worked problems) Marsden, J.E. and Tromba, A.J.. Vector Calculus. W.H. Freeman (or any standard text on Vector calculus) In modeling we are generally concerned with how physical properties change in space and time. Therefore we need a general mathematical description of both the variables of interest and their spatial and temporal variations. Vector calculus provides just that framework….

Definitions of basic operations vector dot product a· b =a i b i =| a || b |cos* (A.1.1) is a scalar that records the amount of vector a that lies in the direction of vector b (and vice versa). *is the smallest angle between the two vectors. vector cross productc = a × b is a vector that is perpendicular to the plane spanned by vectors a and b . The direction that c points in is determined by the right hand rule. Note a × b = b × a . The cross product is most easily calculated as the determinant of the matrix c = * * * * * * * i j k a x a y a z b x b y b z * * * * * * * (A.1.2) or c = (a y b z a z b y ) i (a x b z a z b x ) j+ (a x b y a y b x ) k (A.1.3) or in index notation asc i =* ijk a j b k where* ijk is the horrid permutation symbol. tensor vector dot product is a vector formed by matrix multiplication of a tensor and a vector c =D· a . In the case of stress, the force acting on a plane with normal vector n is simply f =*· n . Each component of the vector is most easily calculated in index notation withc i =D ij a j with summation implied over repeated indices (i.e. c 1 =D 11 a 1 +D 12 a 2 +D 13 a 3 and soon fori=2,3….

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