弹性力学第二章数学基础演示幻灯片.ppt

君子务本，本立道生;四喜;第二章 数学基础;第一节 标量和矢量;二、矢量的表示 大小和方向确定分量 A is completely defined by its magnitude A and by its three direction anglesθ1 , θ2 and θ3 矢量A在三个坐标轴上的投影（分量）;分量（投影）确定矢量 已知分量，矢量的大小和方向可由几何关系得到;三、坐标变换（Coordinate Transformation）; Using the above range convention, these equations may be written more compactly as;记; We may achieve a further simplification by adopting the summation convention requiring that twice-repeated subscripts in an expression always imply summation over the range 1-3. In this case, we have;两个矢量的标量积（Scalar Product of two vectors） The scalar product of two vectors A and B is expressible as;两个矢量的矢量积（Vector Product of Two Vectors） The vector product of two vectors A and B is to be a third vector C perpendicular to A and B; If the symbol eijk is defined as follows: eijk = +1 for i = 1, j = 2, k = 3 or any even number of permutations of this arrangement (e.g., e312 ) eijk = -1 for odd permutations of i = 1, j = 2, k = 3 (e.g., e132 ) eijk = 0 for two or more indices equal (e.g., e113 ) the components of vector C can be written as;标量三重积（Scalar Triple Product） The scalar triple product or box product [A B C] is a scalar product of two vectors, in which any vector is a vector product of other two vectors, i.e.;第二节 笛卡尔张量;二阶笛卡尔张量 Similarly, a Cartesian tensor of order two is defined as a quantity having nine components Tij whose transformation between primed and unprimed coordinate axes is governed by the equations;高阶笛卡尔张量 Third- and higher- order Cartesian tensors are defined analogously. 零阶笛卡尔张量 A Cartesian tensor of zeroth order is defined to be any quantity that is unchanged under coordinate transformation, that is, a scalar.; If Aij and Bij denote components of two second-order tensors, the addition or subtraction of these tensors is defined to be a third tensor of second order having components Cij given by;Multiplication of Cartesian Tens