Appendix - Appendix A VECTORS TENSORS AND MATRIX NOTATION...

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Appendix A VECTORS, TENSORS AND MATRIX NOTATION The objective of this section is to review some of the vector operations that you have already covered in your MATH and ENGR courses. For more details and examples you should refer to your calculus text under the chapters on vectors and vector calculus. The following notes will be divided into two parts. The first will review the theory, while the second part will review how to perform various vector operations in Scientific Workplace. These calculations will be illustrated in Cartesian rectangular and cylindrical coordinates. First you will learn the basic operations, then you will be shown some examples, and finally you will work some problems. A.1 Review of Vector and Matrix Operations In Engineering, we represent physical quantities using three different groups of mathematical ob- jects, i.e., scalars, vectors and tensors. A scalar quantity is represented by a real number with some appropriate units (mass, temperature, energy, time, etc.). A vector is an object that has a scalar magnitude and a direction. A vector in three-dimensional space can be described as a linear com- bination of three base vectors that have unit length and point in the positive direction of the three axes; these form the so-called standard orthonormal basis . They are denoted i , j , and k and point parallel to the x -, y -, and z -axes, respectively. Using them, we can write a three-space vector a in the following form a = a 1 i + a 2 j + a 3 k (A.1) where a 1 , a 2 , a 3 are the scalar components of the vector a with respect to the standard orthonormal basis. Note that in printed text, lower-case roman boldface letters are generally used to represent vectors, and subscripted lower-case italic letters represent their components. In handwriting, the vector a is often written as ¯ a , a or a . A given vector a can be expressed in matrix form as a 3 × 1 column matrix whose entries are the components of the vector: [ a ] = a 1 a 2 a 3 (A.2) 357
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358 APPENDIX A. VECTORS, TENSORS AND MATRIX NOTATION We normally think of a vector as a column matrix, but a vector may also be written in matrix notation as a 1 × 3 row matrix: [ a ] = a 1 a 2 a 3 (A.3) Addition of vectors is defined component-wise by ( a + b ) i = a i + b i for all i . (A.4) Multiplication of a vector by a scalar is defined component-wise by ( c a ) i = c · a i for all i . (A.5) The difference a b is simply a + ( 1) b . Analogous definitions hold for general matrices. The above definitions arise from their geometrical usefulness and from obvious analogy to oper- ations on the real numbers. How to define a useful form of multiplication of one vector by another is not so obvious. We define three products of vectors: the dot product (or scalar product), the cross product (or vector product) and the
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