By Igor Gumowski

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Extra resources for Recurrences and Discrete Dynamic Systems

Example text

45) The matrix form is often convenient for algebraic manipulation. m have n indices and the tensor product has n factors. 47) From this more general viewpoint, a scalar can be interpreted as a tensor of rank zero, since it has a single component which is unchanged in a coordinate transformation; a vector is considered as a tensor of rank one. Most of the tensors we encounter in this book will be of rank zero (scalars), one (vectors), or two. We will also discuss some fourth-rank tensors due to their importance in continuum mechanics.

153) where ds is the length of dxi . 155) Cartesian Tensors 39 Note that ni ui is negative at A and positive at B. 158) is valid for the inﬁnitesimal cylinder considered. 150). We remark that the theorem of Gauss is still valid for multiply connected region. In that case, we have to make branch cuts to prove the theorem. Some special cases of the theorem are listed below: 1. 7 Contour and surface. 151). 152) by ekij . 2. 165) In the two-dimensional case, the theorem changes the surface integral into the line integral and vice versa.

References 1. , Introduction to the Mechanics of a Continuous Medium, PrenticeHall, Englewood Cliffs, NJ, 1969. 2. , Continuum Mechanics, Longman, London and New York, 1980. 3. , Introduction to Continuum Mechanics, Pergamon Press, New York, 1974. 4. , First Course in Continuum Mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1994. 5. , Continuum Mechanics, George Allen & Unwin, London, 1976. © 2005 by Chapman & Hall/CRC Press Cartesian Tensors 41 Problems (1) The equations of motion are ∂Tsr + ρbr = ρar ∂xs Write out the equations in conventional notations, that is, use x = x1 , y = x2 and z = x3 .