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Chapter 1 Introduction - Grassmann Algebra

Chapter 1 BackgroundThe mathematical representation of physical entitiesThree of the more important mathematical systems for representing the entities of contemporaryengineering and physical science are the (three-dimensional) vector Algebra , the more generaltensor Algebra , and geometric Algebra . Grassmann Algebra is more general than vector Algebra ,overlaps aspects of the tensor Algebra , and underpins geometric Algebra . It predates all three. Inthis book we will show that it is only via Grassmann Algebra that many of the geometric andphysical entities commonly used in the engineering and physical sciences may be representedmathematically in a way which correctly models their pertinent properties and leads straightfor-wardly to principal results. As a case in point we may take the concept of force. It is well known that a force is not satisfacto-rily represented by a (free) vector, yet contemporary practice is still to use a (free) vector calcu -lus for this task.

rily represented by a (free) vector, yet contemporary practice is still to use a (free) vector calcu-lus for this task. The deficiency may be made up for by verbal appendages to the mathematical statements: for example ‘where the force f acts along the line through the point P’. Such verbal

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