Abstract
Dyadic product and tensors are introduced in the context of bilinear forms before extending this scheme to arbitrary but finite dimensions. Afterwards, tensor product spaces are defined. The exterior product is motivated within this chapter by the aim to generalize the notion of volume for arbitrary dimensions and to overcome the limitations implied by the cross product of conventional vector calculus. Within this context, symmetric and skew-symmetric tensors, as well as a generalized version of the Kronecker symbol, are discussed. Furthermore, basic aspects of the so-called star-operator are examined. The latter relates spaces of alternating tensors of equal dimension based on the existence of an inner product.
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References
Jänich K (2001) Vector analysis. Undergraduate texts in mathematics. Springer, New York
Lee M (2002) Introduction to smooth manifolds. Graduate texts in mathematics. Springer, Berlin
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Mühlich, U. (2017). Tensor Algebra. In: Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds. Solid Mechanics and Its Applications, vol 230. Springer, Cham. https://doi.org/10.1007/978-3-319-56264-3_4
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DOI: https://doi.org/10.1007/978-3-319-56264-3_4
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