Abstract
Until around 1970s, tensors were almost completely synonymous with (general) relativity except for a minor use in hydrodynamics. Students of physics did not need to study tensors until they took a course in the general theory of relativity. Then they would read the introductory chapter on tensor algebra and analysis, solve a few problems to condition themselves for index “gymnastics”, read through the book, learn some basic facts about relativity, and finally abandon it (unless they became relativists).
Today, with the advent of gauge theories of fundamental particles, the realization that gauge fields are to be thought of as geometrical objects, and the widespread belief that all fundamental interactions (including gravity) are different manifestations of the same superforce, the picture has changed drastically.
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Notes
- 1.
We denote vectors by roman boldface, and tensors of higher rank by sans serif bold letters.
- 2.
Here, we are assuming that
acts on an object (such as v) by “pairing it up” with an appropriate factor of which 
is composed (such as \({\pmb{\epsilon}}^{j}\)). - 3.
When there is no risk of confusion, we shall delete
from 
, it being understood that all tensors are defined on some given underlying vector space. - 4.
The use of
in 
is by convention. Since a member of 
acts on p
dual vectors, it is more natural to use 
. - 5.
The reader should be warned that different authors may use different numerical coefficients in the definition of the exterior product.
- 6.
Note that
is the extension of the pullback operator introduced at the end of Chap. 2. - 7.
See also Problem 5.37.
- 8.
It turns out to be more natural to consider J as a 3-form. However, such a fine distinction is not of any consequence for the present discussion.
acts on an object (such as v) by “pairing it up” with an appropriate factor of which 
is composed (such as 
from 
, it being understood that all tensors are defined on some given underlying vector space.
in 
is by convention. Since a member of 
acts on p
dual vectors, it is more natural to use 
.
is the extension of the pullback operator introduced at the end of Chap. References
Abraham, R., Marsden, J., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, 2nd edn. Springer, Berlin (1988)
Bishop, R., Goldberg, S.: Tensor Analysis on Manifolds. Dover, New York (1980)
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Hassani, S. (2013). Tensors. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01195-0_26
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DOI: https://doi.org/10.1007/978-3-319-01195-0_26
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