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Phrak
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When do we use tensor densities rather than tensors?
bcrowell said:For area, volume, and angular momentum:
http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html#Section4.6
Tensors are mathematical objects that describe geometric quantities, such as vectors and matrices, and how they transform under different coordinate systems. Tensor densities, on the other hand, are a more general concept that includes tensors, but also take into account the scaling of coordinate systems. This allows tensor densities to be used in situations where tensors alone are not sufficient.
Tensor densities are particularly useful in physics and engineering, where the coordinate system may change and the scaling of the system is important. For example, in fluid dynamics, tensor densities are used to describe the velocity field, which varies with position and time, and is affected by the density of the fluid.
Yes, tensor densities are also used in differential geometry, where they are used to define geometric objects that are independent of the choice of coordinates. They are also used in general relativity, where they play a crucial role in describing the curvature of spacetime.
This depends on the specific problem you are trying to solve. If the coordinate system is fixed and the scaling of the system is not important, then tensors alone may be sufficient. However, if the coordinate system is changing or the scaling is important, then tensor densities are needed.
Yes, tensors are defined as multilinear maps that transform according to certain rules under a change of coordinates. Tensor densities, on the other hand, are defined as multilinear maps that transform according to a different set of rules that take into account the scaling of the coordinate system. Additionally, the number of independent components of a tensor density is different from that of a tensor of the same rank, which can be an important consideration in certain applications.