A little clarification on Cartesian tensor notation

In summary, on page 192 of Goldstein's 2nd edition, it is explained that in a Cartesian three-dimensional space, a tensor of the Nth rank can be defined as a quantity with 3^N components that transform under an orthogonal transformation of coordinates. The author also clarifies that the tensor is a function of the position in space and after changing coordinates, every point is re-labelled with a new set of position coordinates. The prime in the argument signifies the change in coordinates.
  • #1
Kashmir
468
74
Goldstein pg 192, 2 edIn a Cartesian three-dimensional space, a tensor ##\mathrm{T}## of the ##N## th rank may be defined for our purposes as a quantity having ##3^{N}## components ##T_{i j k}##.. (with ##N## indices) that transform under an orthogonal transformation of coordinates, ##\mathbf{A}##) according to the following scheme:*
##
T_{i j k \ldots}^{\prime} \left(\mathbf{x}^{\prime}\right)=a_{i l} a_{j m} a_{k n} \ldots T_{l m n \ldots}(\mathbf{x})
##

I've just a small doubt here:

What do the ##\mathbf{x}## mean here, are they vectors that multiply ##T## or is the author using them to signify the coordinate system in which the components of ##T## are calculated?
 
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  • #2
In general the tensor is a function of the position ##\mathbf{x} \in \mathbf{R}^3## in space (e.g. stress, strain, electrical conductivity, etc. all depend on the position within the material). After changing coordinates, every point ##p## has been re-labelled by a new set of position coordinates ##\mathbf{x} \rightarrow \mathbf{x}' = \mathbf{A} \mathbf{x}##; hence why you also put a prime on the argument.
 
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  • #3
ergospherical said:
In general the tensor is a function of the position ##\mathbf{x} \in \mathbf{R}^3## in space (e.g. stress, strain, electrical conductivity, etc. all depend on the position within the material). After changing coordinates, every point ##p## has been re-labelled by a new set of position coordinates ##\mathbf{x} \rightarrow \mathbf{x}' = \mathbf{A} \mathbf{x}##; hence why you also put a prime on the argument.
Thank you. Makes it perfectly clear! I :)
 
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Related to A little clarification on Cartesian tensor notation

1. What is Cartesian tensor notation?

Cartesian tensor notation is a mathematical notation used to represent tensors, which are mathematical objects that describe physical quantities such as force, stress, and strain. It is based on the Cartesian coordinate system, which uses three perpendicular axes to represent three-dimensional space.

2. How is Cartesian tensor notation written?

Cartesian tensor notation uses superscripts and subscripts to represent the components of a tensor. The superscripts indicate the direction of the vector, while the subscripts indicate the order of the tensor. For example, a second-order tensor would have two superscripts and two subscripts.

3. What is the advantage of using Cartesian tensor notation?

Cartesian tensor notation allows for a concise and systematic representation of tensors, making it easier to perform calculations and manipulate equations. It also allows for a clear distinction between covariant and contravariant components of a tensor, which is important in many areas of physics and engineering.

4. Are there any limitations to using Cartesian tensor notation?

One limitation of Cartesian tensor notation is that it is only applicable to tensors in three-dimensional space. It also does not take into account the curvature of space, which is important in areas such as general relativity. In these cases, alternative notations, such as Einstein notation, may be more suitable.

5. How is Cartesian tensor notation used in scientific research?

Cartesian tensor notation is widely used in many fields of science and engineering, including mechanics, electromagnetism, and fluid dynamics. It is used to describe physical phenomena and solve complex equations, making it an essential tool in scientific research and analysis.

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