I A little clarification on Cartesian tensor notation

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In Cartesian three-dimensional space, a tensor of rank N is defined by having 3^N components that transform under orthogonal coordinate transformations. The notation ##\mathbf{x}## represents the position coordinates in which the tensor components are calculated, not vectors multiplying the tensor. Tensors are functions of position in space, meaning their values depend on the specific location within a material. When coordinates change, the points are relabeled, leading to the notation ##\mathbf{x}' = \mathbf{A} \mathbf{x}##. This clarification emphasizes the relationship between tensor components and their dependence on the coordinate system.
Kashmir
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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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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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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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This has been discussed many times on PF, and will likely come up again, so the video might come handy. Previous threads: https://www.physicsforums.com/threads/is-a-treadmill-incline-just-a-marketing-gimmick.937725/ https://www.physicsforums.com/threads/work-done-running-on-an-inclined-treadmill.927825/ https://www.physicsforums.com/threads/how-do-we-calculate-the-energy-we-used-to-do-something.1052162/
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