- #1
grzz
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I am learning about tensors.
Is gαβAβ the same as Aβgαβ ?
Thanks for any help.
Is gαβAβ the same as Aβgαβ ?
Thanks for any help.
Bαβ is not tensor, it is the component of a tensor. The components of a tensor are real or complex numbers. They commute.grzz said:But then is
BαβAγ equal to Aγ Bαβ ?
grzz said:But then is
BαβAγ equal to Aγ Bαβ ?
[itex]A_{\beta\alpha}B^\gamma[/itex] is equal to both the [itex]{}_{\beta\alpha}{}^\gamma[/itex] component of the tensor [itex]A\otimes B[/itex], and the [itex]{}^\gamma{}_{\beta\alpha}[/itex] component of the tensor [itex]B\otimes A[/itex].grzz said:Thanks for the help.
Since [itex]\alpha[/itex] is repeated in g[itex]_{}\beta_{}\alpha[/itex]A[itex]^{}\alpha[/itex] then it was clear to me that this is a sum and the g[itex]_{}\beta_{}\alpha[/itex] and the A[itex]^{}\alpha[/itex] are numbers and so commute.
But I thought that A[itex]_{}\beta_{}\alpha[/itex]B[itex]^{}\gamma[/itex] represented the product of two tensors. From the little I know I thought that sometimes a tensor is represented by one of its components. That is why I said that the second example may not commute.
Tensors are mathematical objects that are used to represent physical quantities with multiple dimensions. They are difficult to understand because they involve abstract mathematical concepts and can have complex properties that are not easily visualized.
Tensors are used in a variety of scientific fields such as physics, engineering, and computer science. They are particularly useful in fields that deal with complex systems or data with multiple dimensions, such as fluid dynamics, machine learning, and quantum mechanics.
Some common challenges in working with tensors include understanding their properties and operations, manipulating and visualizing them, and handling large amounts of data represented by tensors.
Yes, there are many resources available such as textbooks, online courses, and tutorials that can help with understanding tensors. Additionally, there are various software packages and libraries that can assist with working with tensors.
One example of a real-world application of tensors is in image recognition and computer vision. Tensors are used to represent images and their features, and machine learning algorithms can be applied to these tensors to recognize objects in images.