Vector & Tensor Transformation in Physics

In summary, the conversation discusses the transformation of a vector from one coordinate system to another, and the use of Jacobian matrices to describe these changes. It also touches on the implications of covariant and contravariant vectors in differential geometry, as well as the transformation of tensors using Jacobians.
  • #1
nigelscott
135
4
I'm not sure if this belongs in physics or here. Consider the transformation of a vector from one coordinate system to the other. I can write:

Vn = (∂yn/∂xm)Vm - contravariant form

Vn = (∂xm/∂yn)Vm - covariant form

In each case are the partials equivalent to the Jacobean matrices? Also, what about the case of a tensor

Tmn = (∂xr/∂ym)(∂xs/∂yn)Trs

Is the transformation just the product of 2 Jacobeans?
 
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  • #2
Yes, the Jacobian is used to describe coordinate changes. Is that what you

were asking?
 
  • #3
Yes. I wasn't quite sure if there were any implications associated with covariant versus contravariant vectors in differential geometry. Also, in the case of a tensor, I wasn't sure if the transformation was the product of the Jacobean with it's inverse or transpose.
 

1. What is the difference between a vector and a tensor?

A vector is a mathematical quantity that has both magnitude and direction, while a tensor is a mathematical object that describes the relationships between multiple vectors or other quantities. In physics, tensors are used to describe the properties of physical systems that involve multiple dimensions or directions.

2. How are vectors and tensors transformed in physics?

Vectors and tensors can be transformed using mathematical operations such as rotations, translations, and scaling. These transformations are used to describe physical phenomena in different coordinate systems and to analyze the effects of forces and other external influences on a system.

3. What is the significance of tensor transformation in physics?

Tensor transformation is crucial in physics because it allows us to analyze physical systems in different reference frames and to understand how they behave under different conditions. This is especially important in fields such as relativity and quantum mechanics, where the laws of physics can appear to change depending on the observer's frame of reference.

4. Can tensors be represented graphically?

Yes, tensors can be represented graphically using diagrams or matrices. These visual representations can help to visualize the relationships between different quantities and to perform calculations involving tensor operations.

5. How are tensor transformations used in real-world applications?

Tensor transformations are used in a variety of real-world applications, including engineering, computer graphics, and geophysics. They are also essential in fields such as fluid dynamics, electromagnetism, and solid mechanics, where they are used to analyze the behavior of complex systems and to predict the effects of external forces on physical objects.

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