- #1
ehrenfest
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Homework Statement
I am trying to show that the components of the covariant derivative [tex] \del_b v^a are the mixed components of a rank-2 tensor.
If I scan in my calculations, will someone have a look at them?
Covariant Derivative: Proving Rank-2 Tensor Components
- Thread starter ehrenfest
- Start date
In summary, the conversation discusses the attempt to prove the components of the covariant derivative are the mixed components of a rank-2 tensor. The person is asking for help by uploading their calculations and asking if someone can look at them. They mention that they have included the correct transformation for a mixed second rank tensor, but are unsure why the other term needs to vanish. They also inquire if others can view the uploaded file before it is approved.
I am trying to show that the components of the covariant derivative [tex] \del_b v^a are the mixed components of a rank-2 tensor.
If I scan in my calculations, will someone have a look at them?
- #2
ehrenfest
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pretty pretty please
- #3
cristo
Staff Emeritus
Science Advisor
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If you do scan your work and upload it, then I'm sure someone will look at your work.
- #4
ehrenfest
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Here are my calculations. It took me forever to scan them and put them in a file small enough to upload. :(
Anyway. The "stuff" is the correct transformation for a mixed second rank tensor which means the other term (the one that I wrote out) needs to vanish. The problem is that I do not see why it vanishes. Does anyone else?
By the way--can other people not click on the attachment and see it under it gets approval?
Anyway. The "stuff" is the correct transformation for a mixed second rank tensor which means the other term (the one that I wrote out) needs to vanish. The problem is that I do not see why it vanishes. Does anyone else?
By the way--can other people not click on the attachment and see it under it gets approval?
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Related to Covariant Derivative: Proving Rank-2 Tensor Components
1. What is a covariant derivative?
A covariant derivative is a mathematical tool used in differential geometry to measure the rate of change of a tensor field along a given direction. It takes into account the curvature of the space in which the tensor field is defined.
2. What is the significance of proving rank-2 tensor components?
Rank-2 tensor components are important because they represent the second-order derivatives of a tensor field, which can provide valuable information about the behavior and properties of the field. Proving these components is necessary for a more complete understanding of the tensor field and its applications.
3. How do you prove rank-2 tensor components using a covariant derivative?
To prove rank-2 tensor components using a covariant derivative, we use the definition of the covariant derivative and apply it to the components of the tensor field. This involves calculating the partial derivatives of the field with respect to each coordinate, as well as the Christoffel symbols which account for the curvature of the space.
4. Can a covariant derivative be used to prove rank-2 tensor components in any space?
Yes, a covariant derivative can be used to prove rank-2 tensor components in any space, as long as the space is smooth and has a well-defined metric. This includes Euclidean spaces, curved spaces, and even abstract spaces such as Lie groups.
5. What are some practical applications of proving rank-2 tensor components using a covariant derivative?
Proving rank-2 tensor components using a covariant derivative has many practical applications in physics and engineering. It is used in general relativity to describe the curvature of spacetime, in fluid mechanics to study the flow of fluids, and in materials science to analyze the stress and strain of materials. It is also used in computer graphics to model the deformation of objects.
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