# Curl of the transpose of a gradient of a vector: demonstration of an identity

• traianus
In summary, the conversation is about understanding tensors in continuum mechanics and specifically discussing the definition of the curl of a tensor. The individual has attempted to demonstrate an identity with indicial notation and has attached their attempt, asking for feedback and suggestions. They also ask for help in understanding the vector product being used and for suggestions on where to post their question. They also mention confusion about the definitions of \nabla(\nabla\times\mathbf{u}) and \nabla \mathbf{u}, and provide a link to another demonstration. The conversation ends with the individual asking for further input and explaining that they have reached out to an expert for clarification on conflicting definitions.
traianus
I would like to demonstrate an identity with the INDICIAL NOTATION. I have attached my attempt. Please let me know where I made mistakes. Any suggestion? I am trying to understand tensors all by myself because they are the keys in continuum mechanics
Thanks

#### Attachments

• TENSOR_IDENTITY.pdf
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Last edited:
What sort of vector product are you using here $\hat{e}_i \hat{e}_j$?

tensor product

Is this question so difficult? Please help me: I am trying to learn tensors and I would like to know what my mistake is. Thanks!

Any suggestion?

Can anyone suggest a forum to post my question? Thanks

?

I don't really understand what is meant by
$$\nabla(\nabla\times\mathbf{u})$$
and
$$\nabla \mathbf{u}$$.

For example, if $$\mathbf{u}=u_j\hat{e}_j$$, then $$\nabla \mathbf{u}=(\partial_i\hat{e}_i)(u_j\hat{e}_j)=\partial_iu_j\hat{e}_i\hat{e}_j$$.

But what is $$\hat{e}_i\hat{e}_j$$; the inner product between the unit basis vectors? Then the result would be a scalar instead of a vector.

Last edited:
Any other input?

?

You should at least explain how you define $\nabla u$ when u is a vector.

The problem is at the very bottom line in the definition of a curl of a tensor. I found 2 definitions which contradict to each other. Mine is one of them. I will email the authors.

I asked an expert. The question was not trivial. After a while I found out that there are different definitions of curl of a tensor.

## 1. What is the "Curl of the transpose of a gradient of a vector"?

The "Curl of the transpose of a gradient of a vector" is a mathematical operation that combines the concepts of gradient and curl to describe the change in a vector field over a given region.

## 2. What is the purpose of demonstrating an identity involving the "Curl of the transpose of a gradient of a vector"?

Demonstrating an identity involving the "Curl of the transpose of a gradient of a vector" can help to establish a relationship between different mathematical operations and can be useful in simplifying calculations in vector analysis.

## 3. How is the identity involving the "Curl of the transpose of a gradient of a vector" derived?

The identity involving the "Curl of the transpose of a gradient of a vector" can be derived using the properties of vector calculus, such as the product rule, chain rule, and divergence theorem.

## 4. What are some real-world applications of the "Curl of the transpose of a gradient of a vector" identity?

The "Curl of the transpose of a gradient of a vector" identity has applications in many fields, including fluid mechanics, electromagnetism, and quantum mechanics. It is used to describe the behavior of physical systems and can aid in solving problems related to the flow of fluids, the behavior of electric and magnetic fields, and the motion of particles.

## 5. Are there any limitations to using the "Curl of the transpose of a gradient of a vector" identity?

Like any mathematical identity, the "Curl of the transpose of a gradient of a vector" identity may have limitations in certain situations. It is important to carefully consider the assumptions and conditions under which the identity holds in order to use it effectively in calculations.

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